| align:start position:0% |
| |
| foreign |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| for every symmetry for every continuous |
|
|
| align:start position:0% |
| for every symmetry for every continuous |
| |
|
|
| align:start position:0% |
| for every symmetry for every continuous |
| symmetry there's a conserved current |
|
|
| align:start position:0% |
| symmetry there's a conserved current |
| |
|
|
| align:start position:0% |
| symmetry there's a conserved current |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and then we also started talking about |
|
|
| align:start position:0% |
| and then we also started talking about |
| |
|
|
| align:start position:0% |
| and then we also started talking about |
| uh relativistic quantum mechanics |
|
|
| align:start position:0% |
| uh relativistic quantum mechanics |
| |
|
|
| align:start position:0% |
| uh relativistic quantum mechanics |
| uh um how we want to unify |
|
|
| align:start position:0% |
| uh um how we want to unify |
| |
|
|
| align:start position:0% |
| uh um how we want to unify |
| special relativity and quantum mechanics |
|
|
| align:start position:0% |
| special relativity and quantum mechanics |
| |
|
|
| align:start position:0% |
| special relativity and quantum mechanics |
| okay so the most immediate idea for that |
|
|
| align:start position:0% |
| okay so the most immediate idea for that |
| |
|
|
| align:start position:0% |
| okay so the most immediate idea for that |
| is uh what's called the right basic |
|
|
| align:start position:0% |
| is uh what's called the right basic |
| |
|
|
| align:start position:0% |
| is uh what's called the right basic |
| quantum mechanics and the most immediate |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| generalization of the Schrodinger |
|
|
| align:start position:0% |
| generalization of the Schrodinger |
| |
|
|
| align:start position:0% |
| generalization of the Schrodinger |
| equation |
|
|
| align:start position:0% |
| equation |
| |
|
|
| align:start position:0% |
| equation |
| so if you have |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so at the end of last lecture we talked |
|
|
| align:start position:0% |
| so at the end of last lecture we talked |
| |
|
|
| align:start position:0% |
| so at the end of last lecture we talked |
| about say the most immediate realization |
|
|
| align:start position:0% |
| about say the most immediate realization |
| |
|
|
| align:start position:0% |
| about say the most immediate realization |
| of the Schrodinger equation |
|
|
| align:start position:0% |
| of the Schrodinger equation |
| |
|
|
| align:start position:0% |
| of the Schrodinger equation |
| which uh so the |
|
|
| align:start position:0% |
| which uh so the |
| |
|
|
| align:start position:0% |
| which uh so the |
| so if you have e square e equal to say P |
|
|
| align:start position:0% |
| so if you have e square e equal to say P |
| |
|
|
| align:start position:0% |
| so if you have e square e equal to say P |
| Square divided by 2m |
|
|
| align:start position:0% |
| Square divided by 2m |
| |
|
|
| align:start position:0% |
| Square divided by 2m |
| and then you go to non-registic |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| quantum mechanics shielding the equation |
|
|
| align:start position:0% |
| quantum mechanics shielding the equation |
| |
|
|
| align:start position:0% |
| quantum mechanics shielding the equation |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and now if you have e squared equal to p |
|
|
| align:start position:0% |
| and now if you have e squared equal to p |
| |
|
|
| align:start position:0% |
| and now if you have e squared equal to p |
| squared plus M Square |
|
|
| align:start position:0% |
| squared plus M Square |
| |
|
|
| align:start position:0% |
| squared plus M Square |
| for relativistic particle |
|
|
| align:start position:0% |
| for relativistic particle |
| |
|
|
| align:start position:0% |
| for relativistic particle |
| and then you get What's called the |
|
|
| align:start position:0% |
| and then you get What's called the |
| |
|
|
| align:start position:0% |
| and then you get What's called the |
| client golden equation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and again this PSI has the imputation of |
|
|
| align:start position:0% |
| and again this PSI has the imputation of |
| |
|
|
| align:start position:0% |
| and again this PSI has the imputation of |
| the uh of the wave function so this |
|
|
| align:start position:0% |
| the uh of the wave function so this |
| |
|
|
| align:start position:0% |
| the uh of the wave function so this |
| describes and then this |
|
|
| align:start position:0% |
| describes and then this |
| |
|
|
| align:start position:0% |
| describes and then this |
| so if you as a generalization of this |
|
|
| align:start position:0% |
| so if you as a generalization of this |
| |
|
|
| align:start position:0% |
| so if you as a generalization of this |
| then this means to describe the |
|
|
| align:start position:0% |
| then this means to describe the |
| |
|
|
| align:start position:0% |
| then this means to describe the |
| um |
|
|
| align:start position:0% |
| um |
| |
|
|
| align:start position:0% |
| um |
| the quantum mechanics of a relativistic |
|
|
| align:start position:0% |
| the quantum mechanics of a relativistic |
| |
|
|
| align:start position:0% |
| the quantum mechanics of a relativistic |
| free particle say of mass m okay of mass |
|
|
| align:start position:0% |
| free particle say of mass m okay of mass |
| |
|
|
| align:start position:0% |
| free particle say of mass m okay of mass |
| m |
|
|
| align:start position:0% |
| m |
| |
|
|
| align:start position:0% |
| m |
| so here the PSI |
|
|
| align:start position:0% |
| so here the PSI |
| |
|
|
| align:start position:0% |
| so here the PSI |
| TX |
|
|
| align:start position:0% |
| TX |
| |
|
|
| align:start position:0% |
| TX |
| is the wave function |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of a relativistic particle |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| relativistic particle |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of mass n |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and we also notice that this equation |
|
|
| align:start position:0% |
| and we also notice that this equation |
| |
|
|
| align:start position:0% |
| and we also notice that this equation |
| actually is the same |
|
|
| align:start position:0% |
| actually is the same |
| |
|
|
| align:start position:0% |
| actually is the same |
| as the simplest field series equation so |
|
|
| align:start position:0% |
| as the simplest field series equation so |
| |
|
|
| align:start position:0% |
| as the simplest field series equation so |
| we also talked about |
|
|
| align:start position:0% |
| we also talked about |
| |
|
|
| align:start position:0% |
| we also talked about |
| a simplest |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| scalar field Theory |
|
|
| align:start position:0% |
| scalar field Theory |
| |
|
|
| align:start position:0% |
| scalar field Theory |
| classical so here is a a simple simple a |
|
|
| align:start position:0% |
| classical so here is a a simple simple a |
| |
|
|
| align:start position:0% |
| classical so here is a a simple simple a |
| classic simple scalar field Theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this series you can write down an |
|
|
| align:start position:0% |
| so this series you can write down an |
| |
|
|
| align:start position:0% |
| so this series you can write down an |
| action |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of the form |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is the simplest |
|
|
| align:start position:0% |
| so this is the simplest |
| |
|
|
| align:start position:0% |
| so this is the simplest |
| see we can write down |
|
|
| align:start position:0% |
| see we can write down |
| |
|
|
| align:start position:0% |
| see we can write down |
| and then a relativistic Environ Theory |
|
|
| align:start position:0% |
| and then a relativistic Environ Theory |
| |
|
|
| align:start position:0% |
| and then a relativistic Environ Theory |
| and then equation of motion of this so |
|
|
| align:start position:0% |
| and then equation of motion of this so |
| |
|
|
| align:start position:0% |
| and then equation of motion of this so |
| this is a you view this as a classical |
|
|
| align:start position:0% |
| this is a you view this as a classical |
| |
|
|
| align:start position:0% |
| this is a you view this as a classical |
| field and again this has the equation |
|
|
| align:start position:0% |
| field and again this has the equation |
| |
|
|
| align:start position:0% |
| field and again this has the equation |
| motion you have to see exact the same |
|
|
| align:start position:0% |
| motion you have to see exact the same |
| |
|
|
| align:start position:0% |
| motion you have to see exact the same |
| form as this equation |
|
|
| align:start position:0% |
| form as this equation |
| |
|
|
| align:start position:0% |
| form as this equation |
| so but now here Phi |
|
|
| align:start position:0% |
| so but now here Phi |
| |
|
|
| align:start position:0% |
| so but now here Phi |
| again is the function of TX |
|
|
| align:start position:0% |
| again is the function of TX |
| |
|
|
| align:start position:0% |
| again is the function of TX |
| now has a completely different physical |
|
|
| align:start position:0% |
| now has a completely different physical |
| |
|
|
| align:start position:0% |
| now has a completely different physical |
| interpretation so here is the uh uh this |
|
|
| align:start position:0% |
| interpretation so here is the uh uh this |
| |
|
|
| align:start position:0% |
| interpretation so here is the uh uh this |
| is a classical field |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is a classical field |
|
|
| align:start position:0% |
| so this is a classical field |
| |
|
|
| align:start position:0% |
| so this is a classical field |
| okay so in this case |
|
|
| align:start position:0% |
| okay so in this case |
| |
|
|
| align:start position:0% |
| okay so in this case |
| the interpretation of the X in here |
|
|
| align:start position:0% |
| the interpretation of the X in here |
| |
|
|
| align:start position:0% |
| the interpretation of the X in here |
| and in here is very different okay so so |
|
|
| align:start position:0% |
| and in here is very different okay so so |
| |
|
|
| align:start position:0% |
| and in here is very different okay so so |
| not only Phi and beside the physical |
|
|
| align:start position:0% |
| not only Phi and beside the physical |
| |
|
|
| align:start position:0% |
| not only Phi and beside the physical |
| interpretation are different the |
|
|
| align:start position:0% |
| interpretation are different the |
| |
|
|
| align:start position:0% |
| interpretation are different the |
| physical interpretation of X also here |
|
|
| align:start position:0% |
| physical interpretation of X also here |
| |
|
|
| align:start position:0% |
| physical interpretation of X also here |
| are different here x is just a label |
|
|
| align:start position:0% |
| are different here x is just a label |
| |
|
|
| align:start position:0% |
| are different here x is just a label |
| is a label for the for the location in |
|
|
| align:start position:0% |
| is a label for the for the location in |
| |
|
|
| align:start position:0% |
| is a label for the for the location in |
| the space which which we Define this |
|
|
| align:start position:0% |
| the space which which we Define this |
| |
|
|
| align:start position:0% |
| the space which which we Define this |
| field |
|
|
| align:start position:0% |
| field |
| |
|
|
| align:start position:0% |
| field |
| but here the x is the eigenvalue |
|
|
| align:start position:0% |
| but here the x is the eigenvalue |
| |
|
|
| align:start position:0% |
| but here the x is the eigenvalue |
| of the position |
|
|
| align:start position:0% |
| of the position |
| |
|
|
| align:start position:0% |
| of the position |
| operator for this right basic particle |
|
|
| align:start position:0% |
| operator for this right basic particle |
| |
|
|
| align:start position:0% |
| operator for this right basic particle |
| okay and so they have very different |
|
|
| align:start position:0% |
| okay and so they have very different |
| |
|
|
| align:start position:0% |
| okay and so they have very different |
| physical imputation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and so let me just label this equation |
|
|
| align:start position:0% |
| and so let me just label this equation |
| |
|
|
| align:start position:0% |
| and so let me just label this equation |
| by one |
|
|
| align:start position:0% |
| by one |
| |
|
|
| align:start position:0% |
| by one |
| enable this by two |
|
|
| align:start position:0% |
| enable this by two |
| |
|
|
| align:start position:0% |
| enable this by two |
| and this by 2 Prime |
|
|
| align:start position:0% |
| and this by 2 Prime |
| |
|
|
| align:start position:0% |
| and this by 2 Prime |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we also mentioned that this one has a |
|
|
| align:start position:0% |
| so we also mentioned that this one has a |
| |
|
|
| align:start position:0% |
| so we also mentioned that this one has a |
| the interpretation of this as the wave |
|
|
| align:start position:0% |
| the interpretation of this as the wave |
| |
|
|
| align:start position:0% |
| the interpretation of this as the wave |
| function for relativistic yeah foreign |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so the first he said |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| as you will show in your PSAT 2 |
|
|
| align:start position:0% |
| as you will show in your PSAT 2 |
| |
|
|
| align:start position:0% |
| as you will show in your PSAT 2 |
| uh there's no sensitive |
|
|
| align:start position:0% |
| uh there's no sensitive |
| |
|
|
| align:start position:0% |
| uh there's no sensitive |
| no sensible way |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| to Define |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| a positive definite |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| probability density okay |
|
|
| align:start position:0% |
| probability density okay |
| |
|
|
| align:start position:0% |
| probability density okay |
| so if you want to interpret |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| this has a wave equation |
|
|
| align:start position:0% |
| this has a wave equation |
| |
|
|
| align:start position:0% |
| this has a wave equation |
| now you must have a way then you must |
|
|
| align:start position:0% |
| now you must have a way then you must |
| |
|
|
| align:start position:0% |
| now you must have a way then you must |
| have a probability density because in |
|
|
| align:start position:0% |
| have a probability density because in |
| |
|
|
| align:start position:0% |
| have a probability density because in |
| quantum mechanics probability should be |
|
|
| align:start position:0% |
| quantum mechanics probability should be |
| |
|
|
| align:start position:0% |
| quantum mechanics probability should be |
| conserved okay |
|
|
| align:start position:0% |
| conserved okay |
| |
|
|
| align:start position:0% |
| conserved okay |
| and the second difficulty |
|
|
| align:start position:0% |
| and the second difficulty |
| |
|
|
| align:start position:0% |
| and the second difficulty |
| is that the selective energy state |
|
|
| align:start position:0% |
| is that the selective energy state |
| |
|
|
| align:start position:0% |
| is that the selective energy state |
| because of the square because when you |
|
|
| align:start position:0% |
| because of the square because when you |
| |
|
|
| align:start position:0% |
| because of the square because when you |
| take the square roots then you get the |
|
|
| align:start position:0% |
| take the square roots then you get the |
| |
|
|
| align:start position:0% |
| take the square roots then you get the |
| minus sign and then this negative energy |
|
|
| align:start position:0% |
| minus sign and then this negative energy |
| |
|
|
| align:start position:0% |
| minus sign and then this negative energy |
| states |
|
|
| align:start position:0% |
| states |
| |
|
|
| align:start position:0% |
| states |
| which you cannot avoid in quantum |
|
|
| align:start position:0% |
| which you cannot avoid in quantum |
| |
|
|
| align:start position:0% |
| which you cannot avoid in quantum |
| mechanics |
|
|
| align:start position:0% |
| mechanics |
| |
|
|
| align:start position:0% |
| mechanics |
| even though classically you can just |
|
|
| align:start position:0% |
| even though classically you can just |
| |
|
|
| align:start position:0% |
| even though classically you can just |
| throw them away perhaps okay |
|
|
| align:start position:0% |
| throw them away perhaps okay |
| |
|
|
| align:start position:0% |
| throw them away perhaps okay |
| and the third thing we mentioned at the |
|
|
| align:start position:0% |
| and the third thing we mentioned at the |
| |
|
|
| align:start position:0% |
| and the third thing we mentioned at the |
| end |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| he said for relativistic |
|
|
| align:start position:0% |
| he said for relativistic |
| |
|
|
| align:start position:0% |
| he said for relativistic |
| wave equation |
|
|
| align:start position:0% |
| wave equation |
| |
|
|
| align:start position:0% |
| wave equation |
| you can describe |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| fixed number of particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so the particle number cannot change |
|
|
| align:start position:0% |
| so the particle number cannot change |
| |
|
|
| align:start position:0% |
| so the particle number cannot change |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so so this way we so this equation if |
|
|
| align:start position:0% |
| so so this way we so this equation if |
| |
|
|
| align:start position:0% |
| so so this way we so this equation if |
| you describe a single particle |
|
|
| align:start position:0% |
| you describe a single particle |
| |
|
|
| align:start position:0% |
| you describe a single particle |
| and if you want to describe two |
|
|
| align:start position:0% |
| and if you want to describe two |
| |
|
|
| align:start position:0% |
| and if you want to describe two |
| particles |
|
|
| align:start position:0% |
| particles |
| |
|
|
| align:start position:0% |
| particles |
| then you leave the two write down a |
|
|
| align:start position:0% |
| then you leave the two write down a |
| |
|
|
| align:start position:0% |
| then you leave the two write down a |
| separate equation |
|
|
| align:start position:0% |
| separate equation |
| |
|
|
| align:start position:0% |
| separate equation |
| for different wave function |
|
|
| align:start position:0% |
| for different wave function |
| |
|
|
| align:start position:0% |
| for different wave function |
| so this is for the two particle wave |
|
|
| align:start position:0% |
| so this is for the two particle wave |
| |
|
|
| align:start position:0% |
| so this is for the two particle wave |
| function will be like this okay and Etc |
|
|
| align:start position:0% |
| function will be like this okay and Etc |
| |
|
|
| align:start position:0% |
| function will be like this okay and Etc |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| but this does not really make sense |
|
|
| align:start position:0% |
| but this does not really make sense |
| |
|
|
| align:start position:0% |
| but this does not really make sense |
| in a relativistic system |
|
|
| align:start position:0% |
| in a relativistic system |
| |
|
|
| align:start position:0% |
| in a relativistic system |
| because we know that in the relative is |
|
|
| align:start position:0% |
| because we know that in the relative is |
| |
|
|
| align:start position:0% |
| because we know that in the relative is |
| existing e equal to m c Square |
|
|
| align:start position:0% |
| existing e equal to m c Square |
| |
|
|
| align:start position:0% |
| existing e equal to m c Square |
| in any case you have enough energy then |
|
|
| align:start position:0% |
| in any case you have enough energy then |
| |
|
|
| align:start position:0% |
| in any case you have enough energy then |
| you should be able to create particles |
|
|
| align:start position:0% |
| you should be able to create particles |
| |
|
|
| align:start position:0% |
| you should be able to create particles |
| and then that means the lumbar particles |
|
|
| align:start position:0% |
| and then that means the lumbar particles |
| |
|
|
| align:start position:0% |
| and then that means the lumbar particles |
| in the given process is not conserved |
|
|
| align:start position:0% |
| in the given process is not conserved |
| |
|
|
| align:start position:0% |
| in the given process is not conserved |
| okay so if you want to use your quantum |
|
|
| align:start position:0% |
| okay so if you want to use your quantum |
| |
|
|
| align:start position:0% |
| okay so if you want to use your quantum |
| mechanics describe a process and then |
|
|
| align:start position:0% |
| mechanics describe a process and then |
| |
|
|
| align:start position:0% |
| mechanics describe a process and then |
| that's you cannot have a formalism which |
|
|
| align:start position:0% |
| that's you cannot have a formalism which |
| |
|
|
| align:start position:0% |
| that's you cannot have a formalism which |
| the number of particle is fixed which |
|
|
| align:start position:0% |
| the number of particle is fixed which |
| |
|
|
| align:start position:0% |
| the number of particle is fixed which |
| you cannot change |
|
|
| align:start position:0% |
| you cannot change |
| |
|
|
| align:start position:0% |
| you cannot change |
| and so so this is actually the most |
|
|
| align:start position:0% |
| and so so this is actually the most |
| |
|
|
| align:start position:0% |
| and so so this is actually the most |
| fundamental difficulty |
|
|
| align:start position:0% |
| fundamental difficulty |
| |
|
|
| align:start position:0% |
| fundamental difficulty |
| okay is that you cannot change the |
|
|
| align:start position:0% |
| okay is that you cannot change the |
| |
|
|
| align:start position:0% |
| okay is that you cannot change the |
| number of particles |
|
|
| align:start position:0% |
| number of particles |
| |
|
|
| align:start position:0% |
| number of particles |
| and related to this difficulty |
|
|
| align:start position:0% |
| and related to this difficulty |
| |
|
|
| align:start position:0% |
| and related to this difficulty |
| is this interpretation |
|
|
| align:start position:0% |
| is this interpretation |
| |
|
|
| align:start position:0% |
| is this interpretation |
| here we say now if you wanted to |
|
|
| align:start position:0% |
| here we say now if you wanted to |
| |
|
|
| align:start position:0% |
| here we say now if you wanted to |
| we say in here |
|
|
| align:start position:0% |
| we say in here |
| |
|
|
| align:start position:0% |
| we say in here |
| there's a fundamental asymmetry between |
|
|
| align:start position:0% |
| there's a fundamental asymmetry between |
| |
|
|
| align:start position:0% |
| there's a fundamental asymmetry between |
| the T and X okay also yeah maybe let me |
|
|
| align:start position:0% |
| the T and X okay also yeah maybe let me |
| |
|
|
| align:start position:0% |
| the T and X okay also yeah maybe let me |
| put it as four which is also fundamental |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| no additional difficulty there's a |
|
|
| align:start position:0% |
| no additional difficulty there's a |
| |
|
|
| align:start position:0% |
| no additional difficulty there's a |
| fundamental asymmetry |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| between |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| T and X |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so here in the wave equation T is |
|
|
| align:start position:0% |
| so so here in the wave equation T is |
| |
|
|
| align:start position:0% |
| so so here in the wave equation T is |
| just a parameter |
|
|
| align:start position:0% |
| just a parameter |
| |
|
|
| align:start position:0% |
| just a parameter |
| which we describe the evolution |
|
|
| align:start position:0% |
| which we describe the evolution |
| |
|
|
| align:start position:0% |
| which we describe the evolution |
| but the x is the eigenvalue |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| over Quantum operators |
|
|
| align:start position:0% |
| over Quantum operators |
| |
|
|
| align:start position:0% |
| over Quantum operators |
| eigenvalues of quantum operators |
|
|
| align:start position:0% |
| eigenvalues of quantum operators |
| |
|
|
| align:start position:0% |
| eigenvalues of quantum operators |
| say a corresponding to say hats by |
|
|
| align:start position:0% |
| say a corresponding to say hats by |
| |
|
|
| align:start position:0% |
| say a corresponding to say hats by |
| putting ahead with the load the |
|
|
| align:start position:0% |
| putting ahead with the load the |
| |
|
|
| align:start position:0% |
| putting ahead with the load the |
| corresponding a Quantum operator yeah so |
|
|
| align:start position:0% |
| corresponding a Quantum operator yeah so |
| |
|
|
| align:start position:0% |
| corresponding a Quantum operator yeah so |
| so this is diagonal value of position |
|
|
| align:start position:0% |
| so this is diagonal value of position |
| |
|
|
| align:start position:0% |
| so this is diagonal value of position |
| operators |
|
|
| align:start position:0% |
| operators |
| |
|
|
| align:start position:0% |
| operators |
| and this AC major become even more |
|
|
| align:start position:0% |
| and this AC major become even more |
| |
|
|
| align:start position:0% |
| and this AC major become even more |
| prolonged so if you can see the two |
|
|
| align:start position:0% |
| prolonged so if you can see the two |
| |
|
|
| align:start position:0% |
| prolonged so if you can see the two |
| particles |
|
|
| align:start position:0% |
| particles |
| |
|
|
| align:start position:0% |
| particles |
| okay you have two x here but there's |
|
|
| align:start position:0% |
| okay you have two x here but there's |
| |
|
|
| align:start position:0% |
| okay you have two x here but there's |
| only one t okay |
|
|
| align:start position:0% |
| only one t okay |
| |
|
|
| align:start position:0% |
| only one t okay |
| but |
|
|
| align:start position:0% |
| but |
| |
|
|
| align:start position:0% |
| but |
| so uh so those because of those |
|
|
| align:start position:0% |
| so uh so those because of those |
| |
|
|
| align:start position:0% |
| so uh so those because of those |
| fundamental difficulties |
|
|
| align:start position:0% |
| fundamental difficulties |
| |
|
|
| align:start position:0% |
| fundamental difficulties |
| okay so if you |
|
|
| align:start position:0% |
| okay so if you |
| |
|
|
| align:start position:0% |
| okay so if you |
| connect to this one to four |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we can we conclude |
|
|
| align:start position:0% |
| so we can we conclude |
| |
|
|
| align:start position:0% |
| so we can we conclude |
| that the |
|
|
| align:start position:0% |
| that the |
| |
|
|
| align:start position:0% |
| that the |
| um |
|
|
| align:start position:0% |
| um |
| |
|
|
| align:start position:0% |
| um |
| relativistic |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| quantum mechanics defined in the sense |
|
|
| align:start position:0% |
| quantum mechanics defined in the sense |
| |
|
|
| align:start position:0% |
| quantum mechanics defined in the sense |
| that you write down a wave equation |
|
|
| align:start position:0% |
| that you write down a wave equation |
| |
|
|
| align:start position:0% |
| that you write down a wave equation |
| and for for wave function don't even it |
|
|
| align:start position:0% |
| and for for wave function don't even it |
| |
|
|
| align:start position:0% |
| and for for wave function don't even it |
| does not be a |
|
|
| align:start position:0% |
| does not be a |
| |
|
|
| align:start position:0% |
| does not be a |
| that's not |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| cannot be |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| a fundamental discussion okay |
|
|
| align:start position:0% |
| a fundamental discussion okay |
| |
|
|
| align:start position:0% |
| a fundamental discussion okay |
| but yeah but right this Quantum kind of |
|
|
| align:start position:0% |
| but yeah but right this Quantum kind of |
| |
|
|
| align:start position:0% |
| but yeah but right this Quantum kind of |
| just refers to this kind of wave |
|
|
| align:start position:0% |
| just refers to this kind of wave |
| |
|
|
| align:start position:0% |
| just refers to this kind of wave |
| equation okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so at most |
|
|
| align:start position:0% |
| so at most |
| |
|
|
| align:start position:0% |
| so at most |
| this can be approximate approximation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| at the most |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| this is approximate description |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| in situations |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| say there's no |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| there's no particle Creation with a |
|
|
| align:start position:0% |
| there's no particle Creation with a |
| |
|
|
| align:start position:0% |
| there's no particle Creation with a |
| violation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so in case which is your |
|
|
| align:start position:0% |
| so so in case which is your |
| |
|
|
| align:start position:0% |
| so so in case which is your |
| party Columbo is fixed |
|
|
| align:start position:0% |
| party Columbo is fixed |
| |
|
|
| align:start position:0% |
| party Columbo is fixed |
| and then the then you can use this as |
|
|
| align:start position:0% |
| and then the then you can use this as |
| |
|
|
| align:start position:0% |
| and then the then you can use this as |
| approximation okay but it cannot be a |
|
|
| align:start position:0% |
| approximation okay but it cannot be a |
| |
|
|
| align:start position:0% |
| approximation okay but it cannot be a |
| fundamental description |
|
|
| align:start position:0% |
| fundamental description |
| |
|
|
| align:start position:0% |
| fundamental description |
| for example later we will talk about the |
|
|
| align:start position:0% |
| for example later we will talk about the |
| |
|
|
| align:start position:0% |
| for example later we will talk about the |
| fumionic version of this wave equation |
|
|
| align:start position:0% |
| fumionic version of this wave equation |
| |
|
|
| align:start position:0% |
| fumionic version of this wave equation |
| so this will describe a particle without |
|
|
| align:start position:0% |
| so this will describe a particle without |
| |
|
|
| align:start position:0% |
| so this will describe a particle without |
| spin so later we will describe the the |
|
|
| align:start position:0% |
| spin so later we will describe the the |
| |
|
|
| align:start position:0% |
| spin so later we will describe the the |
| linear equation for electrons for spin |
|
|
| align:start position:0% |
| linear equation for electrons for spin |
| |
|
|
| align:start position:0% |
| linear equation for electrons for spin |
| half and then then that can indeed be |
|
|
| align:start position:0% |
| half and then then that can indeed be |
| |
|
|
| align:start position:0% |
| half and then then that can indeed be |
| used to describe electron in the |
|
|
| align:start position:0% |
| used to describe electron in the |
| |
|
|
| align:start position:0% |
| used to describe electron in the |
| hydrogen atom as how as far as you don't |
|
|
| align:start position:0% |
| hydrogen atom as how as far as you don't |
| |
|
|
| align:start position:0% |
| hydrogen atom as how as far as you don't |
| create new electrons Etc |
|
|
| align:start position:0% |
| create new electrons Etc |
| |
|
|
| align:start position:0% |
| create new electrons Etc |
| anyway so so so write this Quantum |
|
|
| align:start position:0% |
| anyway so so so write this Quantum |
| |
|
|
| align:start position:0% |
| anyway so so so write this Quantum |
| connect only be described as some kind |
|
|
| align:start position:0% |
| connect only be described as some kind |
| |
|
|
| align:start position:0% |
| connect only be described as some kind |
| of consider as approximate description |
|
|
| align:start position:0% |
| of consider as approximate description |
| |
|
|
| align:start position:0% |
| of consider as approximate description |
| okay but now if you want to you unify |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| special relativity and quantum mechanics |
|
|
| align:start position:0% |
| special relativity and quantum mechanics |
| |
|
|
| align:start position:0% |
| special relativity and quantum mechanics |
| together |
|
|
| align:start position:0% |
| together |
| |
|
|
| align:start position:0% |
| together |
| it turns out that the right formulation |
|
|
| align:start position:0% |
| it turns out that the right formulation |
| |
|
|
| align:start position:0% |
| it turns out that the right formulation |
| is just Quantum field Theory okay |
|
|
| align:start position:0% |
| is just Quantum field Theory okay |
| |
|
|
| align:start position:0% |
| is just Quantum field Theory okay |
| so it turns out that the quantum field |
|
|
| align:start position:0% |
| so it turns out that the quantum field |
| |
|
|
| align:start position:0% |
| so it turns out that the quantum field |
| Theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| a corner of your theory |
|
|
| align:start position:0% |
| a corner of your theory |
| |
|
|
| align:start position:0% |
| a corner of your theory |
| addresses these difficulties okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so it turns out |
|
|
| align:start position:0% |
| so it turns out |
| |
|
|
| align:start position:0% |
| so it turns out |
| it turns out the right way so if we want |
|
|
| align:start position:0% |
| it turns out the right way so if we want |
| |
|
|
| align:start position:0% |
| it turns out the right way so if we want |
| to describe quantum mechanics say of |
|
|
| align:start position:0% |
| to describe quantum mechanics say of |
| |
|
|
| align:start position:0% |
| to describe quantum mechanics say of |
| write a specific particles of mass m |
|
|
| align:start position:0% |
| write a specific particles of mass m |
| |
|
|
| align:start position:0% |
| write a specific particles of mass m |
| okay as we want to do here |
|
|
| align:start position:0% |
| okay as we want to do here |
| |
|
|
| align:start position:0% |
| okay as we want to do here |
| it turns out the proper thing to do |
|
|
| align:start position:0% |
| it turns out the proper thing to do |
| |
|
|
| align:start position:0% |
| it turns out the proper thing to do |
| which is a little bit unintuitive at |
|
|
| align:start position:0% |
| which is a little bit unintuitive at |
| |
|
|
| align:start position:0% |
| which is a little bit unintuitive at |
| first sight |
|
|
| align:start position:0% |
| first sight |
| |
|
|
| align:start position:0% |
| first sight |
| is to start with this field Siri okay |
|
|
| align:start position:0% |
| is to start with this field Siri okay |
| |
|
|
| align:start position:0% |
| is to start with this field Siri okay |
| which seemingly have nothing to do with |
|
|
| align:start position:0% |
| which seemingly have nothing to do with |
| |
|
|
| align:start position:0% |
| which seemingly have nothing to do with |
| write a basic particle but to start with |
|
|
| align:start position:0% |
| write a basic particle but to start with |
| |
|
|
| align:start position:0% |
| write a basic particle but to start with |
| this classical field Theory |
|
|
| align:start position:0% |
| this classical field Theory |
| |
|
|
| align:start position:0% |
| this classical field Theory |
| and then contact it okay it hands out |
|
|
| align:start position:0% |
| and then contact it okay it hands out |
| |
|
|
| align:start position:0% |
| and then contact it okay it hands out |
| turns out once you treat this Theory as |
|
|
| align:start position:0% |
| turns out once you treat this Theory as |
| |
|
|
| align:start position:0% |
| turns out once you treat this Theory as |
| a Quantum field Theory and this becomes |
|
|
| align:start position:0% |
| a Quantum field Theory and this becomes |
| |
|
|
| align:start position:0% |
| a Quantum field Theory and this becomes |
| a theory of arbitrary number of |
|
|
| align:start position:0% |
| a theory of arbitrary number of |
| |
|
|
| align:start position:0% |
| a theory of arbitrary number of |
| relativistic particles of mass m |
|
|
| align:start position:0% |
| relativistic particles of mass m |
| |
|
|
| align:start position:0% |
| relativistic particles of mass m |
| okay and so that's the non-intuitive |
|
|
| align:start position:0% |
| okay and so that's the non-intuitive |
| |
|
|
| align:start position:0% |
| okay and so that's the non-intuitive |
| part and and that's the uh one of the |
|
|
| align:start position:0% |
| part and and that's the uh one of the |
| |
|
|
| align:start position:0% |
| part and and that's the uh one of the |
| miracle say of the field theory is that |
|
|
| align:start position:0% |
| miracle say of the field theory is that |
| |
|
|
| align:start position:0% |
| miracle say of the field theory is that |
| automatically give you a formalism |
|
|
| align:start position:0% |
| automatically give you a formalism |
| |
|
|
| align:start position:0% |
| automatically give you a formalism |
| for for treating arbitrary lumbar |
|
|
| align:start position:0% |
| for for treating arbitrary lumbar |
| |
|
|
| align:start position:0% |
| for for treating arbitrary lumbar |
| particles okay |
|
|
| align:start position:0% |
| particles okay |
| |
|
|
| align:start position:0% |
| particles okay |
| and uh um yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| um and also in field Theory |
|
|
| align:start position:0% |
| um and also in field Theory |
| |
|
|
| align:start position:0% |
| um and also in field Theory |
| so both T and X are parameters okay even |
|
|
| align:start position:0% |
| so both T and X are parameters okay even |
| |
|
|
| align:start position:0% |
| so both T and X are parameters okay even |
| though X only labels your uh uh your |
|
|
| align:start position:0% |
| though X only labels your uh uh your |
| |
|
|
| align:start position:0% |
| though X only labels your uh uh your |
| location so both T and X are parameters |
|
|
| align:start position:0% |
| location so both T and X are parameters |
| |
|
|
| align:start position:0% |
| location so both T and X are parameters |
| and so you can easily to make them to be |
|
|
| align:start position:0% |
| and so you can easily to make them to be |
| |
|
|
| align:start position:0% |
| and so you can easily to make them to be |
| on equal ground to be compatible with |
|
|
| align:start position:0% |
| on equal ground to be compatible with |
| |
|
|
| align:start position:0% |
| on equal ground to be compatible with |
| special relativity |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| um |
|
|
| align:start position:0% |
| um |
| |
|
|
| align:start position:0% |
| um |
| good |
|
|
| align:start position:0% |
| good |
| |
|
|
| align:start position:0% |
| good |
| so any questions on this |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so we will see that the uh the |
|
|
| align:start position:0% |
| okay so we will see that the uh the |
| |
|
|
| align:start position:0% |
| okay so we will see that the uh the |
| right framework is quantum field three |
|
|
| align:start position:0% |
| right framework is quantum field three |
| |
|
|
| align:start position:0% |
| right framework is quantum field three |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so finally it's the last motivation for |
|
|
| align:start position:0% |
| so finally it's the last motivation for |
| |
|
|
| align:start position:0% |
| so finally it's the last motivation for |
| Quantum field Theory |
|
|
| align:start position:0% |
| Quantum field Theory |
| |
|
|
| align:start position:0% |
| Quantum field Theory |
| so we quickly uh uh describe the last uh |
|
|
| align:start position:0% |
| so we quickly uh uh describe the last uh |
| |
|
|
| align:start position:0% |
| so we quickly uh uh describe the last uh |
| so the fields here you can also arise |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| as a limit |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of discrete systems |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and this is the uh uh the most relevant |
|
|
| align:start position:0% |
| and this is the uh uh the most relevant |
| |
|
|
| align:start position:0% |
| and this is the uh uh the most relevant |
| for this matter physics for example |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so let's just consider |
|
|
| align:start position:0% |
| so so let's just consider |
| |
|
|
| align:start position:0% |
| so so let's just consider |
| say uh some yeah let's consider 803 |
|
|
| align:start position:0% |
| say uh some yeah let's consider 803 |
| |
|
|
| align:start position:0% |
| say uh some yeah let's consider 803 |
| example okay |
|
|
| align:start position:0% |
| example okay |
| |
|
|
| align:start position:0% |
| example okay |
| so let's imagine you have |
|
|
| align:start position:0% |
| so let's imagine you have |
| |
|
|
| align:start position:0% |
| so let's imagine you have |
| just number of particles a number of the |
|
|
| align:start position:0% |
| just number of particles a number of the |
| |
|
|
| align:start position:0% |
| just number of particles a number of the |
| atoms say on the on the Chain okay and |
|
|
| align:start position:0% |
| atoms say on the on the Chain okay and |
| |
|
|
| align:start position:0% |
| atoms say on the on the Chain okay and |
| then they're connected by some screens |
|
|
| align:start position:0% |
| then they're connected by some screens |
| |
|
|
| align:start position:0% |
| then they're connected by some screens |
| between them |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so so this is the uh uh the pro |
|
|
| align:start position:0% |
| okay so so this is the uh uh the pro |
| |
|
|
| align:start position:0% |
| okay so so this is the uh uh the pro |
| yeah consider this to be infinite okay |
|
|
| align:start position:0% |
| yeah consider this to be infinite okay |
| |
|
|
| align:start position:0% |
| yeah consider this to be infinite okay |
| and the spacing between them say is a |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| item I fixed on some matches points |
|
|
| align:start position:0% |
| item I fixed on some matches points |
| |
|
|
| align:start position:0% |
| item I fixed on some matches points |
| and the lattice facing is a okay |
|
|
| align:start position:0% |
| and the lattice facing is a okay |
| |
|
|
| align:start position:0% |
| and the lattice facing is a okay |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| yeah so we can label the other say |
|
|
| align:start position:0% |
| yeah so we can label the other say |
| |
|
|
| align:start position:0% |
| yeah so we can label the other say |
| each particle by their position for |
|
|
| align:start position:0% |
| each particle by their position for |
| |
|
|
| align:start position:0% |
| each particle by their position for |
| example this is x0 this is X1 this is X2 |
|
|
| align:start position:0% |
| example this is x0 this is X1 this is X2 |
| |
|
|
| align:start position:0% |
| example this is x0 this is X1 this is X2 |
| Etc okay and the typical particle is x n |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| at the location of nth particle is x n |
|
|
| align:start position:0% |
| at the location of nth particle is x n |
| |
|
|
| align:start position:0% |
| at the location of nth particle is x n |
| and so we can also introduce the |
|
|
| align:start position:0% |
| and so we can also introduce the |
| |
|
|
| align:start position:0% |
| and so we can also introduce the |
| deviation |
|
|
| align:start position:0% |
| deviation |
| |
|
|
| align:start position:0% |
| deviation |
| between the equivalent position of each |
|
|
| align:start position:0% |
| between the equivalent position of each |
| |
|
|
| align:start position:0% |
| between the equivalent position of each |
| particle so let's call it n |
|
|
| align:start position:0% |
| particle so let's call it n |
| |
|
|
| align:start position:0% |
| particle so let's call it n |
| okay so now let's consider the Dynamics |
|
|
| align:start position:0% |
| okay so now let's consider the Dynamics |
| |
|
|
| align:start position:0% |
| okay so now let's consider the Dynamics |
| of ether M for this Theory |
|
|
| align:start position:0% |
| of ether M for this Theory |
| |
|
|
| align:start position:0% |
| of ether M for this Theory |
| and so this is the just deviation of the |
|
|
| align:start position:0% |
| and so this is the just deviation of the |
| |
|
|
| align:start position:0% |
| and so this is the just deviation of the |
| nth particle from its equivalent |
|
|
| align:start position:0% |
| nth particle from its equivalent |
| |
|
|
| align:start position:0% |
| nth particle from its equivalent |
| position okay so X and zero is its |
|
|
| align:start position:0% |
| position okay so X and zero is its |
| |
|
|
| align:start position:0% |
| position okay so X and zero is its |
| equivalent position |
|
|
| align:start position:0% |
| equivalent position |
| |
|
|
| align:start position:0% |
| equivalent position |
| so now so now if you write down the |
|
|
| align:start position:0% |
| so now so now if you write down the |
| |
|
|
| align:start position:0% |
| so now so now if you write down the |
| lagrangian for this system |
|
|
| align:start position:0% |
| lagrangian for this system |
| |
|
|
| align:start position:0% |
| lagrangian for this system |
| that we can easily do you just write T |
|
|
| align:start position:0% |
| that we can easily do you just write T |
| |
|
|
| align:start position:0% |
| that we can easily do you just write T |
| minus V the t is the kinetic energy and |
|
|
| align:start position:0% |
| minus V the t is the kinetic energy and |
| |
|
|
| align:start position:0% |
| minus V the t is the kinetic energy and |
| V is the uh um the potential energy so |
|
|
| align:start position:0% |
| V is the uh um the potential energy so |
| |
|
|
| align:start position:0% |
| V is the uh um the potential energy so |
| we can just write it as sum over n |
|
|
| align:start position:0% |
| we can just write it as sum over n |
| |
|
|
| align:start position:0% |
| we can just write it as sum over n |
| over o |
|
|
| align:start position:0% |
| over o |
| |
|
|
| align:start position:0% |
| over o |
| particles and then let's assume they |
|
|
| align:start position:0% |
| particles and then let's assume they |
| |
|
|
| align:start position:0% |
| particles and then let's assume they |
| have the same mass let's write mu |
|
|
| align:start position:0% |
| have the same mass let's write mu |
| |
|
|
| align:start position:0% |
| have the same mass let's write mu |
| ETA n dot Square okay so this is a |
|
|
| align:start position:0% |
| ETA n dot Square okay so this is a |
| |
|
|
| align:start position:0% |
| ETA n dot Square okay so this is a |
| kinetic term so so mu is the mass for |
|
|
| align:start position:0% |
| kinetic term so so mu is the mass for |
| |
|
|
| align:start position:0% |
| kinetic term so so mu is the mass for |
| each particle |
|
|
| align:start position:0% |
| each particle |
| |
|
|
| align:start position:0% |
| each particle |
| and then their their potential yeah |
|
|
| align:start position:0% |
| and then their their potential yeah |
| |
|
|
| align:start position:0% |
| and then their their potential yeah |
| let's assume at each point there is also |
|
|
| align:start position:0% |
| let's assume at each point there is also |
| |
|
|
| align:start position:0% |
| let's assume at each point there is also |
| a yeah let's just yeah then there's some |
|
|
| align:start position:0% |
| a yeah let's just yeah then there's some |
| |
|
|
| align:start position:0% |
| a yeah let's just yeah then there's some |
| uh uh uh interaction because each |
|
|
| align:start position:0% |
| uh uh uh interaction because each |
| |
|
|
| align:start position:0% |
| uh uh uh interaction because each |
| particle are connected by the spring and |
|
|
| align:start position:0% |
| particle are connected by the spring and |
| |
|
|
| align:start position:0% |
| particle are connected by the spring and |
| so they're a harmonic Force |
|
|
| align:start position:0% |
| so they're a harmonic Force |
| |
|
|
| align:start position:0% |
| so they're a harmonic Force |
| between |
|
|
| align:start position:0% |
| between |
| |
|
|
| align:start position:0% |
| between |
| neighboring particles okay |
|
|
| align:start position:0% |
| neighboring particles okay |
| |
|
|
| align:start position:0% |
| neighboring particles okay |
| and now let's imagine also there's a |
|
|
| align:start position:0% |
| and now let's imagine also there's a |
| |
|
|
| align:start position:0% |
| and now let's imagine also there's a |
| harmonic potential which trapped this |
|
|
| align:start position:0% |
| harmonic potential which trapped this |
| |
|
|
| align:start position:0% |
| harmonic potential which trapped this |
| particle itself |
|
|
| align:start position:0% |
| particle itself |
| |
|
|
| align:start position:0% |
| particle itself |
| at each location |
|
|
| align:start position:0% |
| at each location |
| |
|
|
| align:start position:0% |
| at each location |
| okay so this is a very simple uh spring |
|
|
| align:start position:0% |
| okay so this is a very simple uh spring |
| |
|
|
| align:start position:0% |
| okay so this is a very simple uh spring |
| and the particle problem which you |
|
|
| align:start position:0% |
| and the particle problem which you |
| |
|
|
| align:start position:0% |
| and the particle problem which you |
| encounter say in 803 |
|
|
| align:start position:0% |
| encounter say in 803 |
| |
|
|
| align:start position:0% |
| encounter say in 803 |
| okay is this problem clear |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay I I assume most of you have seen |
|
|
| align:start position:0% |
| okay I I assume most of you have seen |
| |
|
|
| align:start position:0% |
| okay I I assume most of you have seen |
| this problem before |
|
|
| align:start position:0% |
| this problem before |
| |
|
|
| align:start position:0% |
| this problem before |
| and the and you're tasking 803 is |
|
|
| align:start position:0% |
| and the and you're tasking 803 is |
| |
|
|
| align:start position:0% |
| and the and you're tasking 803 is |
| actually to uh to find the lower modes |
|
|
| align:start position:0% |
| actually to uh to find the lower modes |
| |
|
|
| align:start position:0% |
| actually to uh to find the lower modes |
| say of this system okay |
|
|
| align:start position:0% |
| say of this system okay |
| |
|
|
| align:start position:0% |
| say of this system okay |
| and in 803 you also describe that we can |
|
|
| align:start position:0% |
| and in 803 you also describe that we can |
| |
|
|
| align:start position:0% |
| and in 803 you also describe that we can |
| in a go to zero limit |
|
|
| align:start position:0% |
| in a go to zero limit |
| |
|
|
| align:start position:0% |
| in a go to zero limit |
| so if the left is spacing is very small |
|
|
| align:start position:0% |
| so if the left is spacing is very small |
| |
|
|
| align:start position:0% |
| so if the left is spacing is very small |
| and if you're only interested in the |
|
|
| align:start position:0% |
| and if you're only interested in the |
| |
|
|
| align:start position:0% |
| and if you're only interested in the |
| behavior of the system at a very large |
|
|
| align:start position:0% |
| behavior of the system at a very large |
| |
|
|
| align:start position:0% |
| behavior of the system at a very large |
| distance say the distance much larger |
|
|
| align:start position:0% |
| distance say the distance much larger |
| |
|
|
| align:start position:0% |
| distance say the distance much larger |
| than a equal imagine lighter than a |
|
|
| align:start position:0% |
| than a equal imagine lighter than a |
| |
|
|
| align:start position:0% |
| than a equal imagine lighter than a |
| then you can essentially choose this |
|
|
| align:start position:0% |
| then you can essentially choose this |
| |
|
|
| align:start position:0% |
| then you can essentially choose this |
| season as a Continuum okay you don't |
|
|
| align:start position:0% |
| season as a Continuum okay you don't |
| |
|
|
| align:start position:0% |
| season as a Continuum okay you don't |
| have to resolve individual particles |
|
|
| align:start position:0% |
| have to resolve individual particles |
| |
|
|
| align:start position:0% |
| have to resolve individual particles |
| and so we can just enable the newer |
|
|
| align:start position:0% |
| and so we can just enable the newer |
| |
|
|
| align:start position:0% |
| and so we can just enable the newer |
| limits so you can choose the chain |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and uh so you |
|
|
| align:start position:0% |
| and uh so you |
| |
|
|
| align:start position:0% |
| and uh so you |
| so each |
|
|
| align:start position:0% |
| so each |
| |
|
|
| align:start position:0% |
| so each |
| and T you replace it |
|
|
| align:start position:0% |
| and T you replace it |
| |
|
|
| align:start position:0% |
| and T you replace it |
| by ETA x t |
|
|
| align:start position:0% |
| by ETA x t |
| |
|
|
| align:start position:0% |
| by ETA x t |
| so X label is position |
|
|
| align:start position:0% |
| so X label is position |
| |
|
|
| align:start position:0% |
| so X label is position |
| okay the X label is position and T |
|
|
| align:start position:0% |
| okay the X label is position and T |
| |
|
|
| align:start position:0% |
| okay the X label is position and T |
| describes the Dynamics okay so so ETA is |
|
|
| align:start position:0% |
| describes the Dynamics okay so so ETA is |
| |
|
|
| align:start position:0% |
| describes the Dynamics okay so so ETA is |
| the deviation |
|
|
| align:start position:0% |
| the deviation |
| |
|
|
| align:start position:0% |
| the deviation |
| at the location X and it's depend on T |
|
|
| align:start position:0% |
| at the location X and it's depend on T |
| |
|
|
| align:start position:0% |
| at the location X and it's depend on T |
| okay so this is the oscillator |
|
|
| align:start position:0% |
| okay so this is the oscillator |
| |
|
|
| align:start position:0% |
| okay so this is the oscillator |
| and then sum over n |
|
|
| align:start position:0% |
| and then sum over n |
| |
|
|
| align:start position:0% |
| and then sum over n |
| in the lagrangian |
|
|
| align:start position:0% |
| in the lagrangian |
| |
|
|
| align:start position:0% |
| in the lagrangian |
| then we can replace it by integral |
|
|
| align:start position:0% |
| then we can replace it by integral |
| |
|
|
| align:start position:0% |
| then we can replace it by integral |
| over DX |
|
|
| align:start position:0% |
| over DX |
| |
|
|
| align:start position:0% |
| over DX |
| okay and now you just choose this as a |
|
|
| align:start position:0% |
| okay and now you just choose this as a |
| |
|
|
| align:start position:0% |
| okay and now you just choose this as a |
| one-dimensional continuum see some |
|
|
| align:start position:0% |
| one-dimensional continuum see some |
| |
|
|
| align:start position:0% |
| one-dimensional continuum see some |
| integration of the X but of course here |
|
|
| align:start position:0% |
| integration of the X but of course here |
| |
|
|
| align:start position:0% |
| integration of the X but of course here |
| there's a label a nothing spacing so so |
|
|
| align:start position:0% |
| there's a label a nothing spacing so so |
| |
|
|
| align:start position:0% |
| there's a label a nothing spacing so so |
| the uh so the implementational uh here |
|
|
| align:start position:0% |
| the uh so the implementational uh here |
| |
|
|
| align:start position:0% |
| the uh so the implementational uh here |
| the element is a so a times |
|
|
| align:start position:0% |
| the element is a so a times |
| |
|
|
| align:start position:0% |
| the element is a so a times |
| the sum Over N you can replace it by DX |
|
|
| align:start position:0% |
| the sum Over N you can replace it by DX |
| |
|
|
| align:start position:0% |
| the sum Over N you can replace it by DX |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| is the lattice spacing |
|
|
| align:start position:0% |
| is the lattice spacing |
| |
|
|
| align:start position:0% |
| is the lattice spacing |
| and now you can just write |
|
|
| align:start position:0% |
| and now you can just write |
| |
|
|
| align:start position:0% |
| and now you can just write |
| this Lagrange in terms of Continuum |
|
|
| align:start position:0% |
| this Lagrange in terms of Continuum |
| |
|
|
| align:start position:0% |
| this Lagrange in terms of Continuum |
| Theory |
|
|
| align:start position:0% |
| Theory |
| |
|
|
| align:start position:0% |
| Theory |
| okay now you can write this in the |
|
|
| align:start position:0% |
| okay now you can write this in the |
| |
|
|
| align:start position:0% |
| okay now you can write this in the |
| ground in a Continuum Theory and then |
|
|
| align:start position:0% |
| ground in a Continuum Theory and then |
| |
|
|
| align:start position:0% |
| ground in a Continuum Theory and then |
| let's just do it |
|
|
| align:start position:0% |
| let's just do it |
| |
|
|
| align:start position:0% |
| let's just do it |
| so um so we can write it yeah let me |
|
|
| align:start position:0% |
| so um so we can write it yeah let me |
| |
|
|
| align:start position:0% |
| so um so we can write it yeah let me |
| just write one more step |
|
|
| align:start position:0% |
| just write one more step |
| |
|
|
| align:start position:0% |
| just write one more step |
| so you can write it as sum over a we |
|
|
| align:start position:0% |
| so you can write it as sum over a we |
| |
|
|
| align:start position:0% |
| so you can write it as sum over a we |
| take the a factor out because of the a |
|
|
| align:start position:0% |
| take the a factor out because of the a |
| |
|
|
| align:start position:0% |
| take the a factor out because of the a |
| fact R have to be changed into |
|
|
| align:start position:0% |
| fact R have to be changed into |
| |
|
|
| align:start position:0% |
| fact R have to be changed into |
| integration |
|
|
| align:start position:0% |
| integration |
| |
|
|
| align:start position:0% |
| integration |
| and then you have one half mu divided by |
|
|
| align:start position:0% |
| and then you have one half mu divided by |
| |
|
|
| align:start position:0% |
| and then you have one half mu divided by |
| a |
|
|
| align:start position:0% |
| a |
| |
|
|
| align:start position:0% |
| a |
| ETA N squared |
|
|
| align:start position:0% |
| ETA N squared |
| |
|
|
| align:start position:0% |
| ETA N squared |
| minus one half Lambda a |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so I just slightly rewrite |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| it's lagranging |
|
|
| align:start position:0% |
| it's lagranging |
| |
|
|
| align:start position:0% |
| it's lagranging |
| so that it is easy to take the Continuum |
|
|
| align:start position:0% |
| so that it is easy to take the Continuum |
| |
|
|
| align:start position:0% |
| so that it is easy to take the Continuum |
| limit |
|
|
| align:start position:0% |
| limit |
| |
|
|
| align:start position:0% |
| limit |
| so we have taken the factor of a out |
|
|
| align:start position:0% |
| so we have taken the factor of a out |
| |
|
|
| align:start position:0% |
| so we have taken the factor of a out |
| but for this term |
|
|
| align:start position:0% |
| but for this term |
| |
|
|
| align:start position:0% |
| but for this term |
| because this contains the difference |
|
|
| align:start position:0% |
| because this contains the difference |
| |
|
|
| align:start position:0% |
| because this contains the difference |
| between the two and we also divided by a |
|
|
| align:start position:0% |
| between the two and we also divided by a |
| |
|
|
| align:start position:0% |
| between the two and we also divided by a |
| in the downstairs and then we need to |
|
|
| align:start position:0% |
| in the downstairs and then we need to |
| |
|
|
| align:start position:0% |
| in the downstairs and then we need to |
| multiply a upstairs and then there's a |
|
|
| align:start position:0% |
| multiply a upstairs and then there's a |
| |
|
|
| align:start position:0% |
| multiply a upstairs and then there's a |
| in the front okay |
|
|
| align:start position:0% |
| in the front okay |
| |
|
|
| align:start position:0% |
| in the front okay |
| another Continuum limit |
|
|
| align:start position:0% |
| another Continuum limit |
| |
|
|
| align:start position:0% |
| another Continuum limit |
| you can just replace this by integral |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and now I just |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| you can just write it as one half |
|
|
| align:start position:0% |
| you can just write it as one half |
| |
|
|
| align:start position:0% |
| you can just write it as one half |
| you tilde |
|
|
| align:start position:0% |
| you tilde |
| |
|
|
| align:start position:0% |
| you tilde |
| is a start Square so now e to n just |
|
|
| align:start position:0% |
| is a start Square so now e to n just |
| |
|
|
| align:start position:0% |
| is a start Square so now e to n just |
| replace it by E to x t |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and here let me call it lamina tilde |
|
|
| align:start position:0% |
| and here let me call it lamina tilde |
| |
|
|
| align:start position:0% |
| and here let me call it lamina tilde |
| partial X is a square |
|
|
| align:start position:0% |
| partial X is a square |
| |
|
|
| align:start position:0% |
| partial X is a square |
| and this term we can just replace it by |
|
|
| align:start position:0% |
| and this term we can just replace it by |
| |
|
|
| align:start position:0% |
| and this term we can just replace it by |
| the derivative of ETA and then this is |
|
|
| align:start position:0% |
| the derivative of ETA and then this is |
| |
|
|
| align:start position:0% |
| the derivative of ETA and then this is |
| just become one half Sigma tilde is a |
|
|
| align:start position:0% |
| just become one half Sigma tilde is a |
| |
|
|
| align:start position:0% |
| just become one half Sigma tilde is a |
| square |
|
|
| align:start position:0% |
| square |
| |
|
|
| align:start position:0% |
| square |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and the uh the MU tilde of course |
|
|
| align:start position:0% |
| and the uh the MU tilde of course |
| |
|
|
| align:start position:0% |
| and the uh the MU tilde of course |
| is Mu divided by a |
|
|
| align:start position:0% |
| is Mu divided by a |
| |
|
|
| align:start position:0% |
| is Mu divided by a |
| number is Lambda times a and sigma tilde |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| this is the sigma divided by a |
|
|
| align:start position:0% |
| this is the sigma divided by a |
| |
|
|
| align:start position:0% |
| this is the sigma divided by a |
| okay so the Continuum limit is that |
|
|
| align:start position:0% |
| okay so the Continuum limit is that |
| |
|
|
| align:start position:0% |
| okay so the Continuum limit is that |
| those quantity has to be fixed okay the |
|
|
| align:start position:0% |
| those quantity has to be fixed okay the |
| |
|
|
| align:start position:0% |
| those quantity has to be fixed okay the |
| tier the quantity has to be fixed and |
|
|
| align:start position:0% |
| tier the quantity has to be fixed and |
| |
|
|
| align:start position:0% |
| tier the quantity has to be fixed and |
| then we have a continuum lagrangea |
|
|
| align:start position:0% |
| then we have a continuum lagrangea |
| |
|
|
| align:start position:0% |
| then we have a continuum lagrangea |
| okay and then we have a classical field |
|
|
| align:start position:0% |
| okay and then we have a classical field |
| |
|
|
| align:start position:0% |
| okay and then we have a classical field |
| Theory |
|
|
| align:start position:0% |
| Theory |
| |
|
|
| align:start position:0% |
| Theory |
| and this series is essentially the same |
|
|
| align:start position:0% |
| and this series is essentially the same |
| |
|
|
| align:start position:0% |
| and this series is essentially the same |
| as |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| as that theory |
|
|
| align:start position:0% |
| as that theory |
| |
|
|
| align:start position:0% |
| as that theory |
| okay so if you take this Factor mu tilde |
|
|
| align:start position:0% |
| okay so if you take this Factor mu tilde |
| |
|
|
| align:start position:0% |
| okay so if you take this Factor mu tilde |
| out |
|
|
| align:start position:0% |
| out |
| |
|
|
| align:start position:0% |
| out |
| okay if we take this Factor mu tilde out |
|
|
| align:start position:0% |
| okay if we take this Factor mu tilde out |
| |
|
|
| align:start position:0% |
| okay if we take this Factor mu tilde out |
| okay so let me just take this part of |
|
|
| align:start position:0% |
| okay so let me just take this part of |
| |
|
|
| align:start position:0% |
| okay so let me just take this part of |
| mutilda out in the front just up to |
|
|
| align:start position:0% |
| mutilda out in the front just up to |
| |
|
|
| align:start position:0% |
| mutilda out in the front just up to |
| overall factor and here is Lambda tilde |
|
|
| align:start position:0% |
| overall factor and here is Lambda tilde |
| |
|
|
| align:start position:0% |
| overall factor and here is Lambda tilde |
| by a divided by mu tier that we could |
|
|
| align:start position:0% |
| by a divided by mu tier that we could |
| |
|
|
| align:start position:0% |
| by a divided by mu tier that we could |
| let's call it V Square |
|
|
| align:start position:0% |
| let's call it V Square |
| |
|
|
| align:start position:0% |
| let's call it V Square |
| and this becomes Sigma material divided |
|
|
| align:start position:0% |
| and this becomes Sigma material divided |
| |
|
|
| align:start position:0% |
| and this becomes Sigma material divided |
| by uh by mu till let's call it m Square |
|
|
| align:start position:0% |
| by uh by mu till let's call it m Square |
| |
|
|
| align:start position:0% |
| by uh by mu till let's call it m Square |
| so the V Square |
|
|
| align:start position:0% |
| so the V Square |
| |
|
|
| align:start position:0% |
| so the V Square |
| is equal to Mu tilde |
|
|
| align:start position:0% |
| is equal to Mu tilde |
| |
|
|
| align:start position:0% |
| is equal to Mu tilde |
| is equal to the Lambda tilde |
|
|
| align:start position:0% |
| is equal to the Lambda tilde |
| |
|
|
| align:start position:0% |
| is equal to the Lambda tilde |
| divided by mu tilde |
|
|
| align:start position:0% |
| divided by mu tilde |
| |
|
|
| align:start position:0% |
| divided by mu tilde |
| and the M Square |
|
|
| align:start position:0% |
| and the M Square |
| |
|
|
| align:start position:0% |
| and the M Square |
| is equal to Sigma material |
|
|
| align:start position:0% |
| is equal to Sigma material |
| |
|
|
| align:start position:0% |
| is equal to Sigma material |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and then this is just essentially |
|
|
| align:start position:0% |
| and then this is just essentially |
| |
|
|
| align:start position:0% |
| and then this is just essentially |
| identical to that theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| when V equal to one okay so when b equal |
|
|
| align:start position:0% |
| when V equal to one okay so when b equal |
| |
|
|
| align:start position:0% |
| when V equal to one okay so when b equal |
| to one |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| become the same |
|
|
| align:start position:0% |
| become the same |
| |
|
|
| align:start position:0% |
| become the same |
| as just two okay equation two |
|
|
| align:start position:0% |
| as just two okay equation two |
| |
|
|
| align:start position:0% |
| as just two okay equation two |
| of course corresponding to relative K is |
|
|
| align:start position:0% |
| of course corresponding to relative K is |
| |
|
|
| align:start position:0% |
| of course corresponding to relative K is |
| a speed of light but in general uh this |
|
|
| align:start position:0% |
| a speed of light but in general uh this |
| |
|
|
| align:start position:0% |
| a speed of light but in general uh this |
| describer can yeah addition I can |
|
|
| align:start position:0% |
| describer can yeah addition I can |
| |
|
|
| align:start position:0% |
| describer can yeah addition I can |
| describe a Contin but in general this is |
|
|
| align:start position:0% |
| describe a Contin but in general this is |
| |
|
|
| align:start position:0% |
| describe a Contin but in general this is |
| a noun in general this can be just a |
|
|
| align:start position:0% |
| a noun in general this can be just a |
| |
|
|
| align:start position:0% |
| a noun in general this can be just a |
| long relativistic field series okay for |
|
|
| align:start position:0% |
| long relativistic field series okay for |
| |
|
|
| align:start position:0% |
| long relativistic field series okay for |
| other values of B okay |
|
|
| align:start position:0% |
| other values of B okay |
| |
|
|
| align:start position:0% |
| other values of B okay |
| foreign |
|
|
| align:start position:0% |
| foreign |
| |
|
|
| align:start position:0% |
| foreign |
| so this is |
|
|
| align:start position:0% |
| so this is |
| |
|
|
| align:start position:0% |
| so this is |
| so even though this example is very |
|
|
| align:start position:0% |
| so even though this example is very |
| |
|
|
| align:start position:0% |
| so even though this example is very |
| simple |
|
|
| align:start position:0% |
| simple |
| |
|
|
| align:start position:0% |
| simple |
| but this is actually a very general way |
|
|
| align:start position:0% |
| but this is actually a very general way |
| |
|
|
| align:start position:0% |
| but this is actually a very general way |
| that we can treat many condensed metal |
|
|
| align:start position:0% |
| that we can treat many condensed metal |
| |
|
|
| align:start position:0% |
| that we can treat many condensed metal |
| systems |
|
|
| align:start position:0% |
| systems |
| |
|
|
| align:start position:0% |
| systems |
| which often in Mobile lattice say |
|
|
| align:start position:0% |
| which often in Mobile lattice say |
| |
|
|
| align:start position:0% |
| which often in Mobile lattice say |
| because because solid you can imagine |
|
|
| align:start position:0% |
| because because solid you can imagine |
| |
|
|
| align:start position:0% |
| because because solid you can imagine |
| all the items on the lattice Etc and if |
|
|
| align:start position:0% |
| all the items on the lattice Etc and if |
| |
|
|
| align:start position:0% |
| all the items on the lattice Etc and if |
| you're only interested in the very |
|
|
| align:start position:0% |
| you're only interested in the very |
| |
|
|
| align:start position:0% |
| you're only interested in the very |
| microscopic Behavior then you can treat |
|
|
| align:start position:0% |
| microscopic Behavior then you can treat |
| |
|
|
| align:start position:0% |
| microscopic Behavior then you can treat |
| solid as a continuum |
|
|
| align:start position:0% |
| solid as a continuum |
| |
|
|
| align:start position:0% |
| solid as a continuum |
| and then and now you can uh now if |
|
|
| align:start position:0% |
| and then and now you can uh now if |
| |
|
|
| align:start position:0% |
| and then and now you can uh now if |
| you're interested in quantum mechanics |
|
|
| align:start position:0% |
| you're interested in quantum mechanics |
| |
|
|
| align:start position:0% |
| you're interested in quantum mechanics |
| of such a system then the quantum field |
|
|
| align:start position:0% |
| of such a system then the quantum field |
| |
|
|
| align:start position:0% |
| of such a system then the quantum field |
| theory that naturally Rises okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay good any questions on this example |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| sorry so the limit yeah yeah yeah |
|
|
| align:start position:0% |
| sorry so the limit yeah yeah yeah |
| |
|
|
| align:start position:0% |
| sorry so the limit yeah yeah yeah |
| like same windows view yeah yeah so what |
|
|
| align:start position:0% |
| like same windows view yeah yeah so what |
| |
|
|
| align:start position:0% |
| like same windows view yeah yeah so what |
| is that physically |
|
|
| align:start position:0% |
| is that physically |
| |
|
|
| align:start position:0% |
| is that physically |
| strength of yeah yeah it is |
|
|
| align:start position:0% |
| strength of yeah yeah it is |
| |
|
|
| align:start position:0% |
| strength of yeah yeah it is |
| corresponding to the the case that the |
|
|
| align:start position:0% |
| corresponding to the the case that the |
| |
|
|
| align:start position:0% |
| corresponding to the the case that the |
| um uh uh yeah it just tells you that |
|
|
| align:start position:0% |
| um uh uh yeah it just tells you that |
| |
|
|
| align:start position:0% |
| um uh uh yeah it just tells you that |
| relativistic limit is special happens at |
|
|
| align:start position:0% |
| relativistic limit is special happens at |
| |
|
|
| align:start position:0% |
| relativistic limit is special happens at |
| very special points but I guess why why |
|
|
| align:start position:0% |
| very special points but I guess why why |
| |
|
|
| align:start position:0% |
| very special points but I guess why why |
| is that the relative like to me lenders |
|
|
| align:start position:0% |
| is that the relative like to me lenders |
| |
|
|
| align:start position:0% |
| is that the relative like to me lenders |
| like the strength of your spring and |
|
|
| align:start position:0% |
| like the strength of your spring and |
| |
|
|
| align:start position:0% |
| like the strength of your spring and |
| then yeah use your mask right you've got |
|
|
| align:start position:0% |
| then yeah use your mask right you've got |
| |
|
|
| align:start position:0% |
| then yeah use your mask right you've got |
| to be comparable how does that yeah |
|
|
| align:start position:0% |
| to be comparable how does that yeah |
| |
|
|
| align:start position:0% |
| to be comparable how does that yeah |
| there's not much you can read from here |
|
|
| align:start position:0% |
| there's not much you can read from here |
| |
|
|
| align:start position:0% |
| there's not much you can read from here |
| yeah yeah uh it it just like when you |
|
|
| align:start position:0% |
| yeah yeah uh it it just like when you |
| |
|
|
| align:start position:0% |
| yeah yeah uh it it just like when you |
| choose some special parameters then you |
|
|
| align:start position:0% |
| choose some special parameters then you |
| |
|
|
| align:start position:0% |
| choose some special parameters then you |
| can uh have a relativistic limit |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| other questions yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah so these are all um scalars right |
|
|
| align:start position:0% |
| yeah so these are all um scalars right |
| |
|
|
| align:start position:0% |
| yeah so these are all um scalars right |
| I could you can also have uh you can |
|
|
| align:start position:0% |
| I could you can also have uh you can |
| |
|
|
| align:start position:0% |
| I could you can also have uh you can |
| also you mean uh you can also have |
|
|
| align:start position:0% |
| also you mean uh you can also have |
| |
|
|
| align:start position:0% |
| also you mean uh you can also have |
| tensors or vectors yeah |
|
|
| align:start position:0% |
| tensors or vectors yeah |
| |
|
|
| align:start position:0% |
| tensors or vectors yeah |
| yeah |
|
|
| align:start position:0% |
| yeah |
| |
|
|
| align:start position:0% |
| yeah |
| so like what would you treat with this |
|
|
| align:start position:0% |
| so like what would you treat with this |
| |
|
|
| align:start position:0% |
| so like what would you treat with this |
| pronouns oh you can cheat yeah for |
|
|
| align:start position:0% |
| pronouns oh you can cheat yeah for |
| |
|
|
| align:start position:0% |
| pronouns oh you can cheat yeah for |
| example you can treat four rounds you |
|
|
| align:start position:0% |
| example you can treat four rounds you |
| |
|
|
| align:start position:0% |
| example you can treat four rounds you |
| can also choose spins |
|
|
| align:start position:0% |
| can also choose spins |
| |
|
|
| align:start position:0% |
| can also choose spins |
| and say for example if you have an icing |
|
|
| align:start position:0% |
| and say for example if you have an icing |
| |
|
|
| align:start position:0% |
| and say for example if you have an icing |
| model just consider lattice of spins |
|
|
| align:start position:0% |
| model just consider lattice of spins |
| |
|
|
| align:start position:0% |
| model just consider lattice of spins |
| and then the average Spin and then you |
|
|
| align:start position:0% |
| and then the average Spin and then you |
| |
|
|
| align:start position:0% |
| and then the average Spin and then you |
| can treat it as a scale of field and |
|
|
| align:start position:0% |
| can treat it as a scale of field and |
| |
|
|
| align:start position:0% |
| can treat it as a scale of field and |
| then again you can write down a field |
|
|
| align:start position:0% |
| then again you can write down a field |
| |
|
|
| align:start position:0% |
| then again you can write down a field |
| Theory yeah and actually the the |
|
|
| align:start position:0% |
| Theory yeah and actually the the |
| |
|
|
| align:start position:0% |
| Theory yeah and actually the the |
| Breakthrough |
|
|
| align:start position:0% |
| Breakthrough |
| |
|
|
| align:start position:0% |
| Breakthrough |
| of the phase transition |
|
|
| align:start position:0% |
| of the phase transition |
| |
|
|
| align:start position:0% |
| of the phase transition |
| metaphysics to to understand what phase |
|
|
| align:start position:0% |
| metaphysics to to understand what phase |
| |
|
|
| align:start position:0% |
| metaphysics to to understand what phase |
| transition is really about and describe |
|
|
| align:start position:0% |
| transition is really about and describe |
| |
|
|
| align:start position:0% |
| transition is really about and describe |
| the behavior near the phase transition |
|
|
| align:start position:0% |
| the behavior near the phase transition |
| |
|
|
| align:start position:0% |
| the behavior near the phase transition |
| and precisely coincided with the |
|
|
| align:start position:0% |
| and precisely coincided with the |
| |
|
|
| align:start position:0% |
| and precisely coincided with the |
| development of field Theory and uh yeah |
|
|
| align:start position:0% |
| development of field Theory and uh yeah |
| |
|
|
| align:start position:0% |
| development of field Theory and uh yeah |
| actually increased our understanding of |
|
|
| align:start position:0% |
| actually increased our understanding of |
| |
|
|
| align:start position:0% |
| actually increased our understanding of |
| quantum field Theory yeah |
|
|
| align:start position:0% |
| quantum field Theory yeah |
| |
|
|
| align:start position:0% |
| quantum field Theory yeah |
| foreign |
|
|
| align:start position:0% |
| foreign |
| |
|
|
| align:start position:0% |
| foreign |
| other questions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay good |
|
|
| align:start position:0% |
| okay good |
| |
|
|
| align:start position:0% |
| okay good |
| just to summarize |
|
|
| align:start position:0% |
| just to summarize |
| |
|
|
| align:start position:0% |
| just to summarize |
| what we have discussed so far |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| or path |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| to |
|
|
| align:start position:0% |
| to |
| |
|
|
| align:start position:0% |
| to |
| qft okay |
|
|
| align:start position:0% |
| qft okay |
| |
|
|
| align:start position:0% |
| qft okay |
| so we have described three parts three |
|
|
| align:start position:0% |
| so we have described three parts three |
| |
|
|
| align:start position:0% |
| so we have described three parts three |
| parts but they're pretty General first |
|
|
| align:start position:0% |
| parts but they're pretty General first |
| |
|
|
| align:start position:0% |
| parts but they're pretty General first |
| you say the quantum that line we offer |
|
|
| align:start position:0% |
| you say the quantum that line we offer |
| |
|
|
| align:start position:0% |
| you say the quantum that line we offer |
| increasing Quantum Dynamics |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of some classical fields |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| say such as |
|
|
| align:start position:0% |
| say such as |
| |
|
|
| align:start position:0% |
| say such as |
| say Electric magnetic field |
|
|
| align:start position:0% |
| say Electric magnetic field |
| |
|
|
| align:start position:0% |
| say Electric magnetic field |
| or space-time metric if you are |
|
|
| align:start position:0% |
| or space-time metric if you are |
| |
|
|
| align:start position:0% |
| or space-time metric if you are |
| interested in gravity |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| Etc okay so so so in this case we |
|
|
| align:start position:0% |
| Etc okay so so so in this case we |
| |
|
|
| align:start position:0% |
| Etc okay so so so in this case we |
| already have the classical Fields Theory |
|
|
| align:start position:0% |
| already have the classical Fields Theory |
| |
|
|
| align:start position:0% |
| already have the classical Fields Theory |
| but we know the word is quantum and we |
|
|
| align:start position:0% |
| but we know the word is quantum and we |
| |
|
|
| align:start position:0% |
| but we know the word is quantum and we |
| want to understand what's the quantum |
|
|
| align:start position:0% |
| want to understand what's the quantum |
| |
|
|
| align:start position:0% |
| want to understand what's the quantum |
| version of it |
|
|
| align:start position:0% |
| version of it |
| |
|
|
| align:start position:0% |
| version of it |
| and the second is that it unifies |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| special relativity |
|
|
| align:start position:0% |
| special relativity |
| |
|
|
| align:start position:0% |
| special relativity |
| plus quantum mechanics okay so you read |
|
|
| align:start position:0% |
| plus quantum mechanics okay so you read |
| |
|
|
| align:start position:0% |
| plus quantum mechanics okay so you read |
| the field Theory to to unify them and |
|
|
| align:start position:0% |
| the field Theory to to unify them and |
| |
|
|
| align:start position:0% |
| the field Theory to to unify them and |
| the Third Way |
|
|
| align:start position:0% |
| the Third Way |
| |
|
|
| align:start position:0% |
| the Third Way |
| he said is the a large distance |
|
|
| align:start position:0% |
| he said is the a large distance |
| |
|
|
| align:start position:0% |
| he said is the a large distance |
| description |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of discrete systems |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| um |
|
|
| align:start position:0% |
| um |
| |
|
|
| align:start position:0% |
| um |
| yeah just combine all three elements |
|
|
| align:start position:0% |
| yeah just combine all three elements |
| |
|
|
| align:start position:0% |
| yeah just combine all three elements |
| together the cover many many areas of |
|
|
| align:start position:0% |
| together the cover many many areas of |
| |
|
|
| align:start position:0% |
| together the cover many many areas of |
| physics okay they cover many many areas |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good so so now we can just |
|
|
| align:start position:0% |
| good so so now we can just |
| |
|
|
| align:start position:0% |
| good so so now we can just |
| say a little bit about the plan for the |
|
|
| align:start position:0% |
| say a little bit about the plan for the |
| |
|
|
| align:start position:0% |
| say a little bit about the plan for the |
| whole semester |
|
|
| align:start position:0% |
| whole semester |
| |
|
|
| align:start position:0% |
| whole semester |
| okay so here is the plan |
|
|
| align:start position:0% |
| okay so here is the plan |
| |
|
|
| align:start position:0% |
| okay so here is the plan |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| so this is like just rephrase of the |
|
|
| align:start position:0% |
| so this is like just rephrase of the |
| |
|
|
| align:start position:0% |
| so this is like just rephrase of the |
| outline |
|
|
| align:start position:0% |
| outline |
| |
|
|
| align:start position:0% |
| outline |
| which uh |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so the first thing we do in chapter |
|
|
| align:start position:0% |
| so so the first thing we do in chapter |
| |
|
|
| align:start position:0% |
| so so the first thing we do in chapter |
| two so here is chapter one chapter two |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| we discussed the simplest field Theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| just Express this equation two okay the |
|
|
| align:start position:0% |
| just Express this equation two okay the |
| |
|
|
| align:start position:0% |
| just Express this equation two okay the |
| theory of Q okay 2 and the two prime |
|
|
| align:start position:0% |
| theory of Q okay 2 and the two prime |
| |
|
|
| align:start position:0% |
| theory of Q okay 2 and the two prime |
| yeah a a prime is is equation motion so |
|
|
| align:start position:0% |
| yeah a a prime is is equation motion so |
| |
|
|
| align:start position:0% |
| yeah a a prime is is equation motion so |
| so yeah we in physics we always start |
|
|
| align:start position:0% |
| so yeah we in physics we always start |
| |
|
|
| align:start position:0% |
| so yeah we in physics we always start |
| with a simplistic example okay we always |
|
|
| align:start position:0% |
| with a simplistic example okay we always |
| |
|
|
| align:start position:0% |
| with a simplistic example okay we always |
| start with simplistic example and uh and |
|
|
| align:start position:0% |
| start with simplistic example and uh and |
| |
|
|
| align:start position:0% |
| start with simplistic example and uh and |
| so that's the uh is the one we will |
|
|
| align:start position:0% |
| so that's the uh is the one we will |
| |
|
|
| align:start position:0% |
| so that's the uh is the one we will |
| start with |
|
|
| align:start position:0% |
| start with |
| |
|
|
| align:start position:0% |
| start with |
| so what we will see is that this |
|
|
| align:start position:0% |
| so what we will see is that this |
| |
|
|
| align:start position:0% |
| so what we will see is that this |
| describes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| that field we describe spin is |
|
|
| align:start position:0% |
| that field we describe spin is |
| |
|
|
| align:start position:0% |
| that field we describe spin is |
| there's no |
|
|
| align:start position:0% |
| there's no |
| |
|
|
| align:start position:0% |
| there's no |
| free |
|
|
| align:start position:0% |
| free |
| |
|
|
| align:start position:0% |
| free |
| massive particles okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so we will see when we quantize |
|
|
| align:start position:0% |
| okay so we will see when we quantize |
| |
|
|
| align:start position:0% |
| okay so we will see when we quantize |
| that theory two and then we get the |
|
|
| align:start position:0% |
| that theory two and then we get the |
| |
|
|
| align:start position:0% |
| that theory two and then we get the |
| theory of |
|
|
| align:start position:0% |
| theory of |
| |
|
|
| align:start position:0% |
| theory of |
| three Methodist or three spin is massive |
|
|
| align:start position:0% |
| three Methodist or three spin is massive |
| |
|
|
| align:start position:0% |
| three Methodist or three spin is massive |
| particles okay |
|
|
| align:start position:0% |
| particles okay |
| |
|
|
| align:start position:0% |
| particles okay |
| so you say oh that's a little bit boring |
|
|
| align:start position:0% |
| so you say oh that's a little bit boring |
| |
|
|
| align:start position:0% |
| so you say oh that's a little bit boring |
| because in this series three the |
|
|
| align:start position:0% |
| because in this series three the |
| |
|
|
| align:start position:0% |
| because in this series three the |
| particle by three means they don't |
|
|
| align:start position:0% |
| particle by three means they don't |
| |
|
|
| align:start position:0% |
| particle by three means they don't |
| intact okay the particle they just don't |
|
|
| align:start position:0% |
| intact okay the particle they just don't |
| |
|
|
| align:start position:0% |
| intact okay the particle they just don't |
| intact |
|
|
| align:start position:0% |
| intact |
| |
|
|
| align:start position:0% |
| intact |
| and then in chapter 3 we will add |
|
|
| align:start position:0% |
| and then in chapter 3 we will add |
| |
|
|
| align:start position:0% |
| and then in chapter 3 we will add |
| interactions we will describe |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| how to |
|
|
| align:start position:0% |
| how to |
| |
|
|
| align:start position:0% |
| how to |
| treat |
|
|
| align:start position:0% |
| treat |
| |
|
|
| align:start position:0% |
| treat |
| interactions okay |
|
|
| align:start position:0% |
| interactions okay |
| |
|
|
| align:start position:0% |
| interactions okay |
| so we will introduce interactions |
|
|
| align:start position:0% |
| so we will introduce interactions |
| |
|
|
| align:start position:0% |
| so we will introduce interactions |
| and tell you how to treat the |
|
|
| align:start position:0% |
| and tell you how to treat the |
| |
|
|
| align:start position:0% |
| and tell you how to treat the |
| interactions between those particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| then in chapter four |
|
|
| align:start position:0% |
| then in chapter four |
| |
|
|
| align:start position:0% |
| then in chapter four |
| we go to the real physics |
|
|
| align:start position:0% |
| we go to the real physics |
| |
|
|
| align:start position:0% |
| we go to the real physics |
| so this scale of fields |
|
|
| align:start position:0% |
| so this scale of fields |
| |
|
|
| align:start position:0% |
| so this scale of fields |
| is also real say for example can be used |
|
|
| align:start position:0% |
| is also real say for example can be used |
| |
|
|
| align:start position:0% |
| is also real say for example can be used |
| to describe the Hicks |
|
|
| align:start position:0% |
| to describe the Hicks |
| |
|
|
| align:start position:0% |
| to describe the Hicks |
| okay but the heat exposure maybe it's a |
|
|
| align:start position:0% |
| okay but the heat exposure maybe it's a |
| |
|
|
| align:start position:0% |
| okay but the heat exposure maybe it's a |
| little bit far from what we uh you |
|
|
| align:start position:0% |
| little bit far from what we uh you |
| |
|
|
| align:start position:0% |
| little bit far from what we uh you |
| normally think about so in chapter four |
|
|
| align:start position:0% |
| normally think about so in chapter four |
| |
|
|
| align:start position:0% |
| normally think about so in chapter four |
| we will go to something which is much |
|
|
| align:start position:0% |
| we will go to something which is much |
| |
|
|
| align:start position:0% |
| we will go to something which is much |
| closer we'll talk about the theory of |
|
|
| align:start position:0% |
| closer we'll talk about the theory of |
| |
|
|
| align:start position:0% |
| closer we'll talk about the theory of |
| electron so this is called the rock |
|
|
| align:start position:0% |
| electron so this is called the rock |
| |
|
|
| align:start position:0% |
| electron so this is called the rock |
| theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this Theory describes three |
|
|
| align:start position:0% |
| so this Theory describes three |
| |
|
|
| align:start position:0% |
| so this Theory describes three |
| but spin half |
|
|
| align:start position:0% |
| but spin half |
| |
|
|
| align:start position:0% |
| but spin half |
| particles okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is a theory of electrons |
|
|
| align:start position:0% |
| so this is a theory of electrons |
| |
|
|
| align:start position:0% |
| so this is a theory of electrons |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| when we like these interactions okay so |
|
|
| align:start position:0% |
| when we like these interactions okay so |
| |
|
|
| align:start position:0% |
| when we like these interactions okay so |
| this is the three spring hard particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then |
|
|
| align:start position:0% |
| |
| |
|
|
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| |
| [Applause] |
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| we move on to the maximals here |
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| align:start position:0% |
| we move on to the maximals here |
| |
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| align:start position:0% |
| we move on to the maximals here |
| Maxwell's Theory |
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| so this is the theory of the quantum |
|
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| align:start position:0% |
| so this is the theory of the quantum |
| |
|
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| align:start position:0% |
| so this is the theory of the quantum |
| electric and magnetic field okay |
|
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| align:start position:0% |
| |
| |
|
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| align:start position:0% |
| |
| so we quantize the Maxwell Theory |
|
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| align:start position:0% |
| so we quantize the Maxwell Theory |
| |
|
|
| align:start position:0% |
| so we quantize the Maxwell Theory |
| say without Source the vacuum box real |
|
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| align:start position:0% |
| say without Source the vacuum box real |
| |
|
|
| align:start position:0% |
| say without Source the vacuum box real |
| Siri and you find you get free again |
|
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| align:start position:0% |
| Siri and you find you get free again |
| |
|
|
| align:start position:0% |
| Siri and you find you get free again |
| there's no interaction |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| is |
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| particle |
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| align:start position:0% |
| particle |
| |
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| align:start position:0% |
| particle |
| you get a series of |
|
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| align:start position:0% |
| you get a series of |
| |
|
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| align:start position:0% |
| you get a series of |
| massive spring one particle |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| so this is what we call the photon |
|
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| align:start position:0% |
| so this is what we call the photon |
| |
|
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| align:start position:0% |
| so this is what we call the photon |
| okay so this is the Quantum for |
|
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| align:start position:0% |
| okay so this is the Quantum for |
| |
|
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| align:start position:0% |
| okay so this is the Quantum for |
| electromagnetic field |
|
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| align:start position:0% |
| electromagnetic field |
| |
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| align:start position:0% |
| electromagnetic field |
| okay |
|
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| align:start position:0% |
| okay |
| |
|
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| align:start position:0% |
| okay |
| and then first and then |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| sorry did I so this should be chapter |
|
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| align:start position:0% |
| sorry did I so this should be chapter |
| |
|
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| align:start position:0% |
| sorry did I so this should be chapter |
| five now I think I lost my account |
|
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| align:start position:0% |
| five now I think I lost my account |
| |
|
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| align:start position:0% |
| five now I think I lost my account |
| so now go to chapter six |
|
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| align:start position:0% |
| so now go to chapter six |
| |
|
|
| align:start position:0% |
| so now go to chapter six |
| we combine the four and five together |
|
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| align:start position:0% |
| we combine the four and five together |
| |
|
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| align:start position:0% |
| we combine the four and five together |
| okay |
|
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| align:start position:0% |
| okay |
| |
|
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| align:start position:0% |
| okay |
| combine |
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| electrons |
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| align:start position:0% |
| electrons |
| |
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| align:start position:0% |
| electrons |
| so Photon normally we if we don't |
|
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| align:start position:0% |
| so Photon normally we if we don't |
| |
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| align:start position:0% |
| so Photon normally we if we don't |
| developed by gamma |
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| align:start position:0% |
| developed by gamma |
| |
|
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| align:start position:0% |
| developed by gamma |
| combine the theory of electron and the |
|
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| align:start position:0% |
| combine the theory of electron and the |
| |
|
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| align:start position:0% |
| combine the theory of electron and the |
| photon together |
|
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| align:start position:0% |
| photon together |
| |
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| align:start position:0% |
| photon together |
| and then plus interactions |
|
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| |
| |
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| align:start position:0% |
| |
| between them |
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| align:start position:0% |
| between them |
| |
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| align:start position:0% |
| between them |
| and then we get the so-called |
|
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| align:start position:0% |
| and then we get the so-called |
| |
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| align:start position:0% |
| and then we get the so-called |
| quantum electrodynamics |
|
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| align:start position:0% |
| |
| |
|
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| align:start position:0% |
| |
| so this is called QED |
|
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| align:start position:0% |
| |
| |
|
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| align:start position:0% |
| |
| so QD is very general essentially covers |
|
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| align:start position:0% |
| so QD is very general essentially covers |
| |
|
|
| align:start position:0% |
| so QD is very general essentially covers |
| all the quantum phenomena uh yeah a |
|
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| align:start position:0% |
| all the quantum phenomena uh yeah a |
| |
|
|
| align:start position:0% |
| all the quantum phenomena uh yeah a |
| microscopic phenomena up to say big |
|
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| align:start position:0% |
| microscopic phenomena up to say big |
| |
|
|
| align:start position:0% |
| microscopic phenomena up to say big |
| interactions and strong interactions if |
|
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| align:start position:0% |
| interactions and strong interactions if |
| |
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| align:start position:0% |
| interactions and strong interactions if |
| you could don't go inside the nucleus |
|
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| align:start position:0% |
| you could don't go inside the nucleus |
| |
|
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| align:start position:0% |
| you could don't go inside the nucleus |
| and uh uh or don't go to a very high |
|
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| align:start position:0% |
| and uh uh or don't go to a very high |
| |
|
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| align:start position:0% |
| and uh uh or don't go to a very high |
| energy and think that's covers |
|
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| align:start position:0% |
| energy and think that's covers |
| |
|
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| align:start position:0% |
| energy and think that's covers |
| essentially most of the physics |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| yeah yeah and and then the |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| end of this our our course |
|
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| align:start position:0% |
| end of this our our course |
| |
|
|
| align:start position:0% |
| end of this our our course |
| so do you have any questions on this |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so this is a road map yes |
|
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| align:start position:0% |
| |
| |
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| align:start position:0% |
| |
| good other questions |
|
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| align:start position:0% |
| good other questions |
| |
|
|
| align:start position:0% |
| good other questions |
| yes |
|
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| align:start position:0% |
| |
| |
|
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| align:start position:0% |
| |
| sorry |
|
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| align:start position:0% |
| |
| |
|
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| align:start position:0% |
| |
| yeah it's also Mercedes spin one but |
|
|
| align:start position:0% |
| yeah it's also Mercedes spin one but |
| |
|
|
| align:start position:0% |
| yeah it's also Mercedes spin one but |
| actually the interact with themselves |
|
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| align:start position:0% |
| actually the interact with themselves |
| |
|
|
| align:start position:0% |
| actually the interact with themselves |
| and so good one is different so so glue |
|
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| align:start position:0% |
| and so good one is different so so glue |
| |
|
|
| align:start position:0% |
| and so good one is different so so glue |
| on to describe gluons you have to wait |
|
|
| align:start position:0% |
| on to describe gluons you have to wait |
| |
|
|
| align:start position:0% |
| on to describe gluons you have to wait |
| for Quantum field City too |
|
|
| align:start position:0% |
| for Quantum field City too |
| |
|
|
| align:start position:0% |
| for Quantum field City too |
| and uh so so the thing about the photon |
|
|
| align:start position:0% |
| and uh so so the thing about the photon |
| |
|
|
| align:start position:0% |
| and uh so so the thing about the photon |
| is that the photons don't interact with |
|
|
| align:start position:0% |
| is that the photons don't interact with |
| |
|
|
| align:start position:0% |
| is that the photons don't interact with |
| itself but the gluons interact with |
|
|
| align:start position:0% |
| itself but the gluons interact with |
| |
|
|
| align:start position:0% |
| itself but the gluons interact with |
| itself okay yeah so so essentially we |
|
|
| align:start position:0% |
| itself okay yeah so so essentially we |
| |
|
|
| align:start position:0% |
| itself okay yeah so so essentially we |
| treat everything |
|
|
| align:start position:0% |
| treat everything |
| |
|
|
| align:start position:0% |
| treat everything |
| except gloves yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| other questions |
|
|
| align:start position:0% |
| other questions |
| |
|
|
| align:start position:0% |
| other questions |
| other questions |
|
|
| align:start position:0% |
| other questions |
| |
|
|
| align:start position:0% |
| other questions |
| okay good |
|
|
| align:start position:0% |
| okay good |
| |
|
|
| align:start position:0% |
| okay good |
| so now we can just move to chapter two |
|
|
| align:start position:0% |
| so now we can just move to chapter two |
| |
|
|
| align:start position:0% |
| so now we can just move to chapter two |
| now we are talking about this series |
|
|
| align:start position:0% |
| now we are talking about this series |
| |
|
|
| align:start position:0% |
| now we are talking about this series |
| okay so so actually I should not erase |
|
|
| align:start position:0% |
| okay so so actually I should not erase |
| |
|
|
| align:start position:0% |
| okay so so actually I should not erase |
| it |
|
|
| align:start position:0% |
| it |
| |
|
|
| align:start position:0% |
| it |
| so now we talk about this here |
|
|
| align:start position:0% |
| so now we talk about this here |
| |
|
|
| align:start position:0% |
| so now we talk about this here |
| so because because this series describes |
|
|
| align:start position:0% |
| so because because this series describes |
| |
|
|
| align:start position:0% |
| so because because this series describes |
| three particles |
|
|
| align:start position:0% |
| three particles |
| |
|
|
| align:start position:0% |
| three particles |
| so we call it free scalar field CV |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so so so this is the |
|
|
| align:start position:0% |
| okay so so so this is the |
| |
|
|
| align:start position:0% |
| okay so so so this is the |
| Theory we are interested in so so now we |
|
|
| align:start position:0% |
| Theory we are interested in so so now we |
| |
|
|
| align:start position:0% |
| Theory we are interested in so so now we |
| will describe how to quantize this |
|
|
| align:start position:0% |
| will describe how to quantize this |
| |
|
|
| align:start position:0% |
| will describe how to quantize this |
| series okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good |
|
|
| align:start position:0% |
| good |
| |
|
|
| align:start position:0% |
| good |
| so so first |
|
|
| align:start position:0% |
| so so first |
| |
|
|
| align:start position:0% |
| so so first |
| we will quickly |
|
|
| align:start position:0% |
| we will quickly |
| |
|
|
| align:start position:0% |
| we will quickly |
| go through |
|
|
| align:start position:0% |
| go through |
| |
|
|
| align:start position:0% |
| go through |
| the the quantitation of harmonic |
|
|
| align:start position:0% |
| the the quantitation of harmonic |
| |
|
|
| align:start position:0% |
| the the quantitation of harmonic |
| oscillator |
|
|
| align:start position:0% |
| oscillator |
| |
|
|
| align:start position:0% |
| oscillator |
| which you should already have done in |
|
|
| align:start position:0% |
| which you should already have done in |
| |
|
|
| align:start position:0% |
| which you should already have done in |
| your in your preset and so uh so we can |
|
|
| align:start position:0% |
| your in your preset and so uh so we can |
| |
|
|
| align:start position:0% |
| your in your preset and so uh so we can |
| do it relatively fast |
|
|
| align:start position:0% |
| do it relatively fast |
| |
|
|
| align:start position:0% |
| do it relatively fast |
| so |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| the organization |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of pythonic oscillator |
|
|
| align:start position:0% |
| of pythonic oscillator |
| |
|
|
| align:start position:0% |
| of pythonic oscillator |
| in the Heisenberg picture |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we will see that once we understand |
|
|
| align:start position:0% |
| so we will see that once we understand |
| |
|
|
| align:start position:0% |
| so we will see that once we understand |
| this example |
|
|
| align:start position:0% |
| this example |
| |
|
|
| align:start position:0% |
| this example |
| in the right way |
|
|
| align:start position:0% |
| in the right way |
| |
|
|
| align:start position:0% |
| in the right way |
| and then contacting this field Theory |
|
|
| align:start position:0% |
| and then contacting this field Theory |
| |
|
|
| align:start position:0% |
| and then contacting this field Theory |
| becomes trivial |
|
|
| align:start position:0% |
| becomes trivial |
| |
|
|
| align:start position:0% |
| becomes trivial |
| okay and the quantize in this field |
|
|
| align:start position:0% |
| okay and the quantize in this field |
| |
|
|
| align:start position:0% |
| okay and the quantize in this field |
| series become trigger |
|
|
| align:start position:0% |
| series become trigger |
| |
|
|
| align:start position:0% |
| series become trigger |
| okay so so let's start with a harmonic |
|
|
| align:start position:0% |
| okay so so let's start with a harmonic |
| |
|
|
| align:start position:0% |
| okay so so let's start with a harmonic |
| oscillator |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| for Simplicity I take the mass to be one |
|
|
| align:start position:0% |
| for Simplicity I take the mass to be one |
| |
|
|
| align:start position:0% |
| for Simplicity I take the mass to be one |
| and take the frequency to be y okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah let me put the frequency here |
|
|
| align:start position:0% |
| yeah yeah let me put the frequency here |
| |
|
|
| align:start position:0% |
| yeah yeah let me put the frequency here |
| it won't fit |
|
|
| align:start position:0% |
| it won't fit |
| |
|
|
| align:start position:0% |
| it won't fit |
| okay let's take the mass TDY y okay |
|
|
| align:start position:0% |
| okay let's take the mass TDY y okay |
| |
|
|
| align:start position:0% |
| okay let's take the mass TDY y okay |
| and so so for this series so so this is |
|
|
| align:start position:0% |
| and so so for this series so so this is |
| |
|
|
| align:start position:0% |
| and so so for this series so so this is |
| a simple harmonic oscillator which you |
|
|
| align:start position:0% |
| a simple harmonic oscillator which you |
| |
|
|
| align:start position:0% |
| a simple harmonic oscillator which you |
| have seen it |
|
|
| align:start position:0% |
| have seen it |
| |
|
|
| align:start position:0% |
| have seen it |
| uh uh uh maybe for the most of your |
|
|
| align:start position:0% |
| uh uh uh maybe for the most of your |
| |
|
|
| align:start position:0% |
| uh uh uh maybe for the most of your |
| intellectual life |
|
|
| align:start position:0% |
| intellectual life |
| |
|
|
| align:start position:0% |
| intellectual life |
| and the P will be x dot it's a momentum |
|
|
| align:start position:0% |
| and the P will be x dot it's a momentum |
| |
|
|
| align:start position:0% |
| and the P will be x dot it's a momentum |
| the conjugate momentum is x dot and so |
|
|
| align:start position:0% |
| the conjugate momentum is x dot and so |
| |
|
|
| align:start position:0% |
| the conjugate momentum is x dot and so |
| the hamiltonian is the P-Square divided |
|
|
| align:start position:0% |
| the hamiltonian is the P-Square divided |
| |
|
|
| align:start position:0% |
| the hamiltonian is the P-Square divided |
| by two |
|
|
| align:start position:0% |
| by two |
| |
|
|
| align:start position:0% |
| by two |
| of one half Omega Square x squared okay |
|
|
| align:start position:0% |
| of one half Omega Square x squared okay |
| |
|
|
| align:start position:0% |
| of one half Omega Square x squared okay |
| an equation motion |
|
|
| align:start position:0% |
| an equation motion |
| |
|
|
| align:start position:0% |
| an equation motion |
| is X dots double dot equal to x three x |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| so let's first look at this Theory uh |
|
|
| align:start position:0% |
| so let's first look at this Theory uh |
| |
|
|
| align:start position:0% |
| so let's first look at this Theory uh |
| look at harmonic considered as a |
|
|
| align:start position:0% |
| look at harmonic considered as a |
| |
|
|
| align:start position:0% |
| look at harmonic considered as a |
| classical Theory |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so for classical Theory |
|
|
| align:start position:0% |
| so for classical Theory |
| |
|
|
| align:start position:0% |
| so for classical Theory |
| we know how to solve this equation we |
|
|
| align:start position:0% |
| we know how to solve this equation we |
| |
|
|
| align:start position:0% |
| we know how to solve this equation we |
| just need to solve this equation |
|
|
| align:start position:0% |
| just need to solve this equation |
| |
|
|
| align:start position:0% |
| just need to solve this equation |
| so classical solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| just given by x t |
|
|
| align:start position:0% |
| just given by x t |
| |
|
|
| align:start position:0% |
| just given by x t |
| equal to a |
|
|
| align:start position:0% |
| equal to a |
| |
|
|
| align:start position:0% |
| equal to a |
| cosine Omega t |
|
|
| align:start position:0% |
| cosine Omega t |
| |
|
|
| align:start position:0% |
| cosine Omega t |
| plus b sine of Omega t |
|
|
| align:start position:0% |
| plus b sine of Omega t |
| |
|
|
| align:start position:0% |
| plus b sine of Omega t |
| and a and a and b just some integration |
|
|
| align:start position:0% |
| and a and a and b just some integration |
| |
|
|
| align:start position:0% |
| and a and a and b just some integration |
| constant |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and for convenience I can also write it |
|
|
| align:start position:0% |
| and for convenience I can also write it |
| |
|
|
| align:start position:0% |
| and for convenience I can also write it |
| in the complex form |
|
|
| align:start position:0% |
| in the complex form |
| |
|
|
| align:start position:0% |
| in the complex form |
| bracket as following |
|
|
| align:start position:0% |
| bracket as following |
| |
|
|
| align:start position:0% |
| bracket as following |
| equal to a |
|
|
| align:start position:0% |
| equal to a |
| |
|
|
| align:start position:0% |
| equal to a |
| expansion minus sign Omega t |
|
|
| align:start position:0% |
| expansion minus sign Omega t |
| |
|
|
| align:start position:0% |
| expansion minus sign Omega t |
| plus a star |
|
|
| align:start position:0% |
| plus a star |
| |
|
|
| align:start position:0% |
| plus a star |
| expression I Omega T and A is some |
|
|
| align:start position:0% |
| expression I Omega T and A is some |
| |
|
|
| align:start position:0% |
| expression I Omega T and A is some |
| complex constant |
|
|
| align:start position:0% |
| complex constant |
| |
|
|
| align:start position:0% |
| complex constant |
| and again it's a integration constant I |
|
|
| align:start position:0% |
| and again it's a integration constant I |
| |
|
|
| align:start position:0% |
| and again it's a integration constant I |
| just rewrite the integration constant |
|
|
| align:start position:0% |
| just rewrite the integration constant |
| |
|
|
| align:start position:0% |
| just rewrite the integration constant |
| slightly differently |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| now these are just integration constants |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now |
|
|
| align:start position:0% |
| so now |
| |
|
|
| align:start position:0% |
| so now |
| yeah so this is a complete solution of |
|
|
| align:start position:0% |
| yeah so this is a complete solution of |
| |
|
|
| align:start position:0% |
| yeah so this is a complete solution of |
| the problem |
|
|
| align:start position:0% |
| the problem |
| |
|
|
| align:start position:0% |
| the problem |
| so now let's go to Quantum |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so when we go to Quantum |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then then we replace |
|
|
| align:start position:0% |
| and then then we replace |
| |
|
|
| align:start position:0% |
| and then then we replace |
| then this classical dynamical variable |
|
|
| align:start position:0% |
| then this classical dynamical variable |
| |
|
|
| align:start position:0% |
| then this classical dynamical variable |
| then become |
|
|
| align:start position:0% |
| then become |
| |
|
|
| align:start position:0% |
| then become |
| the the Heisenberg operator |
|
|
| align:start position:0% |
| the the Heisenberg operator |
| |
|
|
| align:start position:0% |
| the the Heisenberg operator |
| becomes the quantum operator in |
|
|
| align:start position:0% |
| becomes the quantum operator in |
| |
|
|
| align:start position:0% |
| becomes the quantum operator in |
| particular in the in the Heisenberg |
|
|
| align:start position:0% |
| particular in the in the Heisenberg |
| |
|
|
| align:start position:0% |
| particular in the in the Heisenberg |
| picture and then then this operator will |
|
|
| align:start position:0% |
| picture and then then this operator will |
| |
|
|
| align:start position:0% |
| picture and then then this operator will |
| depend on time |
|
|
| align:start position:0% |
| depend on time |
| |
|
|
| align:start position:0% |
| depend on time |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and now this equation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| become operating equation okay so now |
|
|
| align:start position:0% |
| become operating equation okay so now |
| |
|
|
| align:start position:0% |
| become operating equation okay so now |
| let's |
|
|
| align:start position:0% |
| let's |
| |
|
|
| align:start position:0% |
| let's |
| maybe I should label my equation |
|
|
| align:start position:0% |
| maybe I should label my equation |
| |
|
|
| align:start position:0% |
| maybe I should label my equation |
| so now this star |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| become an operating equation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| now star is the operating equation |
|
|
| align:start position:0% |
| now star is the operating equation |
| |
|
|
| align:start position:0% |
| now star is the operating equation |
| for X hat |
|
|
| align:start position:0% |
| for X hat |
| |
|
|
| align:start position:0% |
| for X hat |
| so you have exact the same equation as a |
|
|
| align:start position:0% |
| so you have exact the same equation as a |
| |
|
|
| align:start position:0% |
| so you have exact the same equation as a |
| classical equation but not the |
|
|
| align:start position:0% |
| classical equation but not the |
| |
|
|
| align:start position:0% |
| classical equation but not the |
| interpretation is different |
|
|
| align:start position:0% |
| interpretation is different |
| |
|
|
| align:start position:0% |
| interpretation is different |
| and now now the X hat becomes the uh |
|
|
| align:start position:0% |
| and now now the X hat becomes the uh |
| |
|
|
| align:start position:0% |
| and now now the X hat becomes the uh |
| another X become the operating equation |
|
|
| align:start position:0% |
| another X become the operating equation |
| |
|
|
| align:start position:0% |
| another X become the operating equation |
| foreign |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now the solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| now let me call this star star |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this still solves that equation |
|
|
| align:start position:0% |
| so this still solves that equation |
| |
|
|
| align:start position:0% |
| so this still solves that equation |
| okay so this still solves that equation |
|
|
| align:start position:0% |
| okay so this still solves that equation |
| |
|
|
| align:start position:0% |
| okay so this still solves that equation |
| except |
|
|
| align:start position:0% |
| except |
| |
|
|
| align:start position:0% |
| except |
| so these are just C numbers because |
|
|
| align:start position:0% |
| so these are just C numbers because |
| |
|
|
| align:start position:0% |
| so these are just C numbers because |
| because this is a function of T these |
|
|
| align:start position:0% |
| because this is a function of T these |
| |
|
|
| align:start position:0% |
| because this is a function of T these |
| are C numbers |
|
|
| align:start position:0% |
| are C numbers |
| |
|
|
| align:start position:0% |
| are C numbers |
| but now X becomes the uh so now Quantum |
|
|
| align:start position:0% |
| but now X becomes the uh so now Quantum |
| |
|
|
| align:start position:0% |
| but now X becomes the uh so now Quantum |
| mechanically |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this becomes now become the quantum |
|
|
| align:start position:0% |
| so this becomes now become the quantum |
| |
|
|
| align:start position:0% |
| so this becomes now become the quantum |
| solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so now I've had so this still solves the |
|
|
| align:start position:0% |
| so now I've had so this still solves the |
| |
|
|
| align:start position:0% |
| so now I've had so this still solves the |
| equations so mechanically this becomes |
|
|
| align:start position:0% |
| equations so mechanically this becomes |
| |
|
|
| align:start position:0% |
| equations so mechanically this becomes |
| hat |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and so this still solves that equation |
|
|
| align:start position:0% |
| and so this still solves that equation |
| |
|
|
| align:start position:0% |
| and so this still solves that equation |
| but these are C numbers |
|
|
| align:start position:0% |
| but these are C numbers |
| |
|
|
| align:start position:0% |
| but these are C numbers |
| the left hand side is the operator |
|
|
| align:start position:0% |
| the left hand side is the operator |
| |
|
|
| align:start position:0% |
| the left hand side is the operator |
| and it can only be that a hat and B hat |
|
|
| align:start position:0% |
| and it can only be that a hat and B hat |
| |
|
|
| align:start position:0% |
| and it can only be that a hat and B hat |
| are operators and also a must be |
|
|
| align:start position:0% |
| are operators and also a must be |
| |
|
|
| align:start position:0% |
| are operators and also a must be |
| operators and the star will replace it |
|
|
| align:start position:0% |
| operators and the star will replace it |
| |
|
|
| align:start position:0% |
| operators and the star will replace it |
| by diagram |
|
|
| align:start position:0% |
| by diagram |
| |
|
|
| align:start position:0% |
| by diagram |
| okay so now |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| say |
|
|
| align:start position:0% |
| say |
| |
|
|
| align:start position:0% |
| say |
| now a you just go to a hat |
|
|
| align:start position:0% |
| now a you just go to a hat |
| |
|
|
| align:start position:0% |
| now a you just go to a hat |
| and a star goes to a dagger it has |
|
|
| align:start position:0% |
| and a star goes to a dagger it has |
| |
|
|
| align:start position:0% |
| and a star goes to a dagger it has |
| dagger okay now these are |
|
|
| align:start position:0% |
| dagger okay now these are |
| |
|
|
| align:start position:0% |
| dagger okay now these are |
| these are integration constants for the |
|
|
| align:start position:0% |
| these are integration constants for the |
| |
|
|
| align:start position:0% |
| these are integration constants for the |
| operating equations |
|
|
| align:start position:0% |
| operating equations |
| |
|
|
| align:start position:0% |
| operating equations |
| so they are just count now they become |
|
|
| align:start position:0% |
| so they are just count now they become |
| |
|
|
| align:start position:0% |
| so they are just count now they become |
| constant operators |
|
|
| align:start position:0% |
| constant operators |
| |
|
|
| align:start position:0% |
| constant operators |
| okay so so so so so they're just |
|
|
| align:start position:0% |
| okay so so so so so they're just |
| |
|
|
| align:start position:0% |
| okay so so so so so they're just |
| constant |
|
|
| align:start position:0% |
| constant |
| |
|
|
| align:start position:0% |
| constant |
| Quantum operators |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay they're just constant operators |
|
|
| align:start position:0% |
| okay they're just constant operators |
| |
|
|
| align:start position:0% |
| okay they're just constant operators |
| so they are integration constants for |
|
|
| align:start position:0% |
| so they are integration constants for |
| |
|
|
| align:start position:0% |
| so they are integration constants for |
| your for your Quantum operating |
|
|
| align:start position:0% |
| your for your Quantum operating |
| |
|
|
| align:start position:0% |
| your for your Quantum operating |
| equations |
|
|
| align:start position:0% |
| equations |
| |
|
|
| align:start position:0% |
| equations |
| so now the solution |
|
|
| align:start position:0% |
| so now the solution |
| |
|
|
| align:start position:0% |
| so now the solution |
| so the as another uh uh yeah so now this |
|
|
| align:start position:0% |
| so the as another uh uh yeah so now this |
| |
|
|
| align:start position:0% |
| so the as another uh uh yeah so now this |
| is your Quantum solution okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is the form we will often use |
|
|
| align:start position:0% |
| so this is the form we will often use |
| |
|
|
| align:start position:0% |
| so this is the form we will often use |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| you can also use that form but the |
|
|
| align:start position:0% |
| you can also use that form but the |
| |
|
|
| align:start position:0% |
| you can also use that form but the |
| equivalent but this is the form we often |
|
|
| align:start position:0% |
| equivalent but this is the form we often |
| |
|
|
| align:start position:0% |
| equivalent but this is the form we often |
| use okay |
|
|
| align:start position:0% |
| use okay |
| |
|
|
| align:start position:0% |
| use okay |
| you can also from here you take the |
|
|
| align:start position:0% |
| you can also from here you take the |
| |
|
|
| align:start position:0% |
| you can also from here you take the |
| derivative you can find the P so again |
|
|
| align:start position:0% |
| derivative you can find the P so again |
| |
|
|
| align:start position:0% |
| derivative you can find the P so again |
| this is become an operating equation you |
|
|
| align:start position:0% |
| this is become an operating equation you |
| |
|
|
| align:start position:0% |
| this is become an operating equation you |
| take the direction of X hat X and maybe |
|
|
| align:start position:0% |
| take the direction of X hat X and maybe |
| |
|
|
| align:start position:0% |
| take the direction of X hat X and maybe |
| you find P Etc |
|
|
| align:start position:0% |
| you find P Etc |
| |
|
|
| align:start position:0% |
| you find P Etc |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah so you |
|
|
| align:start position:0% |
| yeah so you |
| |
|
|
| align:start position:0% |
| yeah so you |
| the P hats t |
|
|
| align:start position:0% |
| the P hats t |
| |
|
|
| align:start position:0% |
| the P hats t |
| just take a derivative |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay and then you can just |
|
|
| align:start position:0% |
| okay and then you can just |
| |
|
|
| align:start position:0% |
| okay and then you can just |
| walk it out it's very easy |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now |
|
|
| align:start position:0% |
| so now |
| |
|
|
| align:start position:0% |
| so now |
| so this equation just |
|
|
| align:start position:0% |
| so this equation just |
| |
|
|
| align:start position:0% |
| so this equation just |
| so we already solved the quantum Tobin |
|
|
| align:start position:0% |
| so we already solved the quantum Tobin |
| |
|
|
| align:start position:0% |
| so we already solved the quantum Tobin |
| okay so so because we find the full |
|
|
| align:start position:0% |
| okay so so because we find the full |
| |
|
|
| align:start position:0% |
| okay so so because we find the full |
| evolution full solution to the quantum |
|
|
| align:start position:0% |
| evolution full solution to the quantum |
| |
|
|
| align:start position:0% |
| evolution full solution to the quantum |
| operator equation |
|
|
| align:start position:0% |
| operator equation |
| |
|
|
| align:start position:0% |
| operator equation |
| except that we still leads to |
|
|
| align:start position:0% |
| except that we still leads to |
| |
|
|
| align:start position:0% |
| except that we still leads to |
| impose the canonical condition |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is just equal to I |
|
|
| align:start position:0% |
| so this is just equal to I |
| |
|
|
| align:start position:0% |
| so this is just equal to I |
| okay so so the standard so if you plug |
|
|
| align:start position:0% |
| okay so so the standard so if you plug |
| |
|
|
| align:start position:0% |
| okay so so the standard so if you plug |
| in the expression for x |
|
|
| align:start position:0% |
| in the expression for x |
| |
|
|
| align:start position:0% |
| in the expression for x |
| and the T and the P into here |
|
|
| align:start position:0% |
| and the T and the P into here |
| |
|
|
| align:start position:0% |
| and the T and the P into here |
| and then you find |
|
|
| align:start position:0% |
| and then you find |
| |
|
|
| align:start position:0% |
| and then you find |
| that a and a dagger |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| accommodate is equal to one okay so this |
|
|
| align:start position:0% |
| accommodate is equal to one okay so this |
| |
|
|
| align:start position:0% |
| accommodate is equal to one okay so this |
| is your familiar |
|
|
| align:start position:0% |
| is your familiar |
| |
|
|
| align:start position:0% |
| is your familiar |
| a creation Annihilation operator for for |
|
|
| align:start position:0% |
| a creation Annihilation operator for for |
| |
|
|
| align:start position:0% |
| a creation Annihilation operator for for |
| harmonic oscillator |
|
|
| align:start position:0% |
| harmonic oscillator |
| |
|
|
| align:start position:0% |
| harmonic oscillator |
| okay for harmonic oscillator |
|
|
| align:start position:0% |
| okay for harmonic oscillator |
| |
|
|
| align:start position:0% |
| okay for harmonic oscillator |
| and uh |
|
|
| align:start position:0% |
| and uh |
| |
|
|
| align:start position:0% |
| and uh |
| and now we can also use the a to build |
|
|
| align:start position:0% |
| and now we can also use the a to build |
| |
|
|
| align:start position:0% |
| and now we can also use the a to build |
| the Hilbert space so because a are the |
|
|
| align:start position:0% |
| the Hilbert space so because a are the |
| |
|
|
| align:start position:0% |
| the Hilbert space so because a are the |
| yeah because all you operate now |
|
|
| align:start position:0% |
| yeah because all you operate now |
| |
|
|
| align:start position:0% |
| yeah because all you operate now |
| or because X and the T |
|
|
| align:start position:0% |
| or because X and the T |
| |
|
|
| align:start position:0% |
| or because X and the T |
| X and P are expressed in terms of a and |
|
|
| align:start position:0% |
| X and P are expressed in terms of a and |
| |
|
|
| align:start position:0% |
| X and P are expressed in terms of a and |
| a dagger so essentially any operator of |
|
|
| align:start position:0% |
| a dagger so essentially any operator of |
| |
|
|
| align:start position:0% |
| a dagger so essentially any operator of |
| this Theory can all be expressed in |
|
|
| align:start position:0% |
| this Theory can all be expressed in |
| |
|
|
| align:start position:0% |
| this Theory can all be expressed in |
| terms of being a diagram okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then you can just use a and a gagger |
|
|
| align:start position:0% |
| and then you can just use a and a gagger |
| |
|
|
| align:start position:0% |
| and then you can just use a and a gagger |
| because A and negative essentially they |
|
|
| align:start position:0% |
| because A and negative essentially they |
| |
|
|
| align:start position:0% |
| because A and negative essentially they |
| they are fundamental building block of |
|
|
| align:start position:0% |
| they are fundamental building block of |
| |
|
|
| align:start position:0% |
| they are fundamental building block of |
| your full quantum theory and then we can |
|
|
| align:start position:0% |
| your full quantum theory and then we can |
| |
|
|
| align:start position:0% |
| your full quantum theory and then we can |
| also use that to build the helper space |
|
|
| align:start position:0% |
| also use that to build the helper space |
| |
|
|
| align:start position:0% |
| also use that to build the helper space |
| so the healable space |
|
|
| align:start position:0% |
| so the healable space |
| |
|
|
| align:start position:0% |
| so the healable space |
| is defined by the lowest state is |
|
|
| align:start position:0% |
| is defined by the lowest state is |
| |
|
|
| align:start position:0% |
| is defined by the lowest state is |
| evaluated by a |
|
|
| align:start position:0% |
| evaluated by a |
| |
|
|
| align:start position:0% |
| evaluated by a |
| and then and then the in the higher |
|
|
| align:start position:0% |
| and then and then the in the higher |
| |
|
|
| align:start position:0% |
| and then and then the in the higher |
| state |
|
|
| align:start position:0% |
| state |
| |
|
|
| align:start position:0% |
| state |
| uh |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| obtained by acting a dagger |
|
|
| align:start position:0% |
| obtained by acting a dagger |
| |
|
|
| align:start position:0% |
| obtained by acting a dagger |
| on the on the on the ground state okay |
|
|
| align:start position:0% |
| on the on the on the ground state okay |
| |
|
|
| align:start position:0% |
| on the on the on the ground state okay |
| so so so this is your full series okay |
|
|
| align:start position:0% |
| so so so this is your full series okay |
| |
|
|
| align:start position:0% |
| so so so this is your full series okay |
| so this is your 4C |
|
|
| align:start position:0% |
| so this is your 4C |
| |
|
|
| align:start position:0% |
| so this is your 4C |
| and so now |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| uh you can compute anything in this |
|
|
| align:start position:0% |
| uh you can compute anything in this |
| |
|
|
| align:start position:0% |
| uh you can compute anything in this |
| series just miss those knowledge okay |
|
|
| align:start position:0% |
| series just miss those knowledge okay |
| |
|
|
| align:start position:0% |
| series just miss those knowledge okay |
| just with lots of knowledge |
|
|
| align:start position:0% |
| just with lots of knowledge |
| |
|
|
| align:start position:0% |
| just with lots of knowledge |
| so any questions on this regarding the |
|
|
| align:start position:0% |
| so any questions on this regarding the |
| |
|
|
| align:start position:0% |
| so any questions on this regarding the |
| harmonic oscillator |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so so let me just summarize |
|
|
| align:start position:0% |
| so so so let me just summarize |
| |
|
|
| align:start position:0% |
| so so so let me just summarize |
| so this is maybe very familiar |
|
|
| align:start position:0% |
| so this is maybe very familiar |
| |
|
|
| align:start position:0% |
| so this is maybe very familiar |
| but let's summarize the rule we have |
|
|
| align:start position:0% |
| but let's summarize the rule we have |
| |
|
|
| align:start position:0% |
| but let's summarize the rule we have |
| been using Okay summarize the steps |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| in context the harmonic oscillator |
|
|
| align:start position:0% |
| in context the harmonic oscillator |
| |
|
|
| align:start position:0% |
| in context the harmonic oscillator |
| and then the and then the same steps can |
|
|
| align:start position:0% |
| and then the and then the same steps can |
| |
|
|
| align:start position:0% |
| and then the and then the same steps can |
| be used to quantize the the field series |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| steps |
|
|
| align:start position:0% |
| steps |
| |
|
|
| align:start position:0% |
| steps |
| of quantization |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we make it a general |
|
|
| align:start position:0% |
| so we make it a general |
| |
|
|
| align:start position:0% |
| so we make it a general |
| so first |
|
|
| align:start position:0% |
| so first |
| |
|
|
| align:start position:0% |
| so first |
| so the zero step is that the classical |
|
|
| align:start position:0% |
| so the zero step is that the classical |
| |
|
|
| align:start position:0% |
| so the zero step is that the classical |
| equation motion the quantum quantum |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| operator equation |
|
|
| align:start position:0% |
| operator equation |
| |
|
|
| align:start position:0% |
| operator equation |
| okay then the first step |
|
|
| align:start position:0% |
| okay then the first step |
| |
|
|
| align:start position:0% |
| okay then the first step |
| is to find the most General solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| to |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| find the most General solution to |
|
|
| align:start position:0% |
| find the most General solution to |
| |
|
|
| align:start position:0% |
| find the most General solution to |
| classical equation motion |
|
|
| align:start position:0% |
| classical equation motion |
| |
|
|
| align:start position:0% |
| classical equation motion |
| yeah just to equation motion |
|
|
| align:start position:0% |
| yeah just to equation motion |
| |
|
|
| align:start position:0% |
| yeah just to equation motion |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and then |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| when you've got the quantum |
|
|
| align:start position:0% |
| when you've got the quantum |
| |
|
|
| align:start position:0% |
| when you've got the quantum |
| and then you just promote |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| the integration constants |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| in your classical solution in one |
|
|
| align:start position:0% |
| in your classical solution in one |
| |
|
|
| align:start position:0% |
| in your classical solution in one |
| in the step one |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| to |
|
|
| align:start position:0% |
| to |
| |
|
|
| align:start position:0% |
| to |
| constant operators |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| constant Quantum operators okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this gives the uh then then you have |
|
|
| align:start position:0% |
| so this gives the uh then then you have |
| |
|
|
| align:start position:0% |
| so this gives the uh then then you have |
| the full-time evolution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| at the quantum level okay okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| now you know the how the quantum |
|
|
| align:start position:0% |
| now you know the how the quantum |
| |
|
|
| align:start position:0% |
| now you know the how the quantum |
| operator evolves |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| you impose |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| canonical condition |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| s |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so that will tell you |
|
|
| align:start position:0% |
| okay so that will tell you |
| |
|
|
| align:start position:0% |
| okay so that will tell you |
| the commentators between those |
|
|
| align:start position:0% |
| the commentators between those |
| |
|
|
| align:start position:0% |
| the commentators between those |
| integration constant operators okay |
|
|
| align:start position:0% |
| integration constant operators okay |
| |
|
|
| align:start position:0% |
| integration constant operators okay |
| just as we do here |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then |
|
|
| align:start position:0% |
| and then |
| |
|
|
| align:start position:0% |
| and then |
| um |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and then um |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| constant operators |
|
|
| align:start position:0% |
| constant operators |
| |
|
|
| align:start position:0% |
| constant operators |
| in queue |
|
|
| align:start position:0% |
| in queue |
| |
|
|
| align:start position:0% |
| in queue |
| now now you know also load the |
|
|
| align:start position:0% |
| now now you know also load the |
| |
|
|
| align:start position:0% |
| now now you know also load the |
| commutation relation between them among |
|
|
| align:start position:0% |
| commutation relation between them among |
| |
|
|
| align:start position:0% |
| commutation relation between them among |
| them and then now can be used to |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| generate the healable space |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so this step and one two four are very |
|
|
| align:start position:0% |
| so this step and one two four are very |
| |
|
|
| align:start position:0% |
| so this step and one two four are very |
| general |
|
|
| align:start position:0% |
| general |
| |
|
|
| align:start position:0% |
| general |
| and if you can do it |
|
|
| align:start position:0% |
| and if you can do it |
| |
|
|
| align:start position:0% |
| and if you can do it |
| and then you can then you can |
|
|
| align:start position:0% |
| and then you can then you can |
| |
|
|
| align:start position:0% |
| and then you can then you can |
| essentially uh do it applied to gen any |
|
|
| align:start position:0% |
| essentially uh do it applied to gen any |
| |
|
|
| align:start position:0% |
| essentially uh do it applied to gen any |
| system |
|
|
| align:start position:0% |
| system |
| |
|
|
| align:start position:0% |
| system |
| say one degree harmonic acid is one |
|
|
| align:start position:0% |
| say one degree harmonic acid is one |
| |
|
|
| align:start position:0% |
| say one degree harmonic acid is one |
| degrees three then you can apply it two |
|
|
| align:start position:0% |
| degrees three then you can apply it two |
| |
|
|
| align:start position:0% |
| degrees three then you can apply it two |
| degrees three degrees freedom and also |
|
|
| align:start position:0% |
| degrees three degrees freedom and also |
| |
|
|
| align:start position:0% |
| degrees three degrees freedom and also |
| to field series with infinite number |
|
|
| align:start position:0% |
| to field series with infinite number |
| |
|
|
| align:start position:0% |
| to field series with infinite number |
| equals freedom |
|
|
| align:start position:0% |
| equals freedom |
| |
|
|
| align:start position:0% |
| equals freedom |
| okay and now we will apply these two |
|
|
| align:start position:0% |
| okay and now we will apply these two |
| |
|
|
| align:start position:0% |
| okay and now we will apply these two |
| field series yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| uh for this procedure you cannot |
|
|
| align:start position:0% |
| uh for this procedure you cannot |
| |
|
|
| align:start position:0% |
| uh for this procedure you cannot |
| but but uh but you can get the finite uh |
|
|
| align:start position:0% |
| but but uh but you can get the finite uh |
| |
|
|
| align:start position:0% |
| but but uh but you can get the finite uh |
| Dimension yeah |
|
|
| align:start position:0% |
| Dimension yeah |
| |
|
|
| align:start position:0% |
| Dimension yeah |
| because the finite here will space don't |
|
|
| align:start position:0% |
| because the finite here will space don't |
| |
|
|
| align:start position:0% |
| because the finite here will space don't |
| have the classical analog so here we |
|
|
| align:start position:0% |
| have the classical analog so here we |
| |
|
|
| align:start position:0% |
| have the classical analog so here we |
| start with the classical system and then |
|
|
| align:start position:0% |
| start with the classical system and then |
| |
|
|
| align:start position:0% |
| start with the classical system and then |
| we contact it and there's a the the the |
|
|
| align:start position:0% |
| we contact it and there's a the the the |
| |
|
|
| align:start position:0% |
| we contact it and there's a the the the |
| the quantum system with a finite |
|
|
| align:start position:0% |
| the quantum system with a finite |
| |
|
|
| align:start position:0% |
| the quantum system with a finite |
| Dimension healable space they |
|
|
| align:start position:0% |
| Dimension healable space they |
| |
|
|
| align:start position:0% |
| Dimension healable space they |
| essentially intrinsically quantum and uh |
|
|
| align:start position:0% |
| essentially intrinsically quantum and uh |
| |
|
|
| align:start position:0% |
| essentially intrinsically quantum and uh |
| yeah and like spin spin is an intrinsic |
|
|
| align:start position:0% |
| yeah and like spin spin is an intrinsic |
| |
|
|
| align:start position:0% |
| yeah and like spin spin is an intrinsic |
| Quantum scene yeah |
|
|
| align:start position:0% |
| Quantum scene yeah |
| |
|
|
| align:start position:0% |
| Quantum scene yeah |
| yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah yeah it's just because they |
|
|
| align:start position:0% |
| yeah yeah yeah it's just because they |
| |
|
|
| align:start position:0% |
| yeah yeah yeah it's just because they |
| don't have yeah yeah the reason is they |
|
|
| align:start position:0% |
| don't have yeah yeah the reason is they |
| |
|
|
| align:start position:0% |
| don't have yeah yeah the reason is they |
| don't have classical counterpart yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yes um is it always true |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah because because if you think about |
|
|
| align:start position:0% |
| yeah because because if you think about |
| |
|
|
| align:start position:0% |
| yeah because because if you think about |
| this way yeah that's a very good |
|
|
| align:start position:0% |
| this way yeah that's a very good |
| |
|
|
| align:start position:0% |
| this way yeah that's a very good |
| question because |
|
|
| align:start position:0% |
| question because |
| |
|
|
| align:start position:0% |
| question because |
| because let's just look at this harmonic |
|
|
| align:start position:0% |
| because let's just look at this harmonic |
| |
|
|
| align:start position:0% |
| because let's just look at this harmonic |
| oscillator and then you can try to think |
|
|
| align:start position:0% |
| oscillator and then you can try to think |
| |
|
|
| align:start position:0% |
| oscillator and then you can try to think |
| generalize it |
|
|
| align:start position:0% |
| generalize it |
| |
|
|
| align:start position:0% |
| generalize it |
| because |
|
|
| align:start position:0% |
| because |
| |
|
|
| align:start position:0% |
| because |
| they are integrating constant of the X |
|
|
| align:start position:0% |
| they are integrating constant of the X |
| |
|
|
| align:start position:0% |
| they are integrating constant of the X |
| and the p |
|
|
| align:start position:0% |
| and the p |
| |
|
|
| align:start position:0% |
| and the p |
| that any operator in your theory can all |
|
|
| align:start position:0% |
| that any operator in your theory can all |
| |
|
|
| align:start position:0% |
| that any operator in your theory can all |
| be expressed in terms of a and a dagger |
|
|
| align:start position:0% |
| be expressed in terms of a and a dagger |
| |
|
|
| align:start position:0% |
| be expressed in terms of a and a dagger |
| and then and then you'll hear the space |
|
|
| align:start position:0% |
| and then and then you'll hear the space |
| |
|
|
| align:start position:0% |
| and then and then you'll hear the space |
| must begin you must be able to generate |
|
|
| align:start position:0% |
| must begin you must be able to generate |
| |
|
|
| align:start position:0% |
| must begin you must be able to generate |
| the herbal space using them yeah yeah |
|
|
| align:start position:0% |
| the herbal space using them yeah yeah |
| |
|
|
| align:start position:0% |
| the herbal space using them yeah yeah |
| because they are the building product of |
|
|
| align:start position:0% |
| because they are the building product of |
| |
|
|
| align:start position:0% |
| because they are the building product of |
| your of your whole operators yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah the working State here is |
|
|
| align:start position:0% |
| yeah yeah the working State here is |
| |
|
|
| align:start position:0% |
| yeah yeah the working State here is |
| based on uh it's come from the uh uh uh |
|
|
| align:start position:0% |
| based on uh it's come from the uh uh uh |
| |
|
|
| align:start position:0% |
| based on uh it's come from the uh uh uh |
| uh from the energy right so so once we |
|
|
| align:start position:0% |
| uh from the energy right so so once we |
| |
|
|
| align:start position:0% |
| uh from the energy right so so once we |
| solve x and P |
|
|
| align:start position:0% |
| solve x and P |
| |
|
|
| align:start position:0% |
| solve x and P |
| and then you can write to the |
|
|
| align:start position:0% |
| and then you can write to the |
| |
|
|
| align:start position:0% |
| and then you can write to the |
| hamiltonian in terms of X and P and then |
|
|
| align:start position:0% |
| hamiltonian in terms of X and P and then |
| |
|
|
| align:start position:0% |
| hamiltonian in terms of X and P and then |
| you just look for the lowest energy |
|
|
| align:start position:0% |
| you just look for the lowest energy |
| |
|
|
| align:start position:0% |
| you just look for the lowest energy |
| State and then you find the lowest |
|
|
| align:start position:0% |
| State and then you find the lowest |
| |
|
|
| align:start position:0% |
| State and then you find the lowest |
| energy State just satisfy this equation |
|
|
| align:start position:0% |
| energy State just satisfy this equation |
| |
|
|
| align:start position:0% |
| energy State just satisfy this equation |
| yeah and then from there you can find |
|
|
| align:start position:0% |
| yeah and then from there you can find |
| |
|
|
| align:start position:0% |
| yeah and then from there you can find |
| other states |
|
|
| align:start position:0% |
| other states |
| |
|
|
| align:start position:0% |
| other states |
| yeah the same thing we are going to yeah |
|
|
| align:start position:0% |
| yeah the same thing we are going to yeah |
| |
|
|
| align:start position:0% |
| yeah the same thing we are going to yeah |
| the same strategy we are going to use |
|
|
| align:start position:0% |
| the same strategy we are going to use |
| |
|
|
| align:start position:0% |
| the same strategy we are going to use |
| for the uh of a Quantum velocity |
|
|
| align:start position:0% |
| for the uh of a Quantum velocity |
| |
|
|
| align:start position:0% |
| for the uh of a Quantum velocity |
| okay good |
|
|
| align:start position:0% |
| okay good |
| |
|
|
| align:start position:0% |
| okay good |
| okay good so so now become a mechanical |
|
|
| align:start position:0% |
| okay good so so now become a mechanical |
| |
|
|
| align:start position:0% |
| okay good so so now become a mechanical |
| we can just uh apply this to to this |
|
|
| align:start position:0% |
| we can just uh apply this to to this |
| |
|
|
| align:start position:0% |
| we can just uh apply this to to this |
| Theory |
|
|
| align:start position:0% |
| Theory |
| |
|
|
| align:start position:0% |
| Theory |
| okay we can just apply these two three |
|
|
| align:start position:0% |
| okay we can just apply these two three |
| |
|
|
| align:start position:0% |
| okay we can just apply these two three |
| Theory and now let me add here so here |
|
|
| align:start position:0% |
| Theory and now let me add here so here |
| |
|
|
| align:start position:0% |
| Theory and now let me add here so here |
| the canonical momentum density conjugate |
|
|
| align:start position:0% |
| the canonical momentum density conjugate |
| |
|
|
| align:start position:0% |
| the canonical momentum density conjugate |
| to Phi is called the pi before it's just |
|
|
| align:start position:0% |
| to Phi is called the pi before it's just |
| |
|
|
| align:start position:0% |
| to Phi is called the pi before it's just |
| the atom derivative of Phi |
|
|
| align:start position:0% |
| the atom derivative of Phi |
| |
|
|
| align:start position:0% |
| the atom derivative of Phi |
| and the hamiltonian density you can find |
|
|
| align:start position:0% |
| and the hamiltonian density you can find |
| |
|
|
| align:start position:0% |
| and the hamiltonian density you can find |
| it explicitly |
|
|
| align:start position:0% |
| it explicitly |
| |
|
|
| align:start position:0% |
| it explicitly |
| is pi square plus |
|
|
| align:start position:0% |
| is pi square plus |
| |
|
|
| align:start position:0% |
| is pi square plus |
| one half |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and then this is a classical equation |
|
|
| align:start position:0% |
| and then this is a classical equation |
| |
|
|
| align:start position:0% |
| and then this is a classical equation |
| motion okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| now let's just solve this classical |
|
|
| align:start position:0% |
| now let's just solve this classical |
| |
|
|
| align:start position:0% |
| now let's just solve this classical |
| equation motion |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this equation can be |
|
|
| align:start position:0% |
| so this equation can be |
| |
|
|
| align:start position:0% |
| so this equation can be |
| so this equation is easy to solve |
|
|
| align:start position:0% |
| so this equation is easy to solve |
| |
|
|
| align:start position:0% |
| so this equation is easy to solve |
| because the because the translation |
|
|
| align:start position:0% |
| because the because the translation |
| |
|
|
| align:start position:0% |
| because the because the translation |
| symmetry |
|
|
| align:start position:0% |
| symmetry |
| |
|
|
| align:start position:0% |
| symmetry |
| okay you can just do a free transform |
|
|
| align:start position:0% |
| okay you can just do a free transform |
| |
|
|
| align:start position:0% |
| okay you can just do a free transform |
| okay so we can fully transform |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so now let's do the um |
|
|
| align:start position:0% |
| so so now let's do the um |
| |
|
|
| align:start position:0% |
| so so now let's do the um |
| you can freely transform |
|
|
| align:start position:0% |
| you can freely transform |
| |
|
|
| align:start position:0% |
| you can freely transform |
| so 2 Prime |
|
|
| align:start position:0% |
| so 2 Prime |
| |
|
|
| align:start position:0% |
| so 2 Prime |
| to be solved |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so we can just write 5X |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| equal to expenditure |
|
|
| align:start position:0% |
| equal to expenditure |
| |
|
|
| align:start position:0% |
| equal to expenditure |
| minus IET |
|
|
| align:start position:0% |
| minus IET |
| |
|
|
| align:start position:0% |
| minus IET |
| plus I |
|
|
| align:start position:0% |
| plus I |
| |
|
|
| align:start position:0% |
| plus I |
| K dot X |
|
|
| align:start position:0% |
| K dot X |
| |
|
|
| align:start position:0% |
| K dot X |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so and then you you can see that |
|
|
| align:start position:0% |
| okay so and then you you can see that |
| |
|
|
| align:start position:0% |
| okay so and then you you can see that |
| this |
|
|
| align:start position:0% |
| this |
| |
|
|
| align:start position:0% |
| this |
| just solves the |
|
|
| align:start position:0% |
| just solves the |
| |
|
|
| align:start position:0% |
| just solves the |
| so this is provide |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| a basis |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of solutions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| 22 prime okay |
|
|
| align:start position:0% |
| 22 prime okay |
| |
|
|
| align:start position:0% |
| 22 prime okay |
| it's just a plain wave okay just plain |
|
|
| align:start position:0% |
| it's just a plain wave okay just plain |
| |
|
|
| align:start position:0% |
| it's just a plain wave okay just plain |
| weight |
|
|
| align:start position:0% |
| weight |
| |
|
|
| align:start position:0% |
| weight |
| oh |
|
|
| align:start position:0% |
| oh |
| |
|
|
| align:start position:0% |
| oh |
| so now we plug this into there |
|
|
| align:start position:0% |
| so now we plug this into there |
| |
|
|
| align:start position:0% |
| so now we plug this into there |
| then you just get the dispersion |
|
|
| align:start position:0% |
| then you just get the dispersion |
| |
|
|
| align:start position:0% |
| then you just get the dispersion |
| relation |
|
|
| align:start position:0% |
| relation |
| |
|
|
| align:start position:0% |
| relation |
| e Square |
|
|
| align:start position:0% |
| e Square |
| |
|
|
| align:start position:0% |
| e Square |
| should be M squared plus K Square |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so so we'll delote this as |
|
|
| align:start position:0% |
| okay so so we'll delote this as |
| |
|
|
| align:start position:0% |
| okay so so we'll delote this as |
| Omega K Square |
|
|
| align:start position:0% |
| Omega K Square |
| |
|
|
| align:start position:0% |
| Omega K Square |
| so Omega k |
|
|
| align:start position:0% |
| so Omega k |
| |
|
|
| align:start position:0% |
| so Omega k |
| defines to be |
|
|
| align:start position:0% |
| defines to be |
| |
|
|
| align:start position:0% |
| defines to be |
| just the K Square |
|
|
| align:start position:0% |
| just the K Square |
| |
|
|
| align:start position:0% |
| just the K Square |
| plus M squared |
|
|
| align:start position:0% |
| plus M squared |
| |
|
|
| align:start position:0% |
| plus M squared |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so ee when you take the square root of e |
|
|
| align:start position:0% |
| so ee when you take the square root of e |
| |
|
|
| align:start position:0% |
| so ee when you take the square root of e |
| so you can take a plus minus Omega k |
|
|
| align:start position:0% |
| so you can take a plus minus Omega k |
| |
|
|
| align:start position:0% |
| so you can take a plus minus Omega k |
| okay can be plus minus Omega K so we |
|
|
| align:start position:0% |
| okay can be plus minus Omega K so we |
| |
|
|
| align:start position:0% |
| okay can be plus minus Omega K so we |
| normally call the solution |
|
|
| align:start position:0% |
| normally call the solution |
| |
|
|
| align:start position:0% |
| normally call the solution |
| we normally separate |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so for historical reasons okay |
|
|
| align:start position:0% |
| so for historical reasons okay |
| |
|
|
| align:start position:0% |
| so for historical reasons okay |
| we normally quote |
|
|
| align:start position:0% |
| we normally quote |
| |
|
|
| align:start position:0% |
| we normally quote |
| Define UK |
|
|
| align:start position:0% |
| Define UK |
| |
|
|
| align:start position:0% |
| Define UK |
| X to be |
|
|
| align:start position:0% |
| X to be |
| |
|
|
| align:start position:0% |
| X to be |
| experiential matters I Omega k |
|
|
| align:start position:0% |
| experiential matters I Omega k |
| |
|
|
| align:start position:0% |
| experiential matters I Omega k |
| t |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| X so okay now we have inserted the |
|
|
| align:start position:0% |
| X so okay now we have inserted the |
| |
|
|
| align:start position:0% |
| X so okay now we have inserted the |
| positive root of E |
|
|
| align:start position:0% |
| positive root of E |
| |
|
|
| align:start position:0% |
| positive root of E |
| so this is normally called the positive |
|
|
| align:start position:0% |
| so this is normally called the positive |
| |
|
|
| align:start position:0% |
| so this is normally called the positive |
| Energy Solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| even though this |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| even though this name is Elizabeth |
|
|
| align:start position:0% |
| even though this name is Elizabeth |
| |
|
|
| align:start position:0% |
| even though this name is Elizabeth |
| misleading okay so so uh actually this |
|
|
| align:start position:0% |
| misleading okay so so uh actually this |
| |
|
|
| align:start position:0% |
| misleading okay so so uh actually this |
| we don't Define the energy actually uh |
|
|
| align:start position:0% |
| we don't Define the energy actually uh |
| |
|
|
| align:start position:0% |
| we don't Define the energy actually uh |
| yeah later we will see this is not |
|
|
| align:start position:0% |
| yeah later we will see this is not |
| |
|
|
| align:start position:0% |
| yeah later we will see this is not |
| really the energy of a particle and so |
|
|
| align:start position:0% |
| really the energy of a particle and so |
| |
|
|
| align:start position:0% |
| really the energy of a particle and so |
| so this is just a traditional name okay |
|
|
| align:start position:0% |
| so this is just a traditional name okay |
| |
|
|
| align:start position:0% |
| so this is just a traditional name okay |
| this is just a traditional name |
|
|
| align:start position:0% |
| this is just a traditional name |
| |
|
|
| align:start position:0% |
| this is just a traditional name |
| conventional name |
|
|
| align:start position:0% |
| conventional name |
| |
|
|
| align:start position:0% |
| conventional name |
| and then you can Define the compressed |
|
|
| align:start position:0% |
| and then you can Define the compressed |
| |
|
|
| align:start position:0% |
| and then you can Define the compressed |
| conjugate of K |
|
|
| align:start position:0% |
| conjugate of K |
| |
|
|
| align:start position:0% |
| conjugate of K |
| now you have |
|
|
| align:start position:0% |
| now you have |
| |
|
|
| align:start position:0% |
| now you have |
| then corresponding to you have |
|
|
| align:start position:0% |
| then corresponding to you have |
| |
|
|
| align:start position:0% |
| then corresponding to you have |
| minus |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| Omega K in there |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we |
|
|
| align:start position:0% |
| so we |
| |
|
|
| align:start position:0% |
| so we |
| yeah we take this conjugate |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and so this is called the reactive |
|
|
| align:start position:0% |
| and so this is called the reactive |
| |
|
|
| align:start position:0% |
| and so this is called the reactive |
| Energy Solution |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so all together |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| therefore the former company side of |
|
|
| align:start position:0% |
| therefore the former company side of |
| |
|
|
| align:start position:0% |
| therefore the former company side of |
| solutions okay so complete side of |
|
|
| align:start position:0% |
| solutions okay so complete side of |
| |
|
|
| align:start position:0% |
| solutions okay so complete side of |
| solutions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| complete the basis the complete side of |
|
|
| align:start position:0% |
| complete the basis the complete side of |
| |
|
|
| align:start position:0% |
| complete the basis the complete side of |
| yeah complete |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| phases |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| is formed by |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| UK |
|
|
| align:start position:0% |
| UK |
| |
|
|
| align:start position:0% |
| UK |
| and the UK UK star |
|
|
| align:start position:0% |
| and the UK UK star |
| |
|
|
| align:start position:0% |
| and the UK UK star |
| for okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so these are the uh when you these are |
|
|
| align:start position:0% |
| so these are the uh when you these are |
| |
|
|
| align:start position:0% |
| so these are the uh when you these are |
| the complete set of solutions to that |
|
|
| align:start position:0% |
| the complete set of solutions to that |
| |
|
|
| align:start position:0% |
| the complete set of solutions to that |
| wave equation okay so that's just the uh |
|
|
| align:start position:0% |
| wave equation okay so that's just the uh |
| |
|
|
| align:start position:0% |
| wave equation okay so that's just the uh |
| um yeah the uh |
|
|
| align:start position:0% |
| um yeah the uh |
| |
|
|
| align:start position:0% |
| um yeah the uh |
| any questions on this |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is just like a classically this |
|
|
| align:start position:0% |
| so this is just like a classically this |
| |
|
|
| align:start position:0% |
| so this is just like a classically this |
| is like a wave okay just like a plane |
|
|
| align:start position:0% |
| is like a wave okay just like a plane |
| |
|
|
| align:start position:0% |
| is like a wave okay just like a plane |
| wave which you should also have seen in |
|
|
| align:start position:0% |
| wave which you should also have seen in |
| |
|
|
| align:start position:0% |
| wave which you should also have seen in |
| 803 you know |
|
|
| align:start position:0% |
| 803 you know |
| |
|
|
| align:start position:0% |
| 803 you know |
| or |
|
|
| align:start position:0% |
| or |
| |
|
|
| align:start position:0% |
| or |
| good |
|
|
| align:start position:0% |
| good |
| |
|
|
| align:start position:0% |
| good |
| so now we can find that so now we can |
|
|
| align:start position:0% |
| so now we can find that so now we can |
| |
|
|
| align:start position:0% |
| so now we can find that so now we can |
| write down the most General |
|
|
| align:start position:0% |
| write down the most General |
| |
|
|
| align:start position:0% |
| write down the most General |
| so this is a basis |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so these are the counterparts of the |
|
|
| align:start position:0% |
| so these are the counterparts of the |
| |
|
|
| align:start position:0% |
| so these are the counterparts of the |
| exponential plus minus the Omega T here |
|
|
| align:start position:0% |
| exponential plus minus the Omega T here |
| |
|
|
| align:start position:0% |
| exponential plus minus the Omega T here |
| okay so now we can write down the most |
|
|
| align:start position:0% |
| okay so now we can write down the most |
| |
|
|
| align:start position:0% |
| okay so now we can write down the most |
| General solutions by just putting the |
|
|
| align:start position:0% |
| General solutions by just putting the |
| |
|
|
| align:start position:0% |
| General solutions by just putting the |
| integration constant |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| to the most General |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| classical Solutions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so you can just write Phi X |
|
|
| align:start position:0% |
| so you can just write Phi X |
| |
|
|
| align:start position:0% |
| so you can just write Phi X |
| equal to |
|
|
| align:start position:0% |
| equal to |
| |
|
|
| align:start position:0% |
| equal to |
| integrate of all possible value |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| of K because this is for all K so we |
|
|
| align:start position:0% |
| of K because this is for all K so we |
| |
|
|
| align:start position:0% |
| of K because this is for all K so we |
| just sum up all of them |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and uh so we so |
|
|
| align:start position:0% |
| and uh so we so |
| |
|
|
| align:start position:0% |
| and uh so we so |
| so this factor is for for just for |
|
|
| align:start position:0% |
| so this factor is for for just for |
| |
|
|
| align:start position:0% |
| so this factor is for for just for |
| convenience okay it's just a convention |
|
|
| align:start position:0% |
| convenience okay it's just a convention |
| |
|
|
| align:start position:0% |
| convenience okay it's just a convention |
| you don't have to put it here it's just |
|
|
| align:start position:0% |
| you don't have to put it here it's just |
| |
|
|
| align:start position:0% |
| you don't have to put it here it's just |
| a convention |
|
|
| align:start position:0% |
| a convention |
| |
|
|
| align:start position:0% |
| a convention |
| and then and then we have |
|
|
| align:start position:0% |
| and then and then we have |
| |
|
|
| align:start position:0% |
| and then and then we have |
| a k |
|
|
| align:start position:0% |
| a k |
| |
|
|
| align:start position:0% |
| a k |
| u k |
|
|
| align:start position:0% |
| u k |
| |
|
|
| align:start position:0% |
| u k |
| plus a k star |
|
|
| align:start position:0% |
| plus a k star |
| |
|
|
| align:start position:0% |
| plus a k star |
| UK stop |
|
|
| align:start position:0% |
| UK stop |
| |
|
|
| align:start position:0% |
| UK stop |
| okay so this is just the most General |
|
|
| align:start position:0% |
| okay so this is just the most General |
| |
|
|
| align:start position:0% |
| okay so this is just the most General |
| set of solutions |
|
|
| align:start position:0% |
| set of solutions |
| |
|
|
| align:start position:0% |
| set of solutions |
| with AK and the AK Star as integration |
|
|
| align:start position:0% |
| with AK and the AK Star as integration |
| |
|
|
| align:start position:0% |
| with AK and the AK Star as integration |
| constant |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this is a full set of the integration |
|
|
| align:start position:0% |
| so this is a full set of the integration |
| |
|
|
| align:start position:0% |
| so this is a full set of the integration |
| constant okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now when you go to Quantum level |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now we can just follow the rule |
|
|
| align:start position:0% |
| so now we can just follow the rule |
| |
|
|
| align:start position:0% |
| so now we can just follow the rule |
| okay we find the the most General |
|
|
| align:start position:0% |
| okay we find the the most General |
| |
|
|
| align:start position:0% |
| okay we find the the most General |
| classical solution |
|
|
| align:start position:0% |
| classical solution |
| |
|
|
| align:start position:0% |
| classical solution |
| and in the column level we just promote |
|
|
| align:start position:0% |
| and in the column level we just promote |
| |
|
|
| align:start position:0% |
| and in the column level we just promote |
| the distribute operator |
|
|
| align:start position:0% |
| the distribute operator |
| |
|
|
| align:start position:0% |
| the distribute operator |
| you just put a hat there |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and change this to dagger |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so now this becomes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| your basis of quantum operators |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so these are the cons the full set of |
|
|
| align:start position:0% |
| so these are the cons the full set of |
| |
|
|
| align:start position:0% |
| so these are the cons the full set of |
| constant Quantum operators |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and this solves your Theory on |
|
|
| align:start position:0% |
| and this solves your Theory on |
| |
|
|
| align:start position:0% |
| and this solves your Theory on |
| okay so uh so this solves the the |
|
|
| align:start position:0% |
| okay so uh so this solves the the |
| |
|
|
| align:start position:0% |
| okay so uh so this solves the the |
| operator equation |
|
|
| align:start position:0% |
| operator equation |
| |
|
|
| align:start position:0% |
| operator equation |
| and dissolves the operating equation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now the Legacy |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| is to impose the canonical |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| commutation relation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so firstly we have to uh now we have to |
|
|
| align:start position:0% |
| so firstly we have to uh now we have to |
| |
|
|
| align:start position:0% |
| so firstly we have to uh now we have to |
| do a little bit thinking okay |
|
|
| align:start position:0% |
| do a little bit thinking okay |
| |
|
|
| align:start position:0% |
| do a little bit thinking okay |
| so so far you just uh straightforward |
|
|
| align:start position:0% |
| so so far you just uh straightforward |
| |
|
|
| align:start position:0% |
| so so far you just uh straightforward |
| but now we have to do a little bit |
|
|
| align:start position:0% |
| but now we have to do a little bit |
| |
|
|
| align:start position:0% |
| but now we have to do a little bit |
| thinking |
|
|
| align:start position:0% |
| thinking |
| |
|
|
| align:start position:0% |
| thinking |
| so so for |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| for finite |
|
|
| align:start position:0% |
| for finite |
| |
|
|
| align:start position:0% |
| for finite |
| for harmonic oscillator or for |
|
|
| align:start position:0% |
| for harmonic oscillator or for |
| |
|
|
| align:start position:0% |
| for harmonic oscillator or for |
| Quantum system of a single variable |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| you just have X you you just have this |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| you just have these and now we need to |
|
|
| align:start position:0% |
| you just have these and now we need to |
| |
|
|
| align:start position:0% |
| you just have these and now we need to |
| come up |
|
|
| align:start position:0% |
| come up |
| |
|
|
| align:start position:0% |
| come up |
| with the generalization of these two |
|
|
| align:start position:0% |
| with the generalization of these two |
| |
|
|
| align:start position:0% |
| with the generalization of these two |
| field Theory |
|
|
| align:start position:0% |
| field Theory |
| |
|
|
| align:start position:0% |
| field Theory |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so so we need to come up with the |
|
|
| align:start position:0% |
| so so we need to come up with the |
| |
|
|
| align:start position:0% |
| so so we need to come up with the |
| generation of that to field City with |
|
|
| align:start position:0% |
| generation of that to field City with |
| |
|
|
| align:start position:0% |
| generation of that to field City with |
| corresponding to |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| by X |
|
|
| align:start position:0% |
| by X |
| |
|
|
| align:start position:0% |
| by X |
| so now let me |
|
|
| align:start position:0% |
| so now let me |
| |
|
|
| align:start position:0% |
| so now let me |
| make it the time and the special |
|
|
| align:start position:0% |
| make it the time and the special |
| |
|
|
| align:start position:0% |
| make it the time and the special |
| coordinate separate |
|
|
| align:start position:0% |
| coordinate separate |
| |
|
|
| align:start position:0% |
| coordinate separate |
| and these conjugate the momentum is Phi |
|
|
| align:start position:0% |
| and these conjugate the momentum is Phi |
| |
|
|
| align:start position:0% |
| and these conjugate the momentum is Phi |
| it is pi okay |
|
|
| align:start position:0% |
| it is pi okay |
| |
|
|
| align:start position:0% |
| it is pi okay |
| conjugate the momentum density |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so we should do them at the same time |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| remember |
|
|
| align:start position:0% |
| remember |
| |
|
|
| align:start position:0% |
| remember |
| T is the is the same Evolution operator |
|
|
| align:start position:0% |
| T is the is the same Evolution operator |
| |
|
|
| align:start position:0% |
| T is the is the same Evolution operator |
| so they have to be valued at the same |
|
|
| align:start position:0% |
| so they have to be valued at the same |
| |
|
|
| align:start position:0% |
| so they have to be valued at the same |
| time as it's a equal time a canonical of |
|
|
| align:start position:0% |
| time as it's a equal time a canonical of |
| |
|
|
| align:start position:0% |
| time as it's a equal time a canonical of |
| quantitative condition is always at the |
|
|
| align:start position:0% |
| quantitative condition is always at the |
| |
|
|
| align:start position:0% |
| quantitative condition is always at the |
| equal time |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| but the x is a label of operators so so |
|
|
| align:start position:0% |
| but the x is a label of operators so so |
| |
|
|
| align:start position:0% |
| but the x is a label of operators so so |
| actually they don't have to be the same |
|
|
| align:start position:0% |
| actually they don't have to be the same |
| |
|
|
| align:start position:0% |
| actually they don't have to be the same |
| okay so here can be X here can be X |
|
|
| align:start position:0% |
| okay so here can be X here can be X |
| |
|
|
| align:start position:0% |
| okay so here can be X here can be X |
| Prime okay so now we have to come up |
|
|
| align:start position:0% |
| Prime okay so now we have to come up |
| |
|
|
| align:start position:0% |
| Prime okay so now we have to come up |
| with the generation of what is this |
|
|
| align:start position:0% |
| with the generation of what is this |
| |
|
|
| align:start position:0% |
| with the generation of what is this |
| quantity for filter okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so now we just need to do a little |
|
|
| align:start position:0% |
| so so now we just need to do a little |
| |
|
|
| align:start position:0% |
| so so now we just need to do a little |
| bit guesswork okay you can easily guess |
|
|
| align:start position:0% |
| bit guesswork okay you can easily guess |
| |
|
|
| align:start position:0% |
| bit guesswork okay you can easily guess |
| it |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so before we do that you have any |
|
|
| align:start position:0% |
| so before we do that you have any |
| |
|
|
| align:start position:0% |
| so before we do that you have any |
| questions on this yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah X is always so X is always |
|
|
| align:start position:0% |
| yeah yeah X is always so X is always |
| |
|
|
| align:start position:0% |
| yeah yeah X is always so X is always |
| here it's always just the label of the |
|
|
| align:start position:0% |
| here it's always just the label of the |
| |
|
|
| align:start position:0% |
| here it's always just the label of the |
| spatial location right yes yeah it's a |
|
|
| align:start position:0% |
| spatial location right yes yeah it's a |
| |
|
|
| align:start position:0% |
| spatial location right yes yeah it's a |
| label for the yeah yeah it's your field |
|
|
| align:start position:0% |
| label for the yeah yeah it's your field |
| |
|
|
| align:start position:0% |
| label for the yeah yeah it's your field |
| Theory enable yes so |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| operators has been constant in time is |
|
|
| align:start position:0% |
| operators has been constant in time is |
| |
|
|
| align:start position:0% |
| operators has been constant in time is |
| there any way that you can get them |
|
|
| align:start position:0% |
| there any way that you can get them |
| |
|
|
| align:start position:0% |
| there any way that you can get them |
| where it's like the evolution is more |
|
|
| align:start position:0% |
| where it's like the evolution is more |
| |
|
|
| align:start position:0% |
| where it's like the evolution is more |
| complex rather than just a constant |
|
|
| align:start position:0% |
| complex rather than just a constant |
| |
|
|
| align:start position:0% |
| complex rather than just a constant |
| operator and it |
|
|
| align:start position:0% |
| operator and it |
| |
|
|
| align:start position:0% |
| operator and it |
| yeah so so normally if you have second |
|
|
| align:start position:0% |
| yeah so so normally if you have second |
| |
|
|
| align:start position:0% |
| yeah so so normally if you have second |
| order differential equation |
|
|
| align:start position:0% |
| order differential equation |
| |
|
|
| align:start position:0% |
| order differential equation |
| you always have some integration |
|
|
| align:start position:0% |
| you always have some integration |
| |
|
|
| align:start position:0% |
| you always have some integration |
| constant yeah that's it yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah |
|
|
| align:start position:0% |
| yeah |
| |
|
|
| align:start position:0% |
| yeah |
| yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| other questions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| directly into the operator |
|
|
| align:start position:0% |
| directly into the operator |
| |
|
|
| align:start position:0% |
| directly into the operator |
| right yeah yeah so that's a very good |
|
|
| align:start position:0% |
| right yeah yeah so that's a very good |
| |
|
|
| align:start position:0% |
| right yeah yeah so that's a very good |
| the uh that's a very good uh |
|
|
| align:start position:0% |
| the uh that's a very good uh |
| |
|
|
| align:start position:0% |
| the uh that's a very good uh |
| [Music] |
|
|
| align:start position:0% |
| [Music] |
| |
|
|
| align:start position:0% |
| [Music] |
| questions |
|
|
| align:start position:0% |
| questions |
| |
|
|
| align:start position:0% |
| questions |
| so that's just extension |
|
|
| align:start position:0% |
| so that's just extension |
| |
|
|
| align:start position:0% |
| so that's just extension |
| of our usual procedure for the quantum |
|
|
| align:start position:0% |
| of our usual procedure for the quantum |
| |
|
|
| align:start position:0% |
| of our usual procedure for the quantum |
| mechanics |
|
|
| align:start position:0% |
| mechanics |
| |
|
|
| align:start position:0% |
| mechanics |
| so so the usual procedure when you even |
|
|
| align:start position:0% |
| so so the usual procedure when you even |
| |
|
|
| align:start position:0% |
| so so the usual procedure when you even |
| just for harmonic also the for single |
|
|
| align:start position:0% |
| just for harmonic also the for single |
| |
|
|
| align:start position:0% |
| just for harmonic also the for single |
| variable system you have this |
|
|
| align:start position:0% |
| variable system you have this |
| |
|
|
| align:start position:0% |
| variable system you have this |
| correspondence between the classical |
|
|
| align:start position:0% |
| correspondence between the classical |
| |
|
|
| align:start position:0% |
| correspondence between the classical |
| system the quantum system when you |
|
|
| align:start position:0% |
| system the quantum system when you |
| |
|
|
| align:start position:0% |
| system the quantum system when you |
| quantize the classical system then the |
|
|
| align:start position:0% |
| quantize the classical system then the |
| |
|
|
| align:start position:0% |
| quantize the classical system then the |
| classical equation working becomes a |
|
|
| align:start position:0% |
| classical equation working becomes a |
| |
|
|
| align:start position:0% |
| classical equation working becomes a |
| Quantum operating equation uh here we |
|
|
| align:start position:0% |
| Quantum operating equation uh here we |
| |
|
|
| align:start position:0% |
| Quantum operating equation uh here we |
| just use the same rule because Quantum |
|
|
| align:start position:0% |
| just use the same rule because Quantum |
| |
|
|
| align:start position:0% |
| just use the same rule because Quantum |
| field is just the theory of infinite |
|
|
| align:start position:0% |
| field is just the theory of infinite |
| |
|
|
| align:start position:0% |
| field is just the theory of infinite |
| number of degrees freedoms we are not |
|
|
| align:start position:0% |
| number of degrees freedoms we are not |
| |
|
|
| align:start position:0% |
| number of degrees freedoms we are not |
| changing the rule of quantum mechanics |
|
|
| align:start position:0% |
| changing the rule of quantum mechanics |
| |
|
|
| align:start position:0% |
| changing the rule of quantum mechanics |
| and so that's why we just again just |
|
|
| align:start position:0% |
| and so that's why we just again just |
| |
|
|
| align:start position:0% |
| and so that's why we just again just |
| promote the classical equation into the |
|
|
| align:start position:0% |
| promote the classical equation into the |
| |
|
|
| align:start position:0% |
| promote the classical equation into the |
| operating equation |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| other questions |
|
|
| align:start position:0% |
| other questions |
| |
|
|
| align:start position:0% |
| other questions |
| yes |
|
|
| align:start position:0% |
| yes |
| |
|
|
| align:start position:0% |
| yes |
| one way of understanding like the |
|
|
| align:start position:0% |
| one way of understanding like the |
| |
|
|
| align:start position:0% |
| one way of understanding like the |
| Heisenberg equations for the |
|
|
| align:start position:0% |
| Heisenberg equations for the |
| |
|
|
| align:start position:0% |
| Heisenberg equations for the |
| Quantum mechanic system brackets for |
|
|
| align:start position:0% |
| Quantum mechanic system brackets for |
| |
|
|
| align:start position:0% |
| Quantum mechanic system brackets for |
| like the classroom is there something |
|
|
| align:start position:0% |
| like the classroom is there something |
| |
|
|
| align:start position:0% |
| like the classroom is there something |
| like that for field Theory as well |
|
|
| align:start position:0% |
| like that for field Theory as well |
| |
|
|
| align:start position:0% |
| like that for field Theory as well |
| um yeah yeah yeah there is yeah so so |
|
|
| align:start position:0% |
| um yeah yeah yeah there is yeah so so |
| |
|
|
| align:start position:0% |
| um yeah yeah yeah there is yeah so so |
| classically you can Define the possum |
|
|
| align:start position:0% |
| classically you can Define the possum |
| |
|
|
| align:start position:0% |
| classically you can Define the possum |
| bracket |
|
|
| align:start position:0% |
| bracket |
| |
|
|
| align:start position:0% |
| bracket |
| between the uh between the classical |
|
|
| align:start position:0% |
| between the uh between the classical |
| |
|
|
| align:start position:0% |
| between the uh between the classical |
| field variables and then and then |
|
|
| align:start position:0% |
| field variables and then and then |
| |
|
|
| align:start position:0% |
| field variables and then and then |
| Quantum mechanically just become commit |
|
|
| align:start position:0% |
| Quantum mechanically just become commit |
| |
|
|
| align:start position:0% |
| Quantum mechanically just become commit |
| uh uh uh quantum commutators |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah you can also do that that's |
|
|
| align:start position:0% |
| yeah yeah you can also do that that's |
| |
|
|
| align:start position:0% |
| yeah yeah you can also do that that's |
| right yeah so so one way to come to this |
|
|
| align:start position:0% |
| right yeah so so one way to come to this |
| |
|
|
| align:start position:0% |
| right yeah so so one way to come to this |
| is you first describe first you need to |
|
|
| align:start position:0% |
| is you first describe first you need to |
| |
|
|
| align:start position:0% |
| is you first describe first you need to |
| generalize |
|
|
| align:start position:0% |
| generalize |
| |
|
|
| align:start position:0% |
| generalize |
| your standards possible brackets |
|
|
| align:start position:0% |
| your standards possible brackets |
| |
|
|
| align:start position:0% |
| your standards possible brackets |
| for final number because freedom to |
|
|
| align:start position:0% |
| for final number because freedom to |
| |
|
|
| align:start position:0% |
| for final number because freedom to |
| classical field Theory and then you can |
|
|
| align:start position:0% |
| classical field Theory and then you can |
| |
|
|
| align:start position:0% |
| classical field Theory and then you can |
| just generalize that to the uh to |
|
|
| align:start position:0% |
| just generalize that to the uh to |
| |
|
|
| align:start position:0% |
| just generalize that to the uh to |
| Quantum yeah indeed uh that's one route |
|
|
| align:start position:0% |
| Quantum yeah indeed uh that's one route |
| |
|
|
| align:start position:0% |
| Quantum yeah indeed uh that's one route |
| of doing it yeah |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay other questions |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good so we'll be discussed the answer |
|
|
| align:start position:0% |
| good so we'll be discussed the answer |
| |
|
|
| align:start position:0% |
| good so we'll be discussed the answer |
| okay the answer is very easy to guess |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so remember |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so um |
|
|
| align:start position:0% |
| so um |
| |
|
|
| align:start position:0% |
| so um |
| if you have a single X and P |
|
|
| align:start position:0% |
| if you have a single X and P |
| |
|
|
| align:start position:0% |
| if you have a single X and P |
| that's what you have okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| but if you have more than one particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| if you have more than one particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| say just hint |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| say you have multiple particles |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| system in quantum mechanics and then you |
|
|
| align:start position:0% |
| system in quantum mechanics and then you |
| |
|
|
| align:start position:0% |
| system in quantum mechanics and then you |
| have x a |
|
|
| align:start position:0% |
| have x a |
| |
|
|
| align:start position:0% |
| have x a |
| and the PA as your dynamical variable |
|
|
| align:start position:0% |
| and the PA as your dynamical variable |
| |
|
|
| align:start position:0% |
| and the PA as your dynamical variable |
| so a equal to 1 to say ends it says the |
|
|
| align:start position:0% |
| so a equal to 1 to say ends it says the |
| |
|
|
| align:start position:0% |
| so a equal to 1 to say ends it says the |
| lumbar particles okay |
|
|
| align:start position:0% |
| lumbar particles okay |
| |
|
|
| align:start position:0% |
| lumbar particles okay |
| and then your canonical quantization |
|
|
| align:start position:0% |
| and then your canonical quantization |
| |
|
|
| align:start position:0% |
| and then your canonical quantization |
| condition just become x a |
|
|
| align:start position:0% |
| condition just become x a |
| |
|
|
| align:start position:0% |
| condition just become x a |
| t |
|
|
| align:start position:0% |
| t |
| |
|
|
| align:start position:0% |
| t |
| p b t |
|
|
| align:start position:0% |
| p b t |
| |
|
|
| align:start position:0% |
| p b t |
| equal to I Delta ad |
|
|
| align:start position:0% |
| equal to I Delta ad |
| |
|
|
| align:start position:0% |
| equal to I Delta ad |
| and the different XA is commute |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| and different P commute |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay given the P commutes okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so |
|
|
| align:start position:0% |
| so |
| |
|
|
| align:start position:0% |
| so |
| so now this A and B |
|
|
| align:start position:0% |
| so now this A and B |
| |
|
|
| align:start position:0% |
| so now this A and B |
| are essentially just replaced by x and x |
|
|
| align:start position:0% |
| are essentially just replaced by x and x |
| |
|
|
| align:start position:0% |
| are essentially just replaced by x and x |
| Prime |
|
|
| align:start position:0% |
| Prime |
| |
|
|
| align:start position:0% |
| Prime |
| so x and x Prime are just continual of |
|
|
| align:start position:0% |
| so x and x Prime are just continual of |
| |
|
|
| align:start position:0% |
| so x and x Prime are just continual of |
| those A and B okay remember we kind of |
|
|
| align:start position:0% |
| those A and B okay remember we kind of |
| |
|
|
| align:start position:0% |
| those A and B okay remember we kind of |
| emphasize the X and the X Prime are the |
|
|
| align:start position:0% |
| emphasize the X and the X Prime are the |
| |
|
|
| align:start position:0% |
| emphasize the X and the X Prime are the |
| labels of your degrees freedom |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so now you can just guess |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so we must have |
|
|
| align:start position:0% |
| okay so we must have |
| |
|
|
| align:start position:0% |
| okay so we must have |
| the following scene |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so so from here we must have |
|
|
| align:start position:0% |
| so so from here we must have |
| |
|
|
| align:start position:0% |
| so so from here we must have |
| Phi TX |
|
|
| align:start position:0% |
| Phi TX |
| |
|
|
| align:start position:0% |
| Phi TX |
| Phi TX Prime |
|
|
| align:start position:0% |
| Phi TX Prime |
| |
|
|
| align:start position:0% |
| Phi TX Prime |
| must be zero |
|
|
| align:start position:0% |
| must be zero |
| |
|
|
| align:start position:0% |
| must be zero |
| and the pi TX so Pi is the analog of P |
|
|
| align:start position:0% |
| and the pi TX so Pi is the analog of P |
| |
|
|
| align:start position:0% |
| and the pi TX so Pi is the analog of P |
| here |
|
|
| align:start position:0% |
| here |
| |
|
|
| align:start position:0% |
| here |
| yeah so those are operators |
|
|
| align:start position:0% |
| yeah so those are operators |
| |
|
|
| align:start position:0% |
| yeah so those are operators |
| TX Prime |
|
|
| align:start position:0% |
| TX Prime |
| |
|
|
| align:start position:0% |
| TX Prime |
| must be zero |
|
|
| align:start position:0% |
| must be zero |
| |
|
|
| align:start position:0% |
| must be zero |
| and then Phi |
|
|
| align:start position:0% |
| and then Phi |
| |
|
|
| align:start position:0% |
| and then Phi |
| TX |
|
|
| align:start position:0% |
| TX |
| |
|
|
| align:start position:0% |
| TX |
| which pi |
|
|
| align:start position:0% |
| which pi |
| |
|
|
| align:start position:0% |
| which pi |
| TX Prime |
|
|
| align:start position:0% |
| TX Prime |
| |
|
|
| align:start position:0% |
| TX Prime |
| should be something can only be zero but |
|
|
| align:start position:0% |
| should be something can only be zero but |
| |
|
|
| align:start position:0% |
| should be something can only be zero but |
| when X is not equal to X Prime |
|
|
| align:start position:0% |
| when X is not equal to X Prime |
| |
|
|
| align:start position:0% |
| when X is not equal to X Prime |
| can only be long zero when x equal to X |
|
|
| align:start position:0% |
| can only be long zero when x equal to X |
| |
|
|
| align:start position:0% |
| can only be long zero when x equal to X |
| Prime okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay as a generation of this |
|
|
| align:start position:0% |
| okay as a generation of this |
| |
|
|
| align:start position:0% |
| okay as a generation of this |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and uh so so you can now you can guess |
|
|
| align:start position:0% |
| and uh so so you can now you can guess |
| |
|
|
| align:start position:0% |
| and uh so so you can now you can guess |
| so what should this be |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so what |
|
|
| align:start position:0% |
| so what |
| |
|
|
| align:start position:0% |
| so what |
| yeah |
|
|
| align:start position:0% |
| yeah |
| |
|
|
| align:start position:0% |
| yeah |
| yeah I just used to be just directed huh |
|
|
| align:start position:0% |
| yeah I just used to be just directed huh |
| |
|
|
| align:start position:0% |
| yeah I just used to be just directed huh |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| but now you answer the question why has |
|
|
| align:start position:0% |
| but now you answer the question why has |
| |
|
|
| align:start position:0% |
| but now you answer the question why has |
| to be direct Delta or maybe y should not |
|
|
| align:start position:0% |
| to be direct Delta or maybe y should not |
| |
|
|
| align:start position:0% |
| to be direct Delta or maybe y should not |
| be |
|
|
| align:start position:0% |
| be |
| |
|
|
| align:start position:0% |
| be |
| say the derivatives of direct data okay |
|
|
| align:start position:0% |
| say the derivatives of direct data okay |
| |
|
|
| align:start position:0% |
| say the derivatives of direct data okay |
| say y should not be say 100th derivative |
|
|
| align:start position:0% |
| say y should not be say 100th derivative |
| |
|
|
| align:start position:0% |
| say y should not be say 100th derivative |
| of direct Delta |
|
|
| align:start position:0% |
| of direct Delta |
| |
|
|
| align:start position:0% |
| of direct Delta |
| and that question can be addressed just |
|
|
| align:start position:0% |
| and that question can be addressed just |
| |
|
|
| align:start position:0% |
| and that question can be addressed just |
| by a form dimensional analysis |
|
|
| align:start position:0% |
| by a form dimensional analysis |
| |
|
|
| align:start position:0% |
| by a form dimensional analysis |
| so so here we know it's somehow this |
|
|
| align:start position:0% |
| so so here we know it's somehow this |
| |
|
|
| align:start position:0% |
| so so here we know it's somehow this |
| must be really direct Delta and now |
|
|
| align:start position:0% |
| must be really direct Delta and now |
| |
|
|
| align:start position:0% |
| must be really direct Delta and now |
| let's decide |
|
|
| align:start position:0% |
| let's decide |
| |
|
|
| align:start position:0% |
| let's decide |
| so now you can do the dimension do a |
|
|
| align:start position:0% |
| so now you can do the dimension do a |
| |
|
|
| align:start position:0% |
| so now you can do the dimension do a |
| little bit dimensional analysis |
|
|
| align:start position:0% |
| little bit dimensional analysis |
| |
|
|
| align:start position:0% |
| little bit dimensional analysis |
| so so if you just write down the action |
|
|
| align:start position:0% |
| so so if you just write down the action |
| |
|
|
| align:start position:0% |
| so so if you just write down the action |
| yeah the action I have just erased sorry |
|
|
| align:start position:0% |
| yeah the action I have just erased sorry |
| |
|
|
| align:start position:0% |
| yeah the action I have just erased sorry |
| so so if you look back on the action let |
|
|
| align:start position:0% |
| so so if you look back on the action let |
| |
|
|
| align:start position:0% |
| so so if you look back on the action let |
| me just alternate the idea because I'm |
|
|
| align:start position:0% |
| me just alternate the idea because I'm |
| |
|
|
| align:start position:0% |
| me just alternate the idea because I'm |
| sure you can do dimensional analysis |
|
|
| align:start position:0% |
| sure you can do dimensional analysis |
| |
|
|
| align:start position:0% |
| sure you can do dimensional analysis |
| yourself |
|
|
| align:start position:0% |
| yourself |
| |
|
|
| align:start position:0% |
| yourself |
| so if you look at the action |
|
|
| align:start position:0% |
| so if you look at the action |
| |
|
|
| align:start position:0% |
| so if you look at the action |
| so the action is dimension is in the |
|
|
| align:start position:0% |
| so the action is dimension is in the |
| |
|
|
| align:start position:0% |
| so the action is dimension is in the |
| lateral unit we are using |
|
|
| align:start position:0% |
| lateral unit we are using |
| |
|
|
| align:start position:0% |
| lateral unit we are using |
| so so if uh so from that you can deduce |
|
|
| align:start position:0% |
| so so if uh so from that you can deduce |
| |
|
|
| align:start position:0% |
| so so if uh so from that you can deduce |
| the dimension of Phi |
|
|
| align:start position:0% |
| the dimension of Phi |
| |
|
|
| align:start position:0% |
| the dimension of Phi |
| should be 1 over l so one over the lens |
|
|
| align:start position:0% |
| should be 1 over l so one over the lens |
| |
|
|
| align:start position:0% |
| should be 1 over l so one over the lens |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| and from the fact that the pi |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| where is pi or maybe how also erase is |
|
|
| align:start position:0% |
| where is pi or maybe how also erase is |
| |
|
|
| align:start position:0% |
| where is pi or maybe how also erase is |
| equal to Five Dot |
|
|
| align:start position:0% |
| equal to Five Dot |
| |
|
|
| align:start position:0% |
| equal to Five Dot |
| means Pi should be Dimension 1 over L |
|
|
| align:start position:0% |
| means Pi should be Dimension 1 over L |
| |
|
|
| align:start position:0% |
| means Pi should be Dimension 1 over L |
| Square |
|
|
| align:start position:0% |
| Square |
| |
|
|
| align:start position:0% |
| Square |
| okay because you take the derivative one |
|
|
| align:start position:0% |
| okay because you take the derivative one |
| |
|
|
| align:start position:0% |
| okay because you take the derivative one |
| time and then there's a lot of factor of |
|
|
| align:start position:0% |
| time and then there's a lot of factor of |
| |
|
|
| align:start position:0% |
| time and then there's a lot of factor of |
| L |
|
|
| align:start position:0% |
| L |
| |
|
|
| align:start position:0% |
| L |
| then that means on the right hand side |
|
|
| align:start position:0% |
| then that means on the right hand side |
| |
|
|
| align:start position:0% |
| then that means on the right hand side |
| here |
|
|
| align:start position:0% |
| here |
| |
|
|
| align:start position:0% |
| here |
| must be something 1 over L to a cube |
|
|
| align:start position:0% |
| must be something 1 over L to a cube |
| |
|
|
| align:start position:0% |
| must be something 1 over L to a cube |
| okay because there's no other parameters |
|
|
| align:start position:0% |
| okay because there's no other parameters |
| |
|
|
| align:start position:0% |
| okay because there's no other parameters |
| here okay yeah because uh here there |
|
|
| align:start position:0% |
| here okay yeah because uh here there |
| |
|
|
| align:start position:0% |
| here okay yeah because uh here there |
| should be a I okay and if it is a |
|
|
| align:start position:0% |
| should be a I okay and if it is a |
| |
|
|
| align:start position:0% |
| should be a I okay and if it is a |
| dimension a one over Cube then can only |
|
|
| align:start position:0% |
| dimension a one over Cube then can only |
| |
|
|
| align:start position:0% |
| dimension a one over Cube then can only |
| be the data function not 100 derivative |
|
|
| align:start position:0% |
| be the data function not 100 derivative |
| |
|
|
| align:start position:0% |
| be the data function not 100 derivative |
| data function so so so so this thing |
|
|
| align:start position:0% |
| data function so so so so this thing |
| |
|
|
| align:start position:0% |
| data function so so so so this thing |
| should be just a data function okay |
|
|
| align:start position:0% |
| should be just a data function okay |
| |
|
|
| align:start position:0% |
| should be just a data function okay |
| okay so so um |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this you know the convention that |
|
|
| align:start position:0% |
| so this you know the convention that |
| |
|
|
| align:start position:0% |
| so this you know the convention that |
| there should be I |
|
|
| align:start position:0% |
| there should be I |
| |
|
|
| align:start position:0% |
| there should be I |
| and then it should be just the three |
|
|
| align:start position:0% |
| and then it should be just the three |
| |
|
|
| align:start position:0% |
| and then it should be just the three |
| data function |
|
|
| align:start position:0% |
| data function |
| |
|
|
| align:start position:0% |
| data function |
| and this indeed have the dimension one |
|
|
| align:start position:0% |
| and this indeed have the dimension one |
| |
|
|
| align:start position:0% |
| and this indeed have the dimension one |
| of uh L Cube okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| good |
|
|
| align:start position:0% |
| good |
| |
|
|
| align:start position:0% |
| good |
| so now you can just plug |
|
|
| align:start position:0% |
| so now you can just plug |
| |
|
|
| align:start position:0% |
| so now you can just plug |
| so you have the expression for X for Phi |
|
|
| align:start position:0% |
| so you have the expression for X for Phi |
| |
|
|
| align:start position:0% |
| so you have the expression for X for Phi |
| you take the time derivative of this |
|
|
| align:start position:0% |
| you take the time derivative of this |
| |
|
|
| align:start position:0% |
| you take the time derivative of this |
| you get the expression for for pi |
|
|
| align:start position:0% |
| you get the expression for for pi |
| |
|
|
| align:start position:0% |
| you get the expression for for pi |
| and now you can just plug them into here |
|
|
| align:start position:0% |
| and now you can just plug them into here |
| |
|
|
| align:start position:0% |
| and now you can just plug them into here |
| you can just plug them into here okay |
|
|
| align:start position:0% |
| you can just plug them into here okay |
| |
|
|
| align:start position:0% |
| you can just plug them into here okay |
| and then you can find the commutation |
|
|
| align:start position:0% |
| and then you can find the commutation |
| |
|
|
| align:start position:0% |
| and then you can find the commutation |
| relation between those a case |
|
|
| align:start position:0% |
| relation between those a case |
| |
|
|
| align:start position:0% |
| relation between those a case |
| okay and so this is a slightly TD |
|
|
| align:start position:0% |
| okay and so this is a slightly TD |
| |
|
|
| align:start position:0% |
| okay and so this is a slightly TD |
| calculation |
|
|
| align:start position:0% |
| calculation |
| |
|
|
| align:start position:0% |
| calculation |
| which is |
|
|
| align:start position:0% |
| which is |
| |
|
|
| align:start position:0% |
| which is |
| however a little bit fun |
|
|
| align:start position:0% |
| however a little bit fun |
| |
|
|
| align:start position:0% |
| however a little bit fun |
| which of course I will leave you to do |
|
|
| align:start position:0% |
| which of course I will leave you to do |
| |
|
|
| align:start position:0% |
| which of course I will leave you to do |
| so so if you just plug them in and then |
|
|
| align:start position:0% |
| so so if you just plug them in and then |
| |
|
|
| align:start position:0% |
| so so if you just plug them in and then |
| you can deduce |
|
|
| align:start position:0% |
| you can deduce |
| |
|
|
| align:start position:0% |
| you can deduce |
| at the following commutation relation |
|
|
| align:start position:0% |
| at the following commutation relation |
| |
|
|
| align:start position:0% |
| at the following commutation relation |
| between A's |
|
|
| align:start position:0% |
| between A's |
| |
|
|
| align:start position:0% |
| between A's |
| so so this is the I I think this is in P |
|
|
| align:start position:0% |
| so so this is the I I think this is in P |
| |
|
|
| align:start position:0% |
| so so this is the I I think this is in P |
| said two uh but I can still change my |
|
|
| align:start position:0% |
| said two uh but I can still change my |
| |
|
|
| align:start position:0% |
| said two uh but I can still change my |
| mind yeah I wanted to put in pizza two |
|
|
| align:start position:0% |
| mind yeah I wanted to put in pizza two |
| |
|
|
| align:start position:0% |
| mind yeah I wanted to put in pizza two |
| so you find the the commentator between |
|
|
| align:start position:0% |
| so you find the the commentator between |
| |
|
|
| align:start position:0% |
| so you find the the commentator between |
| a |
|
|
| align:start position:0% |
| a |
| |
|
|
| align:start position:0% |
| a |
| and the commentator between a dagger |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah so so now I will suppress the Hat |
|
|
| align:start position:0% |
| yeah so so now I will suppress the Hat |
| |
|
|
| align:start position:0% |
| yeah so so now I will suppress the Hat |
| okay because at the right height I think |
|
|
| align:start position:0% |
| okay because at the right height I think |
| |
|
|
| align:start position:0% |
| okay because at the right height I think |
| over and over I will be too tired |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so these are zero |
|
|
| align:start position:0% |
| so these are zero |
| |
|
|
| align:start position:0% |
| so these are zero |
| okay so the commutation relation between |
|
|
| align:start position:0% |
| okay so the commutation relation between |
| |
|
|
| align:start position:0% |
| okay so the commutation relation between |
| a and zero and a |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| you think |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| between a |
|
|
| align:start position:0% |
| between a |
| |
|
|
| align:start position:0% |
| between a |
| and a dagger |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so this gives you |
|
|
| align:start position:0% |
| so this gives you |
| |
|
|
| align:start position:0% |
| so this gives you |
| two pair Cube |
|
|
| align:start position:0% |
| two pair Cube |
| |
|
|
| align:start position:0% |
| two pair Cube |
| that the function k |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay so this is a three data function |
|
|
| align:start position:0% |
| okay so this is a three data function |
| |
|
|
| align:start position:0% |
| okay so this is a three data function |
| okay okay |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| so again this is a straightforward |
|
|
| align:start position:0% |
| so again this is a straightforward |
| |
|
|
| align:start position:0% |
| so again this is a straightforward |
| generation |
|
|
| align:start position:0% |
| generation |
| |
|
|
| align:start position:0% |
| generation |
| so if you have multiple harmonic |
|
|
| align:start position:0% |
| so if you have multiple harmonic |
| |
|
|
| align:start position:0% |
| so if you have multiple harmonic |
| oscillators |
|
|
| align:start position:0% |
| oscillators |
| |
|
|
| align:start position:0% |
| oscillators |
| so if you have considered the multiple |
|
|
| align:start position:0% |
| so if you have considered the multiple |
| |
|
|
| align:start position:0% |
| so if you have considered the multiple |
| harmonic oscillators before |
|
|
| align:start position:0% |
| harmonic oscillators before |
| |
|
|
| align:start position:0% |
| harmonic oscillators before |
| and then the a between the different |
|
|
| align:start position:0% |
| and then the a between the different |
| |
|
|
| align:start position:0% |
| and then the a between the different |
| harmonicles because K Prime are just |
|
|
| align:start position:0% |
| harmonicles because K Prime are just |
| |
|
|
| align:start position:0% |
| harmonicles because K Prime are just |
| here just corresponding to |
|
|
| align:start position:0% |
| here just corresponding to |
| |
|
|
| align:start position:0% |
| here just corresponding to |
| essentially you have yeah here it just |
|
|
| align:start position:0% |
| essentially you have yeah here it just |
| |
|
|
| align:start position:0% |
| essentially you have yeah here it just |
| is essentially you have infinite number |
|
|
| align:start position:0% |
| is essentially you have infinite number |
| |
|
|
| align:start position:0% |
| is essentially you have infinite number |
| of harmonic oscillators and each one of |
|
|
| align:start position:0% |
| of harmonic oscillators and each one of |
| |
|
|
| align:start position:0% |
| of harmonic oscillators and each one of |
| the enabled by a k okay so this is just |
|
|
| align:start position:0% |
| the enabled by a k okay so this is just |
| |
|
|
| align:start position:0% |
| the enabled by a k okay so this is just |
| like essentially we find yeah let me |
|
|
| align:start position:0% |
| like essentially we find yeah let me |
| |
|
|
| align:start position:0% |
| like essentially we find yeah let me |
| just write it here |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| [Applause] |
|
|
| align:start position:0% |
| [Applause] |
| |
|
|
| align:start position:0% |
| [Applause] |
| so from those commutation relations |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| we conclude |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| conclude |
|
|
| align:start position:0% |
| conclude |
| |
|
|
| align:start position:0% |
| conclude |
| this series will be quantize after we |
|
|
| align:start position:0% |
| this series will be quantize after we |
| |
|
|
| align:start position:0% |
| this series will be quantize after we |
| quantize it |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| become an infinite number |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| independent harmonic oscillators |
|
|
| align:start position:0% |
| independent harmonic oscillators |
| |
|
|
| align:start position:0% |
| independent harmonic oscillators |
| decoupled harmonical status |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| harmonic oscillators labeled by |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| continuous parameter k |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| the K is yeah |
|
|
| align:start position:0% |
| the K is yeah |
| |
|
|
| align:start position:0% |
| the K is yeah |
| okay is the wave Lumber |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| okay |
|
|
| align:start position:0% |
| okay |
| |
|
|
| align:start position:0% |
| okay |
| so for each k |
|
|
| align:start position:0% |
| so for each k |
| |
|
|
| align:start position:0% |
| so for each k |
| there is a |
|
|
| align:start position:0% |
| there is a |
| |
|
|
| align:start position:0% |
| there is a |
| an A and so between between a a |
|
|
| align:start position:0% |
| an A and so between between a a |
| |
|
|
| align:start position:0% |
| an A and so between between a a |
| themselves it's zero between a dag it's |
|
|
| align:start position:0% |
| themselves it's zero between a dag it's |
| |
|
|
| align:start position:0% |
| themselves it's zero between a dag it's |
| zero but a a dagger |
|
|
| align:start position:0% |
| zero but a a dagger |
| |
|
|
| align:start position:0% |
| zero but a a dagger |
| they not equal to zero |
|
|
| align:start position:0% |
| they not equal to zero |
| |
|
|
| align:start position:0% |
| they not equal to zero |
| and uh so this is again the continuing |
|
|
| align:start position:0% |
| and uh so this is again the continuing |
| |
|
|
| align:start position:0% |
| and uh so this is again the continuing |
| generation of one okay this is a |
|
|
| align:start position:0% |
| generation of one okay this is a |
| |
|
|
| align:start position:0% |
| generation of one okay this is a |
| Continuum generation of one uh because |
|
|
| align:start position:0% |
| Continuum generation of one uh because |
| |
|
|
| align:start position:0% |
| Continuum generation of one uh because |
| you have a continuous variables yes |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah then you cannot say for sure |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah yeah yeah |
|
|
| align:start position:0% |
| yeah yeah yeah |
| |
|
|
| align:start position:0% |
| yeah yeah yeah |
| no but but you see the conservation |
|
|
| align:start position:0% |
| no but but you see the conservation |
| |
|
|
| align:start position:0% |
| no but but you see the conservation |
| condition |
|
|
| align:start position:0% |
| condition |
| |
|
|
| align:start position:0% |
| condition |
| is in quantum mechanics quantum |
|
|
| align:start position:0% |
| is in quantum mechanics quantum |
| |
|
|
| align:start position:0% |
| is in quantum mechanics quantum |
| mechanics T and X are not on the same |
|
|
| align:start position:0% |
| mechanics T and X are not on the same |
| |
|
|
| align:start position:0% |
| mechanics T and X are not on the same |
| protein you can require your action to |
|
|
| align:start position:0% |
| protein you can require your action to |
| |
|
|
| align:start position:0% |
| protein you can require your action to |
| be x and t to be on the same 14 once you |
|
|
| align:start position:0% |
| be x and t to be on the same 14 once you |
| |
|
|
| align:start position:0% |
| be x and t to be on the same 14 once you |
| start the quantize your theory and then |
|
|
| align:start position:0% |
| start the quantize your theory and then |
| |
|
|
| align:start position:0% |
| start the quantize your theory and then |
| T will have a pronounced row |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| because I wanted to so I couldn't write |
|
|
| align:start position:0% |
| because I wanted to so I couldn't write |
| |
|
|
| align:start position:0% |
| because I wanted to so I couldn't write |
| that like the commutation relation as |
|
|
| align:start position:0% |
| that like the commutation relation as |
| |
|
|
| align:start position:0% |
| that like the commutation relation as |
| functions in the four Vector x no Delta |
|
|
| align:start position:0% |
| functions in the four Vector x no Delta |
| |
|
|
| align:start position:0% |
| functions in the four Vector x no Delta |
| x no no no no no no no the canonical |
|
|
| align:start position:0% |
| x no no no no no no no the canonical |
| |
|
|
| align:start position:0% |
| x no no no no no no no the canonical |
| computation variation have to be imposed |
|
|
| align:start position:0% |
| computation variation have to be imposed |
| |
|
|
| align:start position:0% |
| computation variation have to be imposed |
| at equal time |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| other questions |
|
|
| align:start position:0% |
| other questions |
| |
|
|
| align:start position:0% |
| other questions |
| good |
|
|
| align:start position:0% |
| good |
| |
|
|
| align:start position:0% |
| good |
| so um |
|
|
| align:start position:0% |
| |
| |
|
|
| align:start position:0% |
| |
| yeah so so so essentially we just get |
|
|
| align:start position:0% |
| yeah so so so essentially we just get |
| |
|
|
| align:start position:0% |
| yeah so so so essentially we just get |
| and now it's just trivial okay so so you |
|
|
| align:start position:0% |
| and now it's just trivial okay so so you |
| |
|
|
| align:start position:0% |
| and now it's just trivial okay so so you |
| can just build up your hillbow space |
|
|
| align:start position:0% |
| can just build up your hillbow space |
| |
|
|
| align:start position:0% |
| can just build up your hillbow space |
| essentially you just have infinite |
|
|
| align:start position:0% |
| essentially you just have infinite |
| |
|
|
| align:start position:0% |
| essentially you just have infinite |
| number harmonical signatures |
|
|
| align:start position:0% |
| number harmonical signatures |
| |
|
|
| align:start position:0% |
| number harmonical signatures |
| okay just have infinite number of |
|
|
| align:start position:0% |
| okay just have infinite number of |
| |
|
|
| align:start position:0% |
| okay just have infinite number of |
| harmonic concentrators |
|
|
| align:start position:0% |
| harmonic concentrators |
| |
|
|
| align:start position:0% |
| harmonic concentrators |
| and there's no surprise you get the |
|
|
| align:start position:0% |
| and there's no surprise you get the |
| |
|
|
| align:start position:0% |
| and there's no surprise you get the |
| infinite number of harmonic oscillators |
|
|
| align:start position:0% |
| infinite number of harmonic oscillators |
| |
|
|
| align:start position:0% |
| infinite number of harmonic oscillators |
| because we mentioned that this field |
|
|
| align:start position:0% |
| because we mentioned that this field |
| |
|
|
| align:start position:0% |
| because we mentioned that this field |
| Theory |
|
|
| align:start position:0% |
| Theory |
| |
|
|
| align:start position:0% |
| Theory |
| can be actually written as a Continuum |
|
|
| align:start position:0% |
| can be actually written as a Continuum |
| |
|
|
| align:start position:0% |
| can be actually written as a Continuum |
| limit |
|
|
| align:start position:0% |
| limit |
| |
|
|
| align:start position:0% |
| limit |
| of these particles on the chain |
|
|
| align:start position:0% |
| of these particles on the chain |
| |
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| align:start position:0% |
| of these particles on the chain |
| which in these eight or three examples |
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| align:start position:0% |
| which in these eight or three examples |
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| align:start position:0% |
| which in these eight or three examples |
| you know that is a harmonic oscillator |
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| align:start position:0% |
| you know that is a harmonic oscillator |
| |
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| align:start position:0% |
| you know that is a harmonic oscillator |
| once you uh find the lower mode they're |
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| align:start position:0% |
| once you uh find the lower mode they're |
| |
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| align:start position:0% |
| once you uh find the lower mode they're |
| all just from a bunch of harmonicles |
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| align:start position:0% |
| all just from a bunch of harmonicles |
| |
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| align:start position:0% |
| all just from a bunch of harmonicles |
| features and this is just a |
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| align:start position:0% |
| features and this is just a |
| |
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| align:start position:0% |
| features and this is just a |
| three-dimensional version of that okay |
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| align:start position:0% |
| three-dimensional version of that okay |
| |
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| align:start position:0% |
| three-dimensional version of that okay |
| and now we will |
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| align:start position:0% |
| and now we will |
| |
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| align:start position:0% |
| and now we will |
| yeah today we are running all the uh |
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| align:start position:0% |
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| align:start position:0% |
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| all the time so next time we will see |
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| align:start position:0% |
| all the time so next time we will see |
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| align:start position:0% |
| all the time so next time we will see |
| that each excitations of the harmonic |
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| align:start position:0% |
| that each excitations of the harmonic |
| |
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| align:start position:0% |
| that each excitations of the harmonic |
| oscillator |
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| align:start position:0% |
| oscillator |
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| align:start position:0% |
| oscillator |
| can be interpreted as a space-time |
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| align:start position:0% |
| can be interpreted as a space-time |
| |
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| align:start position:0% |
| can be interpreted as a space-time |
| particle okay so that's the crossing of |
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| align:start position:0% |
| particle okay so that's the crossing of |
| |
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| align:start position:0% |
| particle okay so that's the crossing of |
| it and now you have this infinite number |
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| align:start position:0% |
| it and now you have this infinite number |
| |
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| align:start position:0% |
| it and now you have this infinite number |
| of harmonic oscillator and now you can |
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| align:start position:0% |
| of harmonic oscillator and now you can |
| |
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| align:start position:0% |
| of harmonic oscillator and now you can |
| act and now you can Define the vacuum |
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| align:start position:0% |
| act and now you can Define the vacuum |
| |
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| align:start position:0% |
| act and now you can Define the vacuum |
| and then actually this creation |
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| align:start position:0% |
| and then actually this creation |
| |
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| align:start position:0% |
| and then actually this creation |
| operators on the vacuum and now you find |
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| align:start position:0% |
| operators on the vacuum and now you find |
| |
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| align:start position:0% |
| operators on the vacuum and now you find |
| each excitation actually equals one into |
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| align:start position:0% |
| each excitation actually equals one into |
| |
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| align:start position:0% |
| each excitation actually equals one into |
| a particle |
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| align:start position:0% |
| a particle |
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| align:start position:0% |
| a particle |
| and and has the uh a corresponding to |
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| align:start position:0% |
| and and has the uh a corresponding to |
| |
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| align:start position:0% |
| and and has the uh a corresponding to |
| relativistic particle and that's how you |
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| align:start position:0% |
| relativistic particle and that's how you |
| |
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| align:start position:0% |
| relativistic particle and that's how you |
| can have actually arbitrary lumbar |
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| align:start position:0% |
| can have actually arbitrary lumbar |
| |
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| align:start position:0% |
| can have actually arbitrary lumbar |
| particles |
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| align:start position:0% |
| particles |
| |
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| align:start position:0% |
| particles |
| in this series and uh yeah because you |
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| align:start position:0% |
| in this series and uh yeah because you |
| |
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| align:start position:0% |
| in this series and uh yeah because you |
| can excite as many times as you want |
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| align:start position:0% |
| can excite as many times as you want |
| |
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| align:start position:0% |
| can excite as many times as you want |
| okay each excitation is a particle |
|
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| align:start position:0% |
| okay each excitation is a particle |
| |
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| align:start position:0% |
| okay each excitation is a particle |
| good good okay so so I think it's a good |
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| align:start position:0% |
| good good okay so so I think it's a good |
| |
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| align:start position:0% |
| good good okay so so I think it's a good |
| time yeah we are two minutes I think |
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| align:start position:0% |
| time yeah we are two minutes I think |
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| align:start position:0% |
| time yeah we are two minutes I think |
| early but I think it's a very good place |
|
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| align:start position:0% |
| early but I think it's a very good place |
| |
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| align:start position:0% |
| early but I think it's a very good place |
| to to break |
