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US_History | Taxes_Smuggling_Prelude_to_Revolution_Crash_Course_US_History_6.txt | Hi, I'm John Green, this is Crash Course: US History, and today we begin discussing the American Revolution. So two things to keep in mind here: One, the American Revolution and the American War for Independence are not the same thing. And two, while I know this will upset some of you, the American Revolution was not r... |
US_History | The_Market_Revolution_Crash_Course_US_History_12.txt | Hi, I'm John Green, this is Crash Course US History and today we return to one of my favorite subjects: economics. Mr. Green, Mr. Green, I don't wanna brag, but economics is actually my best subject. Like, I got the bronze medal at the state academic decathlon tournament...among C students. Yeah, I remember, Me from th... |
US_History | American_Imperialism_Crash_Course_US_History_28.txt | Episode 28: American Imperialism Hi, I’m John Green, this is CrashCourse U.S. History and today we’re gonna talk about a subject near and dear to my white, male heart: imperialism. So, here at CrashCourse we occasionally try to point out that the U.S., much as we hate to admit it, is actually part of a larger world. Mr... |
US_History | George_HW_Bush_and_the_End_of_the_Cold_War_Crash_Course_US_History_44.txt | Hi, I’m John Green, this is CrashCourse U.S. history and we’ve finally done it we have reached the moment where we get to talk about the presidency of George HW Bush. The 2nd most important man named George Bush ever to be President of the United States. A man so fascinating that we did not give him a face. Mr. Green, ... |
US_History | Who_Won_the_American_Revolution_Crash_Course_US_History_7.txt | Hi, I’m John Green; this is Crash Course U.S. History. There are two kinds of revolutions: those where things DO change and those where things don’t change. Like, not to get all Crash Course Mathematics on you or anything, but a Revolution is a 360 degree turn, which leaves you back where you started. That’s what happe... |
US_History | The_Election_of_1860_the_Road_to_Disunion_Crash_Course_US_History_18.txt | Hi I’m John Green; this is Crash Course US History and today we discuss one of the most confusing questions in American history: What caused the Civil War? Just kidding it’s not a confusing question at all: Slavery caused the Civil War. Mr. Green, Mr. Green, but what about, like, states rights and nationalism, economi... |
US_History | The_Civil_War_Part_I_Crash_Course_US_History_20.txt | Hi I’m John Green this is Crash Course US History and today we come at last to the Civil War, the conflict that in many ways created a nation. So here’s what you won’t be getting today. We will not be describing battles and tactics. If that’s your bag, might I suggest Ken Burns or if you prefer books, like 1000 authors... |
US_History | History_of_the_4th_of_July_Crash_Course_US_History_Special.txt | Hi, I'm John Green, this is Crash Course US History, and today we're gonna talk about July 4th, which in the United States is known as Independence Day. This is the day that Americans celebrate our Independence from Great Britain by doing what we do best: blowing stuff up, offering significant discounts on mattresses... |
US_History | Progressive_Presidents_Crash_Course_US_History_29.txt | Hi, I’m John Green, this is CrashCourse U.S. History and today we’re going to finish our discussion of Progressivism, and indulge in a bit of “great man” history. Mr. Green, Mr. Green! Great man history, huh? Well I was born on a sunny, summer morning in 197-- Yeah you’re not great, Me from the Past. Also, you’re a boy... |
US_History | Growth_Cities_and_Immigration_Crash_Course_US_History_25.txt | Hi, I’m John Green, this is CrashCourse U.S. History and today we’re going to continue our extensive look at American capitalism. Mr. Green, Mr. Green, I’m sorry are you saying that I grow up to be a tool of the bourgeoisie… Oh not just a tool of the bourgeoise, Me from the Past, but a card-carrying member of it. I mea... |
US_History | Ford_Carter_and_the_Economic_Malaise_Crash_Course_US_History_42.txt | Hi, I’m John Green, this is Crash Course U.S. History and today we are going to talk about one of the most important periods in American history, the mid-to-late 1970s. Stan why is there nothing on the chalkboard? We can’t find a picture of Gerald Ford somewhere around here? Don’t worry Crash Course fans we got one. Th... |
US_History | 에이지_오브_잭슨_크래쉬_코스_미국_역사_14.txt | Hi I’m John Green. This is Crash Course U.S. History and today, after last week’s bummer on slavery, we turn to a happier topic: the rise of democratization in the U.S. This was also known as the Age of Jackson, no Stan, not that Jackson. No, no, Stan, come’on seriously. No not, no, no, no, no, no, no, no, no, no, no, ... |
US_History | The_Constitution_the_Articles_and_Federalism_Crash_Course_US_History_8.txt | Hi, I’m John Green, this is Crash Course U.S. History, and today we’re going to talk about the United States Constitution. And, in doing so, we’re going to explore how the American style of government became the envy of the entire world, so much so that everyone else copied us. What’s that, Stan? We’re not gonna talk a... |
US_History | The_Cold_War_Crash_Course_US_History_37.txt | Hi I’m John Green; this is Crash Course U.S. history and today we’re gonna talk about the Cold War. The Cold War is called “Cold” because it supposedly never heated up into actual armed conflict, which means, you know, that it wasn’t a war. Mr. Green, Mr. Green, but if the War on Christmas is a war and the War on Drugs... |
US_History | When_is_Thanksgiving_Colonizing_America_Crash_Course_US_History_2.txt | Hi I’m John Green, this is Crash Course US History, and today we're going to tell the story of how a group of plucky English people struck a blow for religious freedom, and founded the greatest, freest and fattest nation the world has ever seen. [Libertage] These Brits entered a barren land containing no people, and qu... |
US_History | Westward_Expansion_Crash_Course_US_History_24.txt | Hi, I’m John Green, this is Crash Course U.S. History and today we leave behind the world of industry and corporations to talk about the Wild Wild West. Spoiler Alert: You You have died of dysentery. And in the process, we’re going to explore how all of us, even those of us who are vegan or eat sustainably-produced foo... |
US_History | The_Civil_War_Part_2_Crash_Course_US_History_21.txt | Hi, I’m John Green; this is Crash Course U.S. History and today we return to...wait, what are we talking about today, Stan? Ah, the Civil War! I can tell because Lincoln’s here. But this week we’re not gonna talk about casualty counts or battles or its generals with their heroic and probably fictional dying declaration... |
US_History | Battles_of_the_Civil_War_Crash_Course_US_History_19.txt | Hi, I’m John Green and this is Crash Course US History. Starting next week, we’re going to be talking about the Civil War. As you may have noticed, Crash Course doesn’t usually focus on military history, because we’re more interested in causes and effects and that kind of stuff, but because some in our audience are lik... |
US_History | The_Industrial_Economy_Crash_Course_US_History_23.txt | Episode 23: The Rise of the Industrial Economy Hi I’m John Green this is Crash Course U.S. History and today we’re going to discuss economics and how a generation of- Mr. Green, Mr. Green, is this going to be one of those boring ones no wars or generals who had cool last words or anything? Alright, Me From The Past, I ... |
US_History | America_in_World_War_I_Crash_Course_US_History_30.txt | Episode 30: America and World War I Hi I’m John Green, this is Crash Course U.S. history and today we’re finally going to make the military history buffs happy. That’s right, today we’re going to talk about how the United States with its superior technology, innovative tactics and remarkable generalship turned the tide... |
US_History | The_Natives_and_the_English_Crash_Course_US_History_3.txt | Hi I'm John Green, this is Crash Course US History and today we're going to talk about one of the worst relationships in American history. No Thought Bubble, not my college girlfriend and me. Mr. Green, Mr. Green! Your relationship with your high school girlfriend? Oh Me From The Past, you and I both know that I didn't... |
US_History | The_Black_Legend_Native_Americans_and_Spaniards_Crash_Course_US_History_1.txt | Hi, I’m John Green and this is Crash Course U.S. History— No, Stan, that’s not gonna work actually. I mean, we’re talking about the 16th century today, when this was neither United nor States. By the way, this globe reflects the fact that I believe that Alaskan statehood is illegitimate. In fact, we’re gonna call this ... |
US_History | The_Quakers_the_Dutch_and_the_Ladies_Crash_Course_US_History_4.txt | Hi, I'm John Green, this is Crash Course US History, and today we're going to cram 150 years of American history into one video. Why? Well, many American history classes don't cover the colonial period at all, because most major American history tests have, like, one question about it. Mr. Green, Mr. Green, so this isn... |
US_History | Slavery_Crash_Course_US_History_13.txt | Hi, I'm John Green, this is Crash Course U.S. History, and today, we're going to talk about slavery, which is not funny. Yeah, so we put a lei on the eagle to try and cheer you up, but let's face it, this is going to be depressing. With slavery, every time you think, like, "Aw, it couldn't have been that bad," it turns... |
US_History | The_Great_Depression_Crash_Course_US_History_33.txt | Hi, I'm John Green, this is Crash Course U.S. history and Herbert Hoover's here, which is never a good sign. Today we're gonna return to two of my favorite topics: economics and inaccurate naming conventions. That's right, we're gonna be talking about the Great Depression, which was only great if you enjoy, like, being... |
US_History | The_Roaring_20s_Crash_Course_US_History_32.txt | Hi, I'm John Green, this is Crash Course US History, and today we're gonna learn about one of the best eras ever: the 1920s. The 20s gave us jazz, movies, radio, making out in cars, illegal liquor, and the 20s also gave us prosperity--although not for everybody-- and gangsters, and a consumer culture based on credit, a... |
US_History | 19th_Century_Reforms_Crash_Course_US_History_15.txt | Hi I’m John Green. This is Crash Course U.S. History and today we finally get to talk about sex. Also some other things. Today we’re gonna discuss religious and moral reform movements in 19th century America, but I promise there will be some sex. Mr. Green, Mr. Green. Is it gonna be about real sex or is it gonna be abl... |
US_History | The_Cold_War_in_Asia_Crash_Course_US_History_38.txt | Hi, I’m John Green, this is CrashCourse US History, and today we’re going to talk about the Cold War again. Really less about the “cold,” more about the “war.” As usual, we’re not going to focus so much on the generals and the tactics, but instead on why the wars were fought and what it all meant. And today we get to v... |
US_History | Women_in_the_19th_Century_Crash_Course_US_History_16.txt | Hi, I’m John Green; this is Crash Course U.S. History and today we’re going to talk about wonder women. Mr. Green, Mr. Green, finally we get to the history of the United States as seen through the lens of Marvel comic superheroes. Oh, Me from the Past, you sniveling little idiot. Wonder Woman is from the DC Universe. A... |
US_History | Terrorism_War_and_Bush_43_Crash_Course_US_History_46.txt | Hi, I’m John Green, this is CrashCourse U.S. history and today we’ve done it! WE’VE FINALLY REACHED THE 21st CENTURY! Today, we boldly go where no history course has gone before, because your teacher ran out of time and never made it to the present. Also, if you’re preparing for the AP test it’s unlikely that today’s v... |
MIT_804_Quantum_Physics_I_Spring_2016 | Photons_and_the_loss_of_determinism.txt | PROFESSOR: Determinism. And it all begins with photons. Einstein reluctantly came up with the idea that light was made of quanta-- quanta of light called photons. Now when you think of photons, we think of a particle. So everybody knew that light was a wave. Maxwell's equations had been so successful. Nevertheless, pho... |
MIT_804_Quantum_Physics_I_Spring_2016 | Widths_and_uncertainties.txt | PROFESSOR: So we go back to the integral. We think of k. We'll write it as k naught plus k tilde. And then we have psi of x0 equal 1 over square root of 2pi e to the ik naught x-- that part goes out-- integral dk tilde phi of k naught plus k tilde e to the ik tilde x dk. OK. So we're doing this integral. And now we're ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Phase_shift_for_a_potential_well.txt | PROFESSOR: So let's do an example where we can calculate from the beginning to the end everything. Now, you have to get accustomed to the idea of even though you can calculate everything, your formulas that you get sometimes are a little big. And you look at them and they may not tell you too much unless you plot them ... |
MIT_804_Quantum_Physics_I_Spring_2016 | The_nature_of_superposition_MachZehnder_interferometer.txt | PROFESSOR: Superposition is very unusual and very interesting. Now we've said about superposition that in classical physics, when we talk about superposition we have electric fields, and you add the electric fields, and the total electric field is the sum of electric fields, and it's an electric field. And there's noth... |
MIT_804_Quantum_Physics_I_Spring_2016 | Commuting_observables_for_angular_momentum.txt | BARTON ZWIEBACH: We want to understand now our observables. So we said these are observables, so can we observe them? Can we have a state in which we say, what is the value of Lx, the value of Ly, and the value of Lz. Well, a little caution is necessary because we have states and we have position and momentum operator ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Translation_operator_Central_potentials.txt | PROFESSOR: The fact is that angular momentum is an observable, and as such it deserves attention. There is an active way of thinking of observables, and we have not developed it that much in this course. But for example, with a momentum operator you've learned that the momentum operator can give you the differential op... |
MIT_804_Quantum_Physics_I_Spring_2016 | Uncertainty_and_eigenstates.txt | PROFESSOR: This definition in which the uncertainty of the permission operator Q in the state psi. It's always important to have a state associated with measuring the uncertainty. Because the uncertainty will be different in different states. So the state should always be there. Sometimes we write it, sometimes we get ... |
MIT_804_Quantum_Physics_I_Spring_2016 | The_wave_for_a_free_particle.txt | PROFESSOR: So what are we trying to do? We're going to try to write a matter wave. We have a particle with energy e and momentum p. e is equal to h bar omega. So you can get the omega of the wave. And p is equal to h bar k. You can get the k of the wave. So de Broglie has told you that's the way to do it. That's the p ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Delta_function_potential_I_Preliminaries.txt | PROFESSOR: Delta function potential. So it's still a one-dimensional potential-- potential is a function of x. We'll write it this way-- minus alpha delta of x, where alpha is positive. So this is a delta function in a negative direction. So if you want to draw the potential-- there's no way to draw really nicely a del... |
MIT_804_Quantum_Physics_I_Spring_2016 | Recursion_relation_for_the_solution.txt | PROFESSOR: This was a differential equation for the energy eigenstates phi. Supposed to be normalizable functions. We looked at this equation and decided we would first clean out the constant. We did that by replacing x by a unit-free coordinate u. For that we needed a constant that carries units of length, and that co... |
MIT_804_Quantum_Physics_I_Spring_2016 | Motion_of_a_wavepacket.txt | PROFESSOR: Let me demonstrate now with plain doing the integral that, really, the shape of this wave is moving with that velocity. So in order to do that, I basically have to do the integral. And of course, if it's a general integral, I cannot do it. So I have to figure out enough about the integral. So here it is. We ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Parseval_identity.txt | PROFESSOR: What we want to understand now is really about momentum space. So we can ask the following question-- what happens to the normalization condition that we have for the wave function when we think in momentum variables? So yes, I will do this first. So let's think of integral dx psi of x star psi of x. Well, t... |
MIT_804_Quantum_Physics_I_Spring_2016 | Waves_on_the_finite_square_well.txt | PROFESSOR: Today's lecture continues the thing we're doing with scattering states. We send in a scattering state. That is an energy eigenstate that cannot be normalized into a step barrier. And we looked at what could happen. And we saw all kinds of interesting things happening. There was reflection and transmission wh... |
MIT_804_Quantum_Physics_I_Spring_2016 | Entanglement.txt | BARTON ZWIEBACH: Let's talk now about entanglement. So we talk about entanglement when we have two non-interacting particles. You don't need a strong interaction between particles to produce entanglement, the particles can be totally non-interacting. Suppose particle 1 can be in any of these states-- u 1, u 2. Let's as... |
MIT_804_Quantum_Physics_I_Spring_2016 | Potentials_that_satisfy_Vx_Vx.txt | PROFESSOR: Now we prove the other thing that we used in order to solve the square well. So this is property number 3. It's so important that I think I should do it here. If a potential is even-- here comes again the careful statement-- the energy eigenstates can be chosen to be even or odd under x goes to minus x. So t... |
MIT_804_Quantum_Physics_I_Spring_2016 | Particle_on_the_forbidden_region.txt | PROFESSOR: We have found in this solution with some energy like this, that there's a decaying exponential over this side. And the question is often asked, well what happens if you tried to measure the particle in the forbidden region? Must be a problem. If you find the particle in the forbidden region, it has energy E ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Expectation_values_on_stationary_states.txt | PROFESSOR: How about the expectation value of the Hamiltonian in a stationary state? You would imagine, somehow it has to do with energy ion states and energy. So let's see what happens. The expectation value of the Hamiltonian on this stationary state. That would be integral dx stationary state Hamiltonian stationary ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Angular_momentum_operators_and_their_algebra.txt | PROFESSOR: So angular momentum, we need to deal with angular momentum, and the inspiration for it is classical. We have L is r cross p. Classically. So let's try to just use that information and write the various operators. And in fact, we're lucky in this case. The operators that we would write inspired by the classic... |
MIT_804_Quantum_Physics_I_Spring_2016 | Interpretation_of_the_wavefunction.txt | PROFESSOR: interpretation of the wave function. --pretation-- the wave function. So you should look at what the inventor said. So what did Schrodinger say? Schrodinger thought that psi represents particles that disintegrate. You have a wave function. And the wave function is spread all over space, so the particle has d... |
MIT_804_Quantum_Physics_I_Spring_2016 | Harmonic_oscillator_Differential_equation.txt | PROFESSOR: Simple harmonic oscillator. So what is there about this simple harmonic oscillator? Well, it's a classical system that you understand perfectly well. An oscillator, a spring with a mass oscillates and has an energy, which is the kinetic energy plus the potential energy, and that's p squared over 2m plus 1/2 ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Threedimensional_Fourier_transforms.txt | PROFESSOR: We got here finally in terms of position and in terms of momentum. So this was not an accident that it worked for position and wave number. It works with position and for momentum. And remember, this phi of p now has interpretation of the weight that is associated with a plane wave of momentum p, and you're ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Galilean_transformation_of_ordinary_waves.txt | BARTON ZWIEBACH: Do normal wave analysis to demonstrate that indeed these things should not quite happen. So for that, so ordinary waves and Galilean transformations. So when you have a wave, as you've probably have seen many times before, the key object in the wave is something called the phaze of the wave. Phaze, the... |
MIT_804_Quantum_Physics_I_Spring_2016 | Associated_Legendre_functions_and_spherical_harmonics.txt | PROFESSOR: We're talking about angular momentum. We've motivated angular momentum as a set of operators that provided observables, things we can measure. Therefore, they are important. But they're particularly important for systems in which you have central potentials. Potentials that depend just on the magnitude of th... |
MIT_804_Quantum_Physics_I_Spring_2016 | Scattered_wave_and_phase_shift.txt | PROFESSOR: Let's look at the magnitude squared of those waves that we've already defined here. We have two solutions, one for no potential and one for a real potential. Both are for some finite range potential. We have phi of x squared is equal to sine squared of kx. And psi of x squared is equal to sine squared of kx ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Quantization_of_the_energy.txt | PROFESSOR: We have to ask what happens here? This series for h of u doesn't seem to stop. You go a 0, a 2, a 4. Well, it could go on forever. And what would happen if it goes on forever? So if it goes on forever, let's calculate what this aj plus 2 over aj as j goes to infinity. Let's see how the coefficients vary as y... |
MIT_804_Quantum_Physics_I_Spring_2016 | Halfwidth_and_time_delay.txt | PROFESSOR: What is gamma? Well, it seems to have a beta in there, alpha. It's a little unclear what gamma means, so let's try to make sense of it. Gamma has units of energy, because this term has units of energy, and therefore the other term must have units of energy as well. So gamma has units of energy. And from a un... |
MIT_804_Quantum_Physics_I_Spring_2016 | Finite_square_well_Setting_up_the_problem.txt | BARTON ZWIEBACH: Finite square well. So this brings us also to a little common aside. So far, we could find every solution. Now we're going to write the equations for the finite square well, and we're not going to be able to find the solution. But we're going to understand the solution. So you're going to enjoy a littl... |
MIT_804_Quantum_Physics_I_Spring_2016 | Is_probability_conserved_Hermiticity_of_the_Hamiltonian.txt | PROFESSOR: Let's do a work check. So main check. If integral psi star x t0, psi x t0 dx is equal to 1 at t equal to t0, as we say there, then it must hold for later times, t greater than t0. This is what we want to check, or verify, or prove. Now, to do it, we're going to take our time. So it's not going to happen in f... |
MIT_804_Quantum_Physics_I_Spring_2016 | Nondegeneracy_of_bound_states_in_1D_Real_solutions.txt | PROFESSOR: You first are facing the calculation of the energy eigenstate with some arbitrary potential. You probably want to know some of the key features of the wave functions you're going to calculate. So in fact, all of today's lecture is going to be devoted to this intuitive, qualitative insights into the nature of... |
MIT_804_Quantum_Physics_I_Spring_2016 | Levinsons_theorem_part_2.txt | PROFESSOR: We have two equations now relating this number of states. And now you can say, oh, OK, so I look at the k line. And I look at the little piece of the k and say, oh, how many states were there with 0 potential, some number, first blackboard. How many states are there now with some potential, some other number... |
MIT_804_Quantum_Physics_I_Spring_2016 | Modelling_a_resonance.txt | PROFESSOR: That's how it looks, a resonance. You can see it basically in the phase shift. And great increase of the phase of almost minutely pi over a very small change of energy. And it should [INAUDIBLE] with a very big [INAUDIBLE].. So this is how it looks. And I want to now proceed, after if there are some question... |
MIT_804_Quantum_Physics_I_Spring_2016 | Three_dimensional_current_and_conservation.txt | Three-dimensional case. Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians-- it's not complicated-- so that you will derive with the current is. And the current must be a very similar formula as this one. And indeed, I'll just write it here. The current is h bar o... |
MIT_804_Quantum_Physics_I_Spring_2016 | Node_Theorem.txt | BARTON ZWIEBACH: The next thing I want to talk about for a few minutes is about the node theorem. Theorem. And it's something we've seen before. We've heard that if you have a one-dimensional potential and you have bound states, the ground state has no nodes. The first excited state has 1 node. Second, 2, 3, 4. All I w... |
MIT_804_Quantum_Physics_I_Spring_2016 | MachZehnder_interferometers_and_beam_splitters.txt | PROFESSOR: Mach-Zehnder-- interferometers. And we have a beam splitter. And the beam coming in, it splits into 2. A mirror-- another mirror. The beams are recombined into another beam splitter. And then, 2 beams come out. One to a detector d0-- and a detector d1. We could put here any kind of devices in between. We cou... |
MIT_804_Quantum_Physics_I_Spring_2016 | Consistency_condition_Particle_on_a_circle.txt | PROFESSOR: Let me do a little exercise using still this manipulation. And I'll confirm the way we think about expectations values. So, suppose exercise. Suppose you have indeed that psi is equal to alpha i psi i. Compute the expectation value of Q in the state of psi. Precisely, the expectation value of this operator w... |
MIT_804_Quantum_Physics_I_Spring_2016 | Algebraic_solution_of_the_harmonic_oscillator.txt | PROFESSOR: SHO algebraically. And we go back to the Hamiltonian, p squared over 2m plus 1/2 m omega squared x hat squared. And what we do is observe that this some sort of sum of squares plus p squared over m-- p squared over m squared omega squared. So the sum of two things squared. Now, the idea that we have now is t... |
MIT_804_Quantum_Physics_I_Spring_2016 | Expectation_value_of_Hermitian_operators.txt | PROFESSOR: Today we'll talk about observables and Hermitian operators. So we've said that an operator, Q, is Hermitian in the language that we've been working so far, if you find that the integral, dx psi 1 Q psi 2, is actually equal to the integral dx of Q, acting this time of Psi 1 all star psi 2. So as you've learne... |
MIT_804_Quantum_Physics_I_Spring_2016 | Momentum_operator_energy_operator_and_a_differential_equation.txt | PROFESSOR: Last time, we talked about the Broglie wavelength. And our conclusion was, at the end of the day, that we could write the plane wave that corresponded to a matter particle, with some momentum, p, and some energy, E. So that was our main result last time, the final form for the wave. So we had psi of x and t ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Resonant_transmission.txt | PROFESSOR: Here is the answer, answer. It's easier apparently to write 1 over T. And 1 over T is equal to 1 plus 1 over 4 V0 squared over E times E plus V0 times sine squared of 2k2a. So the one thing to notice in this formula, it's a little complicated, is that the second term is positive. Because V0 squared is positi... |
MIT_804_Quantum_Physics_I_Spring_2016 | The_general_Schrödinger_equation_x_p_commutator.txt | PROFESSOR: ih bar d psi dt equal E psi where E hat is equal to p squared over 2m, the operator. That is the Schrodinger equation. The free particle Schrodinger equation-- you should realize it's the same thing as this. Because p is h bar over i ddx. And now Schrodinger did the kind of obvious thing to do. He said, well... |
MIT_804_Quantum_Physics_I_Spring_2016 | Commutators_matrices_and_3dimensional_Schrödinger_equation.txt | PROFESSOR: This is very important. This is the beginning of the uncertainty principle, the matrix formulation of quantum mechanics, and all those things. I want to just tabulate the information of matrices. We have an analog, so we have operators. And we think of them as matrices. Then in addition to operators, we have... |
MIT_804_Quantum_Physics_I_Spring_2016 | Fourier_transforms_and_delta_functions.txt | BARTON ZWIEBACH: Today's subject is momentum space. We're going to kind of discover the relevance of momentum space. We've been working with wave functions that tell you the probabilities for finding a particle in a given position and that's sometimes called coordinate space or position space representations of quantum... |
MIT_804_Quantum_Physics_I_Spring_2016 | Defining_uncertainty.txt | PROFESSOR: Uncertainty. When you talk about random variables, random variable Q, we've said that it has values Q1 up to, say, Qn, and probabilities P1 up to Pn, we speak of a standard deviation, delta Q, as the uncertainty, the standard deviation. And how is that standard deviation defined? Well you begin by making sur... |
MIT_804_Quantum_Physics_I_Spring_2016 | Delta_function_potential_I_Solving_for_the_bound_state.txt | PROFESSOR: So, what is the wave function that we have? We must have a wave function now that is symmetric, and built with e to the k x, kappa x, and into the minus kappa x. This is the only possibility. E to the minus of kappa absolute value of x. This is psi of x for x different from 0. This is-- as you can quickly se... |
MIT_804_Quantum_Physics_I_Spring_2016 | Scattering_states_and_the_step_potential.txt | PROFESSOR: Scattering states are energy eigenstates that cannot be normalized. And when you say this cannot be normalized, so what's the use of them? They don't represent particles. Well, it's like they e to the ipx over h bar, those infinite plane waves. Each one by itself cannot be normalized, but you can conserve wa... |
MIT_804_Quantum_Physics_I_Spring_2016 | Time_evolution_of_a_free_particle_wavepacket.txt | PROFESSOR: Time evolution of a free particle wave packet. So, suppose you know psi of x and 0. Suppose you know psi of x and 0. So what do you do next, if you want to calculate psi of x and t? Well, the first step, step one, is calculate phi of k. So you have phi of k is equal 1 over square root of 2 pi integral dx psi... |
MIT_804_Quantum_Physics_I_Spring_2016 | Time_delay_and_resonances.txt | PROFESSOR: Let me begin by introducing the subject. The subject is resonances. And we have seen, actually, a little bit of this in the resonant transmission of the Ramsauer-Townsend effect. Because of a resonance phenomenon within the square well obstacle, somehow, for some particular frequencies, for some particular e... |
MIT_804_Quantum_Physics_I_Spring_2016 | Units_of_h_and_Compton_wavelength_of_particles.txt | BARTON ZWIEBACH: Now that we've introduce h, h is a very important quantity in quantum mechanics. So let's talk a little more about h, its units, and we already put one number that I really wish you will remember. Now let's talk about the units of h and some other things you can do with h. So units of h. So if you have... |
MIT_804_Quantum_Physics_I_Spring_2016 | Interferometer_and_interference.txt | PROFESSOR: And let me I assume, for example, that I'll put the state alpha beta in. Alpha and beta. What do I get out? So you have this state, alpha beta. What do you get out? Well, state comes in and is acted by beam splitter 1. So you must put the beam splitter, 1 matrix. And then it comes the mirrors. And lets assum... |
MIT_804_Quantum_Physics_I_Spring_2016 | RamsauerTownsend_phenomenology.txt | PROFESSOR: If you have potential transmission coefficient for a potential where z0 is equal to 13 pi over 4. That's a square well of certain depth, and we represent it in this way. Remember n must be greater than or equal than 2z0 over pi. So this will be 13/2. And 13/2 means that we can start with n equals 7, 8, 9, an... |
MIT_804_Quantum_Physics_I_Spring_2016 | Orthonormality_of_spherical_harmonics.txt | PROFESSOR: How do we state the issue of normalization? See, the spherical harmonics are functions of theta and phi. So it makes sense that you would integrate over theta and phi-- solid angle. The solid angle is the natural integration. And it's a helpful integration, because if you have solid angle integrals and then ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Energy_below_the_barrier_and_phase_shift.txt | PROFESSOR: Let's do E less than V not. So we're back here. And now of the energy e here is v not is x equal 0. X-axis. And that's the situation. Now you could solve this again. And do your calculations once more. But we can do this in an easier way by trusting the principle of analytic continuation. In this case, it's ... |
MIT_804_Quantum_Physics_I_Spring_2016 | de_Broglies_proposal.txt | PROFESSOR: This is Louis-- L-O-U-I-S d-e Broglie. And this is not hyphenated nor together. They are separate. And the d is not capitalized apparently too. And it's 1924, the photon as a particle is clear, and the photon is also a wave. And Louis de Broglie basically had a great insight in which he said that if this is ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Number_operator_and_commutators.txt | PROFESSOR: Last time we discussed the differential equation. I'll be posting notes very soon. Probably this afternoon, at some time. And last time, we solved the differential equation, we found the energy eigenstates, and then turned into an algebraic analysis in which we factorized the Hamiltonian. Which meant, essent... |
MIT_804_Quantum_Physics_I_Spring_2016 | Eigenfunctions_of_a_Hermitian_operator.txt | PROFESSOR: So here comes the point that this quite fabulous about Hermitian operators. Here is the thing that it really should impress you. It's the fact that any, all Hermitian operators have as many eigenfunctions and eigenvalues as you can possibly need, whatever that means. But they're rich. It's a lot of those sta... |
MIT_804_Quantum_Physics_I_Spring_2016 | Wavepackets.txt | PROFESSOR: In order to learn more about this subject, we must do the wave packets. So this is the place where you really connect this need solution of Schrodinger's equation, the energy eigenstates, to a physical problem. So we'll do our wave packets. So we've been dealing with packets for a while, so I think it's not ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Step_potential_probability_current.txt | PROFESSOR: I've put on the blackboard here the things we were doing last time. We began our study of stationary states that are not normalizable. These are scattering states. Momentum eigenstates were not normalizable, but now we have more interesting states that represent the solutions of the Schrodinger equation, tha... |
MIT_804_Quantum_Physics_I_Spring_2016 | Group_velocity_and_stationary_phase_approximation.txt | PROFESSOR: Velocity. So we assume that we have an omega of k. That's the assumption. There are waves in which, if you give me k, the wavelength, I can tell you what is omega. And it may be as simple as omega equal to kc, but it may be more complicated. In fact, the different waves have different relations. In mechanics... |
MIT_804_Quantum_Physics_I_Spring_2016 | Shape_changes_in_a_wave.txt | PROFESSOR: Next is this phenomenon that when you have a wave packet and it moves it can change shape and get distorted. And that is a very nice phenomenon that takes place in general and causes technological complications. And it's conceptually interesting. So let's discuss it. So it's still wave packets. But now we ha... |
MIT_804_Quantum_Physics_I_Spring_2016 | Linearity_and_nonlinear_theories_Schrödingers_equation.txt | PROFESSOR: So here is of something funny. You might say, OK, what is simpler? A theory that is linear or a theory that is not linear? And the answer, of course, a linear theory is much simpler. General-- Maxwell's equations are linear. Einstein's theory of relativity is very nonlinear, very complicated. How about class... |
MIT_804_Quantum_Physics_I_Spring_2016 | Reflection_and_transmission_coefficients.txt | PROFESSOR: We can write, however, J left as J of A, minus J of B. Where J of A would be h bar k over m A squared. And J of B would be h bar k over m B squared. You see the current that exists to the left of the barrier has two components and-- it's very intuitive. It's the current that would have been brought alone by ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Normalizable_wavefunctions_and_the_question_of_time_evolution.txt | BARTON ZWIEBACH: We were faced last time with a question of interpretation of the Schrodinger wave function. And so to recap the main ideas that we were looking at, we derive this Schrodinger equation, basically derived it from simple ideas-- having operators, energy operator, momentum operator, and exploring how the d... |
MIT_804_Quantum_Physics_I_Spring_2016 | Correspondence_principle_amplitude_as_a_function_of_position.txt | PROFESSOR: There's one more property of this thing that is important, and it's something called the correspondence principle, which is another classical intuition. And it says that the wave function, and it addresses the question of what happens to the amplitude of the wave function. It says that the wave function shou... |
MIT_804_Quantum_Physics_I_Spring_2016 | Scattering_in_1D_Incoming_and_outgoing_waves.txt | BARTON ZWIEBACH: Scattering in dimension. And we will consider a world that is just one dimensional, x. And, in fact, there's an infinite barrier at x equals 0. Infinite barrier, nothing goes beyond there. On the other hand, in here, up to some distance R, there could be a potential V of x. So we will have a potential ... |
MIT_804_Quantum_Physics_I_Spring_2016 | Energy_eigenstates_on_a_generic_symmetric_potential_Shooting_method.txt | PROFESSOR: Here is your potential. It's going to be a smooth, nice potential like that. V of x. x, x. And now, suppose you don't know anything about the energy eigenstates. Now, this potential will be assumed to be symmetric. So here is one thing you can do. You can exploit some things that you know about this potentia... |
MIT_804_Quantum_Physics_I_Spring_2016 | Local_picture_of_the_wavefunction.txt | PROFESSOR: OK, so, local picture. It's all about getting insight into how the way function looks. That's what we'll need to get. These comments now will be pretty useful. For this equation you have one over psi, d second psi, d x squared is minus 2 m over h squared, E minus v of x. Look how I wrote it, I put the psi ba... |
MIT_804_Quantum_Physics_I_Spring_2016 | Excited_states_of_the_harmonic_oscillator.txt | PROFESSOR: Suppose you define now, one state called phi 1 as a dagger acting on phi 0. You could not define any interesting state with a acting on phi 0 because a kills phi 0, so you try phi 0 like this. Now you could ask, OK, what energy does it have? Is it an energy eigenstate? Well it is an energy eigenstate if it's... |
MIT_804_Quantum_Physics_I_Spring_2016 | Infinite_square_well_energy_eigenstates.txt | PROFESSOR: Square well. So what is this problem? This is the problem of having a particle that can actually just move on a segment, like it can move on this eraser, just from the left to the right. It cannot escape here. So the way we represent it is the interval 0 to a on the x-axis. And there's going to be two walls,... |
MIT_804_Quantum_Physics_I_Spring_2016 | Simultaneous_eigenstates_and_quantization_of_angular_momentum.txt | PROFESSOR: Simultaneous eigenstates. So let's begin with that. We decided that we could pick 1 l and l squared, and they would commute. And we could try to find functions that are eigenstates of both. So if we have functions that are eigenstates of those, we'll try to expand in terms of those functions. And all this op... |
MIT_804_Quantum_Physics_I_Spring_2016 | Compton_Scattering.txt | PROFESSOR: So we're building this story. We had the photoelectric effect. But at this moment, Einstein, in the same year that he was talking about general relativity, he came back to the photon. And there there's actually a quote of Einstein's saying, his greatest discoveries, for sure, were special relativity and gene... |
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