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quantum mechanics | List of the basic quantum mechanical variables | https://physics.stackexchange.com/questions/66592/list-of-the-basic-quantum-mechanical-variables | <p>Is there a list of basic quantum variables/attributes that all quantum particles have? </p>
<p>Ex. An electron has charge, position, speed, momentum, etc. Is there a complete list of these variables? </p>
<p>I would figure not all quantum particles share the same set of variables? A photon has position and an elec... | <p>I will collate my comments here:</p>
<p>The <a href="http://en.wikipedia.org/wiki/Standard_Model" rel="nofollow noreferrer">Standard Model</a> of particle physics encapsulates almost all the experimental evidence to date about particles and their interactions. It consists of two branches, the particles and their ob... | 100 |
quantum mechanics | Spectral Decomposition Grover operator | https://physics.stackexchange.com/questions/66813/spectral-decomposition-grover-operator | <p>I am studing the Grover operator. Let be $x_0$ the marked element. Let be $t_f = \pi/2\omega$. The measurement of the register in the computational basis return $x_0$ with probability </p>
<p>$$p_{x_0}(t_f)=\langle x_0|U^{t_f}|D\rangle|=1-1/N$$</p>
<p>Where $|D\rangle$ is a diagonal state state, $U$ the grover ope... | 101 | |
quantum mechanics | Consequences of Quantum Mechanics | https://physics.stackexchange.com/questions/68498/consequences-of-quantum-mechanics | <p>First of all, this question is going to seem a a bit of philosophy but know that vague and purposeless wandering is certainly not what i'm trying to propose here.<br>
Also, the reason i didn't post in philosophy communities is that they certainly know a lot less ( if anything ) of quantum mechanics than most of you ... | <p>On the topic of the <em>actual</em> consequences of QM, <a href="/questions/65397/quantum-mechanics-and-everyday-nature">here</a> is answers with a few things that cannot be explained <em>without</em> QM.</p>
<p>That aside...</p>
<p>There is a golden rule that one should recite before all theoretical physics studi... | 102 |
quantum mechanics | Is a permutation of coordinates or labels really equivalent? | https://physics.stackexchange.com/questions/68673/is-a-permutation-of-coordinates-or-labels-really-equivalent | <p>To construct a N-body anti-symmetric wave function some derivations start with the requirement that the N-body wave function should be anti-symmetric under a permutation of coordinates, other derivations start with the requirement that the total wave function should be anti-symmetric under a permutation of labels or... | <p>You had the following cross term for the case of coordinate permutation:</p>
<p>\begin{align}
\propto \int \textrm{d}x_1 \textrm{d}x_2 \left[ \varphi_{b}^{\dagger}(x_1) \varphi_b(x_2) \right] \left[ \varphi_{a}^{\dagger}(x_2) \varphi_{a}(x_1) \right].
\end{align}</p>
<p>However, this is not how you take the inner ... | 103 |
quantum mechanics | What is the volume of electron? | https://physics.stackexchange.com/questions/68717/what-is-the-volume-of-electron | <p>I know that electron has mass , and that is particle( a body which has only mass and whose size is negligible) but can we ever calculate the volume of the electron . if yes how much it is . if no why?</p>
| 104 | |
quantum mechanics | Quantization for particle in a box problem | https://physics.stackexchange.com/questions/70410/quantization-for-particle-in-a-box-problem | <p>Consider the particle in a box problem in QM. The crux of the reason why QM is able to explain the physical phenomenon is not just the theory but also able to impose boundary conditions which eventually result in quantization. Now in the particle in a 1-d box problem, the wave function is assumed to be zero at the b... | <p>Differentiability of the wave function is only required for finite changes in the potential. If the your potential is infinite (as it is outside the inifinitely deep potential well which you describe) the Hamiltonial is ill-defined anyways.</p>
<p>An other case where you can have an infinite potential is if you ha... | 105 |
quantum mechanics | Potentials in Feynman path integral II | https://physics.stackexchange.com/questions/71667/potentials-in-feynman-path-integral-ii | <p>I am still working on the Feynman path integral, more specifically on the case of a free particle with an infinite potential wall, i.e. the quantum system defined by the Hamiltonian </p>
<p>$$H_1 = \frac{\mathbf{P}^2}{2m} + V(\mathbf{Q})$$</p>
<p>where $V(\mathbf{Q})$ is the potential defined by</p>
<p>$$
V(\math... | 106 | |
quantum mechanics | Where does the change in energy come from when trapping a photon between mirrors? | https://physics.stackexchange.com/questions/71840/where-does-the-change-in-energy-come-from-when-trapping-a-photon-between-mirrors | <p>You have a photon traveling with E=hf and you trap it between two perfectly reflecting mirrors (like a QM particle in a box). The photon has to make a standing wave between the mirrors and its spacial frequency is dependant on the distance between the mirrors, L. Its time frequency, and hence energy are derived from... | <p>When you change the separation between the mirrors you are doing [positive or negative] mechanical work against the radiation pressure of the photon. This work is the source of energy here. The situation is completely analogous to the classical gas (or a single particle) under piston. </p>
<p>More explicitly:</p>
... | 107 |
quantum mechanics | QM - calculating expectation value for velocity of an electron | https://physics.stackexchange.com/questions/73445/qm-calculating-expectation-value-for-velocity-of-an-electron | <p>How do we calculate the expectation value for speed? I have heard that we must first calculate the expectaion value for kinetic energy. Someone please explain a bit what options do we have.</p>
| <p>Calculating the energy eigenvalue, will give you $\langle v^2 \rangle$. This is how it's done:</p>
<p>$$\langle v^2 \rangle = \frac{2}{m}\langle T \rangle=\frac{2}{m}\langle \psi|\hat T|\psi\rangle=\frac{2}{m}\int\psi^*(x)\hat T \psi(x)dx$$</p>
<p>Where you should write $T$(the kinematic energy) as an <a href="htt... | 108 |
quantum mechanics | What is a ket of a vector with a bra of another one? | https://physics.stackexchange.com/questions/74995/what-is-a-ket-of-a-vector-with-a-bra-of-another-one | <p>Suppose we have an orthonormal basis $\{ \psi \}$ in finite dimension of a hilbert space;
What is the butterfly operator of a sum of the $ \psi $, say $\psi_i +\psi_j$ ?</p>
<p>Since by linearity of "taking the dual" we must have, writing $B$ for but..terfly,
$\lvert\psi_i +\psi_j\rangle\langle\psi_i +\psi_j \rver... | 109 | |
quantum mechanics | A projector equal to its own conjugate by a unitary | https://physics.stackexchange.com/questions/75282/a-projector-equal-to-its-own-conjugate-by-a-unitary | <p>For projector $p$, in finite dimension say, some unitaries $u, v$ does $upu^\dagger = vpv^\dagger$ implies $u = v$ ?
Intuitively, can we not say that a unitary is matrix permuting the basis and since $p$ is diagonal then obviously $u$ is $v$ ?
But for an exact proof ?</p>
<p>what if further, $p = upu^\dagger = vpv^... | <p>The answer is no, your result does not follow.</p>
<p>To understand why, suppose we have some $N\times N$ projector $\mathbf{P}$, with a range space $\operatorname{Ran}(\mathbf{P})$ (the subspace that the projector projects into) of dimension $\dim(\operatorname{Ran}(\mathbf{P})) = n$ and kernel (nullspace) $\ker(... | 110 |
quantum mechanics | Does the Nakajima-Zwanzig equation preserve the trace of the projected density matrix? | https://physics.stackexchange.com/questions/77869/does-the-nakajima-zwanzig-equation-preserve-the-trace-of-the-projected-density-m | <p>Looking at the <a href="http://en.wikipedia.org/wiki/Nakajima-Zwanzig_equation" rel="nofollow">Nakajima-Zwanzig equation</a>, wich gives the time evolution of a projection $\cal {P} \rho$ of a full density matrix $\rho$, I am wondering if the trace of $\cal P \rho$ is preserved under time evolution.</p>
<p>The t... | <p>If you use the canonical choice $\mathcal{P}X = \mathrm{Tr}_B(X)\otimes\rho_B$, where $B$ denotes the irrelevant degrees of freedom in the total Hilbert space $\mathbb{H} = \mathbb{H}_A \otimes \mathbb{H}_B$, and $\rho_B$ is an arbitrary reference state on $\mathbb{H}_B$, then $$ \mathrm{Tr} (\mathcal{P} L X )= \mat... | 111 |
quantum mechanics | Probability amplitude in basic quantum mechanics | https://physics.stackexchange.com/questions/79058/probability-amplitude-in-basic-quantum-mechanics | <p>I came across this proportionality statement in my quantum mechanics notebook:
$\psi(x,t)$ is proportional to </p>
<p>$$
\begin{align}
\cos(kx - wt) &= \exp(i(kx-wt)) + \exp(-i(kx-wt)) \\
&= \exp (i(kx-wt))
\end{align}
$$
I looked through most commonly used textbooks for quantum mechanics and I couldn't fi... | <p>The equality $\exp(i(kx-wt)) + \exp(-i(kx-wt)) = \exp (i(kx-wt))$ is incorrect, perhaps you recorded something else incorrectly too?</p>
<p>Generally, wavefunctions that go as $e^{i(kx - \omega t)}$ represent travelling waves and are used in calculating transmission/reflection coefficients for potentials, while a w... | 112 |
quantum mechanics | Hermitian Operators in time and Measurements | https://physics.stackexchange.com/questions/81773/hermitian-operators-in-time-and-measurements | <p>Consider an observable that can be described by a hermitian operator $A$ . No explicit relationship with time is given. What would happen to the probability if the quantity is measured a few days later? </p>
| <p>The time evolution of observables in the Schrodinger picture is determined by the wave function. So, the operator itself can say nothing about the time evolution. To get this information one has to know the Hamiltonian of the system and solve time-dependent Schrodinger equation. In turn, Schrodinger equation require... | 113 |
quantum mechanics | Quantum Regime of Particles in Solids | https://physics.stackexchange.com/questions/81872/quantum-regime-of-particles-in-solids | <p>On my midterm today, I read that when the deBroglie wavelength of a particle exceeds the spacing between the particles in a solid or liquid, the particles begin to behave quantum dynamically. Why is this? I thought a larger deBroglie wavelength implied a less quantum mechanical behavior.</p>
| <p>The <a href="http://en.wikipedia.org/wiki/Debroglie_Wavelength" rel="nofollow noreferrer">de Broglie wavelength</a></p>
<p><img src="https://i.sstatic.net/Vj3Q1.png" alt="debrogliewave"></p>
<blockquote>
<p>where lambda is the wavelength and p the momentum of a particle, E its energy and f the frequency in a pro... | 114 |
quantum mechanics | How to derive the Bethe stopping power formula | https://physics.stackexchange.com/questions/80525/how-to-derive-the-bethe-stopping-power-formula | <p>I need the derivation of Bethe formula for stopping power, but I can't see the corresponding paper to this matter.</p>
<blockquote>
<p>Application of Ordinary Space-Time Concepts in Collision Problems and Relation of Classical Theory to Born's Approximation. E. J. Williams. <a href="http://dx.doi.org/10.1103/RevM... | 115 | |
quantum mechanics | Ehrenfest's theorem on Gaussians | https://physics.stackexchange.com/questions/82002/ehrenfests-theorem-on-gaussians | <p>Considering the free evolution of a Gaussian wave packet, is it possible to use Ehrenfest's theorem to determine the average value of momentum given that of position? </p>
<p>And I imply the simplified version of the theorem, namely $\frac{\mathrm d}{\mathrm d t}\langle x \rangle = \frac{1}{m}\langle p \rangle$, wh... | <p>Yes, you can apply the Ehrenfest's theorem to determine the expectation value of momentum for a given position or to determine the expectation value of position for a given momentum for a Gaussian wave packet! </p>
| 116 |
quantum mechanics | Quantum Box and Quantum Number | https://physics.stackexchange.com/questions/86505/quantum-box-and-quantum-number | <p>How many quantum numbers are needed to describe a stationary state of a particle in a multi-dimensional quantum box (say 73)?</p>
| <p>Generalizing from 1d. For every of the $d$ dimensions you have one independent momentum variable $k_i$ such that $i\in\{1..d\}$. The boundary conditions quantize them. </p>
<p>The <em>energy</em> on the other hand is fixed by a single positive integer, since for a free particle $$ E \propto k_1^2+k_2^2+\cdots k_d^2... | 117 |
quantum mechanics | Decomposition of two particle wavefunction into product of single-particle wavefunctions | https://physics.stackexchange.com/questions/87807/decomposition-of-two-particle-wavefunction-into-product-of-single-particle-wavef | <p>Suppose you prepare a two-particle system such that $\Psi(\vec{r}_1,\vec{r}_2, t_0) = \Psi_1(\vec{r}_1, t_0)\Psi_2(\vec{r}_2, t_0)$.</p>
<p>So then, initially $\Psi(\vec{r}_1,\vec{r}_2,t_0) - \Psi_1(\vec{r}_1, t_0)\Psi_2(\vec{r}_2, t_0) = 0$</p>
<p>But now you let the system evolve in time and $\Psi(\vec{r}_1,\vec... | <p>You might be interested by the correlation between two observables relatives to particles $1$ and $2$, for instance $X_1$ and $X_2$ : </p>
<p>$corr(X_1,X_2) = \dfrac{cov(X_1, X_2)}{\sigma_{X_1}\sigma_{X_2}} = \dfrac{\langle X_1X_2 \rangle - \langle X_1 \rangle \langle X_2 \rangle }{\sqrt{\langle X_1^2 \rangle ... | 118 |
quantum mechanics | Total angular momentum operator forms a complete set? (Clebsch-Gordan coefficients) | https://physics.stackexchange.com/questions/91071/total-angular-momentum-operator-forms-a-complete-set-clebsch-gordan-coefficien | <p>While introducing Clebsch-Gordan coefficients, they state that the operators:
$$ \vec{J_1}^2,\vec{J_2}^2,J_{1z},J_{2z}$$
form a complete set of compatible observables. Which means that there is no degeneracy in their common eigenspaces. </p>
<p>What I wonder is, how it follows that (if the above operators form inde... | <p>It does, yes. The reason for this is central to the mathematical theory which undergirds quantum mechanics.</p>
<p>We are used to thinking about vectors as little arrows with lengths and directions, but that is not complete nor is it always useful. A particle's wave function (and indeed its spin) are also vectors b... | 119 |
quantum mechanics | Does measurement prevent tunneling? | https://physics.stackexchange.com/questions/92520/does-measurement-prevent-tunneling | <p>Does observation collapse the wave function, thus preventing an object from tunneling to a classically forbidden region?</p>
<p>If I understand correctly, observation causes objects to collapse into the state in which they were observed, so there will no longer be a probability of finding them elsewhere. Then if I ... | <p>Prior to measurement you can have a wavefunction that has some contribution beyond a potential barrier. Thus if you were to measure the position of the particle you have some probability to find that the particle would have tunnelled to the other side of the barrier. Observing the particles position doesn't stop the... | 120 |
quantum mechanics | Difference: Fermi wave length vs. phase-breaking length? | https://physics.stackexchange.com/questions/93471/difference-fermi-wave-length-vs-phase-breaking-length | <p>I am reading a quantum transport book, where they often mention: phase breaking length and Fermi wavelength. I have looked up and found that:</p>
<p><strong>Phase breaking length</strong>= length over which electron remains its phase.</p>
<p><strong>Fermi wavelength</strong>= Wavelength associated with the maximum... | <p>You have the correct definition for the two lengths. From them you realise that they are not connected to each other, so both your questions make no real sense. </p>
<p>The Fermi wave-length is a property of a Fermi gas, the phase-breaking length (also called coherence length) is a property of coherent gas. Bosons ... | 121 |
quantum mechanics | Approximating evolution as occurring in a two-dimensional subspace | https://physics.stackexchange.com/questions/93589/approximating-evolution-as-occurring-in-a-two-dimensional-subspace | <p>Suppose you have a quantum system with a Hamiltonian having some number (greater than 2, possibly infinite) of eigenfunctions, and that the system is prepared in the ground state.</p>
<p>When can you approximate it as by two-level system (using just the ground state and first excited state)? Is there some property ... | <p>It depends very much on the hamiltonian and the external potential. There is a huge class of possible situations and of possible behaviours, and many of those do lend themselves quite often to two-level approximations.</p>
<p>The most common, I think, is when you have a weak perturbation which oscillates at the rig... | 122 |
quantum mechanics | When combining two quantum states is there any rules that say its going to be in a mixed or pure state? | https://physics.stackexchange.com/questions/95793/when-combining-two-quantum-states-is-there-any-rules-that-say-its-going-to-be-in | <p>I have two quantum systems, a probe and a target that have been entangled. The probe is prepared in a mixed state and so has the target. Is the combined system therefore a mixed state?</p>
<p>If I had prepared my target in a pure state and probe in a mixed is the resulting combined system still a mixed state?</p>
| <p>So, if I understand you correctly, you have a system with two parts, the probe and the target, i.e. the density matrices $\rho_{probe}$ and $\rho_{target}$ are the reduced density matrices of a state $\rho$ describing the whole system.</p>
<p>Let us first consider the case that you don't have entanglement. If the t... | 123 |
quantum mechanics | Superposition of electron and positron particle states | https://physics.stackexchange.com/questions/31793/superposition-of-electron-and-positron-particle-states | <p>Let $b_k^\dagger ,b_k$ represent the creation and annihilation operators for an electron in state $k$. Let $d_j^\dagger ,d_j$ represent the same for a positron in state $j$. And let
$|0\rangle$ represent the vacuum.</p>
<p>Is it possible to have a state described by $ \left( b_k^\dagger + re^{i\theta} d_k^\dagge... | <p>The Vacuum sector Hilbert space is generated by the action of $U(1)$ invariant operators $\overline{\hat\psi(x)}\Gamma\hat\psi(x)$ (and their Fourier transforms, where $\Gamma$ is an arbitrary Dirac matrix) on the Lorentz and translation invariant vacuum vector, which does not allow the state you ask about to be con... | 124 |
quantum mechanics | Is this paragraph on probabilities of sub atomic partials accurate? | https://physics.stackexchange.com/questions/34035/is-this-paragraph-on-probabilities-of-sub-atomic-partials-accurate | <p>I am working on a concept for something and i want to make sure i understand something clearly before i start on everything else. Note, my project is more about the interactions of elements of complex systems rather than physics. Im just using this paragraph as an example, not doing a project on quantum mechanics.... | <blockquote>
<p>absolute certainty do not exist.</p>
</blockquote>
<p>I'm absolutely certain that I exist and more, I am absolutely certain that Existence exists and that I am aware of it.</p>
<p>It is no more incorrect to say that 1 + 1 = 2 than it is incorrect to say that all bachelors are unmarried men.</p>
<p>... | 125 |
quantum mechanics | Beginning with an arbitrary classical equation for energy, how do I get the QM Hamiltonian? | https://physics.stackexchange.com/questions/35773/beginning-with-an-arbitrary-classical-equation-for-energy-how-do-i-get-the-qm-h | <p>For linear momentum I can use the de Broglie equation, but what about energy in terms of moment of inertia or some other form? </p>
| 126 | |
quantum mechanics | Does a particle lose its (location) wavefunction if its location is measured exactly? | https://physics.stackexchange.com/questions/35827/does-a-particle-lose-its-location-wavefunction-if-its-location-is-measured-exa | <p>As the title says, does a particle lose its location wavefunction if its location is measured exactly (I know this would be impossible in reality)?</p>
<p>Also, in reality, if one measures a particle, does the wavefunction of a particle become something different from original afterwards?</p>
| <p>The particle doesn't "lose" its location wavefunction, rather its wavefunction changes to one which is sharply peaked around the location which results from the measurement. In the Copenhagen interpretation, the wavefunction changes smoothly with time until a measurement, such as this one, is performed, after which... | 127 |
quantum mechanics | Is emission/absorption of a photon lossy? | https://physics.stackexchange.com/questions/35343/is-emission-absorption-of-a-photon-lossy | <p>I recall vaguely that energy is absorbed/radiated in packets called quanta. Quanta were what are now known as photons. </p>
<p>What I'm curious about - Is absorption/radiation vis-a-vis photon lossy? Do the total number of photons exactly match the energy acquired/released?</p>
| <p>It is not a lossy process. For there to be a loss at one place would require there to be a gain somewhere else. When the atom releases the photon there would be recoil of the atom that would contain some energy, but 100% of the energy lost by the atom would be gained by the photon.</p>
| 128 |
quantum mechanics | Energy is quantized | https://physics.stackexchange.com/questions/38433/energy-is-quantized | <p>How can energy be quantized if we can have energy be measured like in 1.56364, 5.7535, 6423.654 kilo joules, with decimals? Thanks</p>
<p>Also isnt it quantization means energy is represented in bit quantities meaning you can not divide, lets say 1 bit of energy</p>
| <p>Typically, in quantum mechanics, bound states are quantized and free/scattering states are not. This is because bound states, by the mere fact that they're constrained to a certain area, will have to satisfy certain boundary conditions, and these conditions won't be able to be satisfied in a continuous range. </p>... | 129 |
quantum mechanics | Something I don't understand in Quantum Mechanics | https://physics.stackexchange.com/questions/39540/something-i-dont-understand-in-quantum-mechanics | <p>I've just started on QM and I'm puzzled with a lot of new ideas in it.</p>
<p>1.On a recent lecture I've attended, there is an equation says:
$\langle q'|\sum q|q\rangle \langle q|q' \rangle =\sum q \delta(q,q')$</p>
<p>I don't understand why $\langle q'|q\rangle \langle q|q' \rangle =\delta (q,q')$</p>
<p>Can yo... | <ol>
<li><p>The equation is true, if $|q\rangle$,$|q'\rangle$ are chosen from an orthonormal set of vectors, such as an eigenbasis of an operator. Then, by definition, $\langle q|q' \rangle = \delta_{q,q'}$ </p></li>
<li><p>$| q \rangle$ just denotes some vector labeled $q$ in some Hilbert space. The dimension eq... | 130 |
quantum mechanics | Is there a quantum state for a large system | https://physics.stackexchange.com/questions/41740/is-there-a-quantum-state-for-a-large-system | <p>My understanding of quantum mechanics is that the state of a system is represented by a vector in multidimensional complex vector space. Is there, in principal, a state vector that represents a large, classical object such as, say, a cheeseburger, at an instant in time? If so, what is the physical meaning of that "s... | <p>Quantum states of macroscopic systems are routinely considered in statistical mechanics. They used to derive both the thermodynamic properties of macroscopic materials and the way they deform and respond to external forces. </p>
<p>However, these macroscopic quantum states are never described by state vectors (pure... | 131 |
quantum mechanics | How can I prove this inequality? | https://physics.stackexchange.com/questions/44218/how-can-i-prove-this-inequality | <p>Prove that
$$ \lambda _{1}\lambda _{2}^{*}\varphi _{1}\varphi _{2}^{*}+\lambda _{1}^{*}\lambda _{2}\varphi _{1}^{*}\varphi _{2}
\leq \left | \lambda _{1} \right |\left | \lambda _{2} \right |\left \{ \left | \varphi _{1} \right |^{2}+\left | \varphi _{2} \right |^{2} \right \} $$
where all symbols are complex numb... | <p>OP's inequality (v3) is obvious if either $\lambda_1=0$ or $\lambda_2=0$, so we may assume that $\lambda_1\neq 0$ and $\lambda_2\neq 0$.</p>
<p>Define
$$\phi_1:=\sqrt{\frac{\lambda_1\lambda_2^*}{|\lambda_1\lambda_2|}}\varphi_1,$$</p>
<p>and</p>
<p>$$\phi_2:=\sqrt{\frac{\lambda_1^*\lambda_2}{|\lambda_1\lambda_2|}... | 132 |
quantum mechanics | Origin of exchage interactions | https://physics.stackexchange.com/questions/46185/origin-of-exchage-interactions | <p>Can someone explain to me the origin of the exchange interaction between two electrically charged spin 1/2 fermions? Quantitative or qualitative accepted. </p>
| <p>The wave function is antisymmetric under exchange of (all) the coordinates of each electron (we'll just call them electrons since that's shorter than "two electrically charged spin 1/2 fermions" and equivalent). We'll write the wave function as:
\begin{align}
\Psi(1,2) &= \psi_1(\mathbf{r}_1)\psi_2(\mathbf{r}_2)... | 133 |
quantum mechanics | Differential of Quantum mean value or expectation value | https://physics.stackexchange.com/questions/46722/differential-of-quantum-mean-value-or-expectation-value | <p>How to take differential of Quantum mean value over hermitian operator (mean or expectation value)?
$$d\langle \hat A\rangle$$
remark:
or time evolution of mean value over operator
$$\frac {d\langle \hat A\rangle}{dt}$$</p>
<blockquote>
<p>what is the problem here?
ok let me talk a little more special in three ... | <p>In fact, you missed another type of force:</p>
<p>In Lagrangian Mechanics there is a scalar potential field V in
which the gradient of V is the force:
$$F=-\nabla V$$</p>
<p>and this is exactly what we dealing with in QM.</p>
<p>$$\frac {d \langle \hat P\rangle}{dt}=\langle \hat F\rangle=\langle -\nabla V\rangle... | 134 |
quantum mechanics | Quantum mechanics in macroscopic systems | https://physics.stackexchange.com/questions/51570/quantum-mechanics-in-macroscopic-systems | <p>I don't understand the superposition principle in quantum mechanics or the collapse of wave-function (I think it's impossible for me to understand it) My question is: </p>
<p>Is it possible to demonstrate the quantum mechanical behaviour (Superposition and wavefunction collapse, etc.) in some macroscopic systems un... | <p>You've asked two questions. Firstly is it possible to see superposition in macroscopic systems? The answer is yes. <a href="http://www.scientificamerican.com/article.cfm?id=quantum-microphone" rel="nofollow noreferrer">This article</a> describes making a tiny "tuning fork" that can be put into a superposition of dif... | 135 |
quantum mechanics | Quantum mechanics, what's possible? | https://physics.stackexchange.com/questions/51627/quantum-mechanics-whats-possible | <p>There is a thread in Physicsforums.com which states due to Quantum Mechanics, if you wait long enough diamonds will appear in your pocket, it also states it's possible for all your atoms to spontaneously re-arrange themselves so you turn into a Boeing airplane. Surely this is fiction?</p>
| <blockquote>
<p>There is a thread in Physicsforums.com which states due to Quantum Mechanics, if you wait long enough diamonds will appear in your pocket, it also states its possible for all your atoms to spontaneously re-arrange themselves so you turn into a Boeing airplane. Surely this is fiction?</p>
</blockquote>... | 136 |
quantum mechanics | Intuitive meaning of the Hilbert Space formalism | https://physics.stackexchange.com/questions/52252/intuitive-meaning-of-the-hilbert-space-formalism | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://physics.stackexchange.com/questions/48469/intuitive-meaning-of-hilbert-space-formalism">Intuitive meaning of Hilbert Space formalism</a> </p>
</blockquote>
<p>I am totally confused about the Hilbert Space formalism of Quantum Mechanics. C... | 137 | |
quantum mechanics | How do particles, such as electrons become visible? | https://physics.stackexchange.com/questions/53422/how-do-particles-such-as-electrons-become-visible | <p>Quantum mechanics says that atoms are invisible - they do not have some specified location, only a probability distribution. So, how can we see them? If there is to be particle-antiparticle annihilation (or other interactions), the particles must have a fixed location, right? So, is this process just random? Is it i... | <p>Firstly, let's address your statement that</p>
<blockquote>
<p>Quantum mechanics says that atoms are invisible.</p>
</blockquote>
<p>I would say that this is misleading <em>at best</em>. Quantum mechanics says more something along the lines of "before we make a measurement of the position of a particle, we can ... | 138 |
quantum mechanics | Given a state function of a particle, can we determine its mass? | https://physics.stackexchange.com/questions/53429/given-a-state-function-of-a-particle-can-we-determine-its-mass | <p>The quantum state of a system is supposed to contain all the information that can be obtained about the system such as its energy, momentum...etc.</p>
<p>So I have 2 questions:</p>
<p>1-If someone gave us a quantum state of a single particle, can we tell of what mass it is?</p>
<p>2-Another question is that, give... | <blockquote>
<p>1-If someone gave us a quantum state of a single particle, can we tell of what mass it is?</p>
</blockquote>
<p>If you know the energy and momentum, yes, you have the mass. </p>
<p><img src="https://i.sstatic.net/8P3Kx.png" alt="mass"></p>
<blockquote>
<p>2-Another question is that, given a quant... | 139 |
quantum mechanics | What happens if an atom absorbs a photon of energy higher than first excited state but lower than second excited state? | https://physics.stackexchange.com/questions/53790/what-happens-if-an-atom-absorbs-a-photon-of-energy-higher-than-first-excited-sta | <p>Since the energy levels of atoms are quantized, I was wondering what happens if an electron is hit by a photon whose energy is higher than electron's first excited state but lower than second excited state. Does it excite to the first excited state? If yes, what happens to the remaining energy?</p>
| <p>Let the electron ground state have energy $E_g$, let the first excited state have energy $E_1$, and let the second excited state have energy $E_2$.</p>
<p>Let the energy of the photon be given by $E_p = hf$.</p>
<p>Now it isn't the energy of the exited states that is important in transitions, but the <strong>energ... | 140 |
quantum mechanics | Force analysis of silver atom in Stern–Gerlach experiment | https://physics.stackexchange.com/questions/59491/force-analysis-of-silver-atom-in-stern-gerlach-experiment | <p>In this experiment we only consider the force at z direction, but $\vec B$ field gradient doesn't exclusively exist at z direction according to Maxwell's equations. So why don't we see the splitting in other directions?</p>
| <p>The point made by Otto Stern in the <a href="http://positron.physik.uni-halle.de/F-Praktikum/PDF/39_zphys1921v7_249_stern.pdf" rel="nofollow">original publication (german)</a> is that the contribution of $\frac{\partial \vec{B}}{\partial x}$ and $\frac{\partial \vec{B}}{\partial y}$ can be neglected when averaging o... | 141 |
quantum mechanics | Indistinguishable particles in quantum mechanics | https://physics.stackexchange.com/questions/59570/indistinguishable-particles-in-quantum-mechanics | <p>If you have two particles of the same species , Quantum mechanics says that $\Phi_{m_{1},x_{1},p_{1},m_{2},x_{2},p_{2}}=\alpha\Phi_{m_{2},x_{2},p_{2},m_{1},x_{1},p_{1}}$
But I don't understand why $\alpha$ doesn't depend on $x$ , $p$ . If $\alpha$ depends on $SO(3)$ invariants as $x^2 , x.p , p^2$ etc then it will... | <p>Let us step back for a moment to answer your question.</p>
<p>We consider a system of $n$ indistinguishable particles. What does that mean ?
Let $S_n$ be the set of all permutations of $n$ elements, and let $\sigma \in S_n$. Then if $P(\sigma)$ is the (unitary) operator representing $\sigma$ on the $n$-particles Hi... | 142 |
quantum mechanics | Is a blackbody real or imagined? | https://physics.stackexchange.com/questions/59894/is-a-blackbody-real-or-imagined | <p>In my reading of blackbody radiation I am always asked to imagine this or that body being a perfect absorber or emitter of radiation, and I am always left with the impression that a blackbody exists only as a theoretical construct. But is it? Or can one be constructed and tested? </p>
| <p>A <a href="http://en.wikipedia.org/wiki/Black_body" rel="nofollow">Blackbody</a> is only theoretical. In other words, it is an ideal one. No such body has been observed with such a perfection in the emissivity. I think the Wiki article holds good. Its first statement:</p>
<blockquote>
<p><em>A blackbody is an ide... | 143 |
quantum mechanics | Free 1d proton in magnetic field | https://physics.stackexchange.com/questions/62061/free-1d-proton-in-magnetic-field | <h2>Question Statement</h2>
<blockquote>
<p>Consider a proton which has spin $1/2$ that is free to move throughout all locations $-\infty<x<\infty$. A magnetic field of constant magnitude $B_{\circ}$ is applied perpendicular to the $x$ axis. Let the Hamiltonian be given by:
$$
\mathcal{H}=\frac{\hat{p}^2_{x}... | <p>One way to think about this is that the particle has two completely separate degrees of freedom, one associated to translations (i.e. position, momentum) and one associated to spin. You are right that in this case the two parts of the Hamiltonian act only on one or the other of these degrees of freedom. The operator... | 144 |
quantum mechanics | Frank Hertz experiment and different jumps | https://physics.stackexchange.com/questions/65514/frank-hertz-experiment-and-different-jumps | <p>Why is it assumed that in this experiment, the jump will be between the second and the first states. Couldn't it be that when the electrons have enough energy, an atom absorbs enough to get to the third state and then jumping only to the second, emiting less energetic light?</p>
<p>Isn't this right? Or is it just t... | <p>You might be interested in <a href="http://grundpraktikum.physik.uni-saarland.de/scripts/What_really_happens.pdf" rel="nofollow">this article</a>, that goes into some detail on what actually happens in the Frank Hertz experiment.</p>
<p>To answer your specific question, once you have electrons ricocheting around th... | 145 |
quantum mechanics | Expectation value - Zetilli vs Griffith | https://physics.stackexchange.com/questions/65779/expectation-value-zetilli-vs-griffith | <p>I know that an inner product between two vectors is defined like: </p>
<p>$$\langle a | b\rangle = {a_1}^\dagger b_1+{a_2}^\dagger b_2+\dots$$</p>
<p>but because a transpose of a component for example $a_1$ is again only $a_1$ the above simplifies to: </p>
<p>$$\langle a | b\rangle = \overline{a_1} b_1+\overline{... | <p>If the wave function $\Psi$ is normalized, then $\langle\Psi|\Psi\rangle$ should equal 1. Griffiths' definition assumes the wave function is already normalized, while Zetilli accounts for all possibilities by dividing out the normalization constant. So if the wave function $\Psi$ is normalized, Zetilli's definition ... | 146 |
quantum mechanics | QM the superposition principle | https://physics.stackexchange.com/questions/66096/qm-the-superposition-principle | <p>In <a href="http://www.goodreads.com/book/show/8514649-quantum-mechanics" rel="nofollow">Zetilli's</a> book author says that we can interpret an inner product $\langle x | \psi(t) \rangle$ as a wave function $\psi (x,t)$ and i understand this.</p>
<p>Next he talks about how a state of the system $|\psi(t)\rangle$ c... | <p><strong>A1:</strong> Yes,you are right. He means to include the time dependace into |ψi⟩ itself. That is; |ψi⟩ denotes |ψi(t)⟩ is understood.</p>
<p><strong>A2:</strong> Why are they not? You mean eigenvectors of the Hamiltonian, right?
Then, just apply the Hamiltonian operator on your |ψi(t)⟩, (which means taking ... | 147 |
quantum mechanics | What is the most general unitary that commutes with a one dimensional projector in a finite dimensional Vector Space | https://physics.stackexchange.com/questions/66097/what-is-the-most-general-unitary-that-commutes-with-a-one-dimensional-projector | <p>Given a Hilbert space of finite dimension $N$ with an orthonormal basis $\mathcal{B}=\{|0\rangle,\ldots,|N-1\rangle \}$ what is the most general unitary operation that commutes with the projector onto one of the basis elements (say $|0\rangle$), \emph{i.e.}, what is the most general $\mathcal{U}$ such that $\mathcal... | <p>As <a href="https://physics.stackexchange.com/a/66100/12623">Lagerbaer pointed out</a>, working with finite dimension can be done in matrix representation. The projector on $|0\rangle$ has only the top left element nonzero, $\hat{\Pi}_0 = \mathrm{diag}(1,0,0,\ldots)$ and we want a unitary $\hat{U}$ that commutes wit... | 148 |
quantum mechanics | Electron-hole symmetry in H and He | https://physics.stackexchange.com/questions/5083/electron-hole-symmetry-in-h-and-he | <p>I'm contemplating particle-hole symmetry, and as an example I am looking at either an electron moving along a hypothetical lattice of hydrogen ions, or a hole moving along a hypothetical lattice of helium atoms. </p>
<p>According to some lecture notes I found, the hopping integral I get when I treat this in a tight... | <p>Yes, a hole with energy $E$ is the same as an electron with a negative energy $E$ missing - that's why it's called a hole and that's how Paul Dirac first encountered it in the relativistic context (in the form of positrons).</p>
<p>A positively-charged positron may look more "particle-like" but one may describe it ... | 149 |
quantum mechanics | Why is it necessary to represent Schrodinger's equation as a partial differential equation? | https://physics.stackexchange.com/questions/5224/why-is-it-necessary-to-represent-schrodingers-equation-as-a-partial-differentia | <p>The Schrodinger equation governs the possible time evolution of a wave function, expressed as a partial differential equation. Isn't this equivalent to the simpler equation</p>
<p>$$\omega = \hbar k^2/2m$$</p>
<p>i.e. any wave function that satisfies this dispersion relation will also satisfy Schrodinger's equati... | <p>Your equation is the right solution to Schrödinger's equation in the momentum-energy representation. However, it's only that simple for Schrödinger's equation with no potential, $V(x,y,z)=0$. </p>
<p>If it's zero, the solution (or, similarly, the reformulation of the equation) is as easy as the algebraic relationsh... | 150 |
quantum mechanics | What are the statistics of three to five bosons? | https://physics.stackexchange.com/questions/7245/what-are-the-statistics-of-three-to-five-bosons | <p>This should be a very easy question. If you look at the bottom of "Identical Particles" in Wikipedia, you see Table 1, which gives you the two particle statistics for bosons, fermions and distinguishable particles. The problem is to extend this table for three, four and five particles, or give an equivalent formul... | <p>Dear Jim, it depends on how many different states - boxes - you have for those particles. If you still have 2 states, just like in the Wikipedia example, then for fermions, the probability is 0 everywhere - it's impossible to put more than 2 fermions to 2 states.</p>
<p>For distinguishable particles, each particle ... | 151 |
quantum mechanics | Charge distribution in positronium | https://physics.stackexchange.com/questions/8944/charge-distribution-in-positronium | <p>Inspired by this: <a href="https://physics.stackexchange.com/questions/8937/electrical-neutrality-of-atoms">Electrical neutrality of atoms</a></p>
<p>If I have a wavefunction of the 'reduced mass coordinate' for a hydrogen like atom made from an electron and a positron, what is the spatial charge distribution?</p>
... | <p>Okay we have the center of mass coordinate $r_{cm} = (r_e + r_p)/2$, and the reduced mass coordinate $r = r_e - r_p$. So given the wavefunction $\psi(r_{cm},r)$ what you are asking is just a change of basis from $|r_{cm},r\rangle$ to $|r_e,r_p\rangle$. So you just need to consider</p>
<p>$$\langle r_e,r_p|r_{cm},... | 152 |
quantum mechanics | Quick question on the ionization energy and the selection rule | https://physics.stackexchange.com/questions/8991/quick-question-on-the-ionization-energy-and-the-selection-rule | <p>So I am looking through my book and it says ".... the order of the excited states is exactly the same order (3p-4s-3d-4p)".</p>
<p>But now I am looking at a question in the book and it says "Is 3d to 4s transition possible? Why or why not?"</p>
<p>My answer to this question is: No it can't be because it doesn't ab... | <p><em>Order of excited states</em> means ordered by their energy: 3p is lower than 4s is lower than 3d is lower than 4p. As you correctly point out, the transition from 4s to 3d is forbidden due to a selection rule concerning the angular momentum. So, 3d being higher in energy than 4s has nothing to with there being o... | 153 |
quantum mechanics | Bell Tests using position measurement | https://physics.stackexchange.com/questions/9170/bell-tests-using-position-measurement | <p>I don't know about all the details of Bell tests using methods like parametric down conversion, but at least in some versions of the EPR paradox you get two photons moving apart in opposite directions. I wonder if you can look for detection coincidences by using photographic plates instead of coincidence counters? T... | <p>Yes, something like that is possible, and they've done it in Anton Zeilinger's lab. It's not a Bell's inequality test, and to get it to work it's more complicated than what you're describing, but: they do see spatial (i.e. position) correlations/interference disappear or not based on measurement of one of the entan... | 154 |
quantum mechanics | Tunneling Rate Constant | https://physics.stackexchange.com/questions/9506/tunneling-rate-constant | <p>I am trying to "decode"/derive an expression for the macroscopic rate constant for the tunneling of protons through a potential energy barrier that I read in a journal article:
$$
k_{\rm tun}(T)=(2\pi\hbar)^{-1}\int_0^{V_{\rm max}} Q(V,T) P_{\rm tun}(V)\ dV.
$$
So basically: the authors say work out the probabilit... | <p>$E=\hbar\omega$ is a totally universal formula that holds for all particles and everywhere in quantum mechanics. Schrödinger's equation guarantees that. The same question was being answered yesterday:</p>
<blockquote>
<p><a href="https://physics.stackexchange.com/questions/9457/gravitational-wave-energy/9460">Gra... | 155 |
quantum mechanics | What are the conditions for decoherence to be irreversible? | https://physics.stackexchange.com/questions/10201/what-are-the-conditions-for-decoherence-to-be-irreversible | <p>Spin echo experiments have been able to reverse the motions of all the molecules in a gas in statistical mechanics in the manner of Loschmidt. The Fermi-Ulam-Pasta model has solutions with a single mode dispersing, only to recohere after quite some time has elapsed. Can the same thing happen for decoherence? What ar... | <p>An article you might be interested in: <a href="http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/RBH97.pdf" rel="nofollow">http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/RBH97.pdf</a></p>
| 156 |
quantum mechanics | Why is $\frac{dx}{dt}=0$ in this average momentum calculation? | https://physics.stackexchange.com/questions/9705/why-is-fracdxdt-0-in-this-average-momentum-calculation | <p>In the following excerpt from S. Gasiorowicz's <em>Quantum Physics</em>, he derives an expression for the average momentum of a free particle. $\psi(x,t)$ is the wave function of a free particle, $\psi^*$ denotes its complex conjugate.</p>
<blockquote>
<p>We try the following: since classically,</p>
<p>$$ p ... | <p>The confusion seems to stem from a) not understanding what kind of objects you are dealing with and b) usual custom of not writing (all) arguments of functions when they are understood.</p>
<p>To clarify a) note that the position operator $\hat x$ <strong>does not</strong> depend on time, and so also its kernel $\l... | 157 |
quantum mechanics | Wave function normalization | https://physics.stackexchange.com/questions/11740/wave-function-normalization | <p>A book by C. J. Ballhausen led me to believe that a quick way to check that I performed step operators properly was by observing that the "wave function should appear normalized," but I have found some issue applying this in practice and believe it is due to my misunderstanding of the underlying physics; I'm trying ... | <p>It was just an arithmetic error:<br>
$$\psi(5,3) = -\sqrt{12/90} (2^-, 1^+, 0^+) + \sqrt{12/90} (2^+,1^-,0^+) + \sqrt{18/90} (2^+, 2^-, -1^+)$$<br>
$$+ \sqrt{12/90} (2^+,1^-, 0^+) - \sqrt{12/90}(2^+,1^+,0^-)$$<br>
needs to be simplified as the second and fourth terms are the same. One has:<br>
$$\psi(5,3) = -\sqrt{1... | 158 |
quantum mechanics | Heisenberg's Uncertainty Forms | https://physics.stackexchange.com/questions/12636/heisenbergs-uncertainty-forms | <p>Can Heisenberg's Uncertainty principle be rewritten in terms of energy density</p>
<p>writing
$$\Delta E \Delta T \geqslant \hbar/2$$
in factors of energy density
$\Delta \sigma \text{ }= \frac{3\Delta E}{4\pi r^3}$</p>
<p>I get
$$\Delta \sigma \frac{3\text{$\Delta $T}}{4\pi r^3} \geqslant \hbar/2$$</p>
<... | <p>It seems that you are missing what $\Delta T$ is here. It is not an interval of time. It is time uncertainty. Which is not what usually called $\Delta T$. The most standard application for this uncertainty principle is a linewidth. From this inequality you know that lifetime in upper state limits lower bound of ene... | 159 |
quantum mechanics | What is the basic postulate on which QM rests | https://physics.stackexchange.com/questions/13639/what-is-the-basic-postulate-on-which-qm-rests | <p>What is the basic postulate on which QM rests. Is it that the position of a particle can only be described only in the probabilistic sense given by the state function $\psi(r)$ ? We can even go ahead and abandon the particle formalism as well. So what is the QM all about ? A probabilistic description of the physical... | <p>Existence of non-compatible observables: measuring one of them (say, coordinate) leads to an unavoidable uncertainty in the result of a subsequent measurement of the other (say, momentum). This is the essence of the Heisenberg uncertainty principle in the kinematics of your system. There is a detailed discussion al... | 160 |
quantum mechanics | weight function and the metric | https://physics.stackexchange.com/questions/15051/weight-function-and-the-metric | <p>The weight function comes from Dirac's book, PRINCIPLES OF QUANTUM MECHANICS. On page 66,he says that sometimes it is more convenient not to normalise the eigenvectors, i.e. $$\langle\xi_1'...\xi_u'|\xi_1''...\xi_u''\rangle=\rho'^{-1}\delta(\xi_1'-\xi_1'')...\delta(\xi_u'-... | <p>Without a separate definition of a volume element, you can just take Dirac's $\rho$ to define a volume element on the configuration space, and it works as such, because of the second formula you wrote.</p>
<p>But in the example, you are transforming rectangular coordinates to different coordinates, the volume form ... | 161 |
quantum mechanics | separation of variables | https://physics.stackexchange.com/questions/15949/separation-of-variables | <p>I'm a math student who's dabbled a little in physics, and one thing I'm a little confused by is separation of variables. Specifically, consider the following simple example: I have a Hamiltonian $H$ which can be written as $H_x + H_y + H_z$ depending only on $x,y$, and $z$ , respectively, and I want to find the eige... | <p>Actually, it's not true that all eigenfunctions are separable. Consider the 3D isotropic harmonic oscillator, whose Hamiltonian is a sum of three 1D SHO Hamiltonians,</p>
<p>$$H = \frac{p^2}{2m} + \frac{m\omega^2 r^2}{2} = \sum_{i\in\{x,y,z\}}\hbar\omega\biggl(a_i^\dagger a_i + \frac{1}{2}\biggr) = H_x + H_y + H_z$... | 162 |
quantum mechanics | functional determinant and WKB approximation | https://physics.stackexchange.com/questions/16477/functional-determinant-and-wkb-approximation | <p>let be a Hamiltonian in one dimension, i would like to evaluate the functional determinant $ det(E-H) $ in onde dimension</p>
<p>i believe that $ det(E-H)= Cexp(iN(E)) $ here $ N(E)$ is the number of energy levels less than a given number 'E'</p>
<p>my steps</p>
<ol>
<li><p>i use the identity $ logDet(E-H)=TrLog... | <p>The formula doesn't work. Most of the manipulations are formally ok, although it is probably best to start right at step 3--- the derivative of the logarithm of the determinant is the (trace of the) Green's function, which is better behaved than the determinant itself.</p>
<p>Step 5 is incorrect--- there is no redu... | 163 |
quantum mechanics | Is decoherence due to coarse graining or coupling with the environment? | https://physics.stackexchange.com/questions/16297/is-decoherence-due-to-coarse-graining-or-coupling-with-the-environment | <p>In the literature, sometimes one reads that decoherence is due to the coupling of the system to the external environment, and sometimes one reads that it is due to coarse graining over the microscopic degrees of freedom. Are these two different cases of decoherence, or is one more fundamental than the other?</p>
| <p>The more conventional way is to describe decoherence as being due to the "coupling to the environmental degrees of freedom" that are traced over. However, the "environmental degrees of freedom" may also include geometrically internal degrees of freedom of a physical system such as a cat – unmeasurably complicated co... | 164 |
quantum mechanics | Does frequentism require exponentially many trials in some cases? | https://physics.stackexchange.com/questions/17189/does-frequentism-require-exponentially-many-trials-in-some-cases | <p>Frequentism is the philosophy that probabilities are statistical in the sense that they give the limiting frequency ratios of outcomes as the number of trials is large enough. For tiny probabilities like exponentially small probabilities, would this require exponentially many trials?</p>
<p>I know, I know, you are ... | <p>One way to address this Question is to respond that <em>frequentism</em> is <em>not</em> the standard interpretation of probability in Physics, so it doesn't have to be saved. See Section 3.3 of <a href="http://plato.stanford.edu/entries/probability-interpret/" rel="nofollow">this Stanford Encyclopedia of Philosophy... | 165 |
quantum mechanics | Pauli Matrices in orthogonal space | https://physics.stackexchange.com/questions/18018/pauli-matrices-in-orthogonal-space | <p>In some literature there is reference to $\tau$ matrices which are the same pauli matrices in an orthogonal space. I have not seen any explicit constructions of this anywhere. Could someone tell me or point to literature on how to find the matrix elements of these $\tau$ matrices. </p>
| <p>Georgi is in Exercises 3D, 3E and 6C using the word <em>orthogonal</em> in a non-standard sense. Basically, he means <em>independent copies of sigma matrices that act in different spaces.</em> In detail, first let us define the $gl(2,\mathbb{C})$ Lie algebra as the span of the sigma matrices and the unit matrix $\si... | 166 |
quantum mechanics | Does an interaction of entangled particles with each-other cause decoherence? | https://physics.stackexchange.com/questions/19464/does-an-interaction-of-entangled-particles-with-each-other-cause-decoherence | <p>I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question.</p>
<p>Given a system of entangled particles (eg, 2 or more electrons), p... | <p>That depends on the interaction. Consider two spins interacting with a Heisenberg type interaction</p>
<p>$$H = -J \vec{S}_1 \cdot \vec{S}_2$$</p>
<p>which basically means that the spins want to be parallel if $J > 0$ and anti-parallel if $J < 0$. </p>
<p>For anti-ferromagnetic coupling, $J < 0$, the gro... | 167 |
quantum mechanics | What are the practical applications of decoherence? | https://physics.stackexchange.com/questions/20400/what-are-the-practical-applications-of-decoherence | <p>Let me clarify this question somewhat. I know decoherence is ubiquitous in nature and explains the emergence of a classical world from quantum physics. My question is really about how a knowledge of how decoherence actually works can be put to use in a practical application. An application we can't design in the abs... | <p>Since very weak interactions are sufficient to significantly decohere a quantum system, <strong>quantum systems can potentially be used as very sensitive force sensors if their decoherence is monitored</strong>. This monitoring can take the form of interferometric measurements in which the fringe visibility is measu... | 168 |
quantum mechanics | How to interpret the derivative in the momentum operator in quantum mechanics? | https://physics.stackexchange.com/questions/23036/how-to-interpret-the-derivative-in-the-momentum-operator-in-quantum-mechanics | <p>Given a stationary 1-D wave function $\psi(x)$, how is the derivative in the momentum operator interpreted?</p>
<p>$$
\int_{-\infty}^\infty \psi^*(x) \hat{p} \psi(x) dx
= \int_{-\infty}^\infty \psi^*(x) (-i\hbar\nabla) \psi(x) dx
$$</p>
<p>Should the integral be interpreted as</p>
<p>$$-i\hbar\int_{-\infty}^\i... | <p>Yes. In mathematics, the symbol $f'$ means
$$ f'(x) = \frac{{\rm d}f}{{\rm d}x} = \lim_{\epsilon\to 0} \frac{f(x+\epsilon)-f(x)}{\epsilon} $$
This notation using ${\rm d}$ was introduced by Leibniz; the notation with the prime was introduced by Lagrange.</p>
<p>You also ask whether there is some problem wit... | 169 |
quantum mechanics | In $H_2^+$, what is the Hamiltonian of the movement of the electron? | https://physics.stackexchange.com/questions/23569/in-h-2-what-is-the-hamiltonian-of-the-movement-of-the-electron | <p>An electron is orbiting two protons. With the Born-Oppenheimer approximation that the protons do not move, I'd write the Hamiltonian of the electron's movement as:</p>
<p>$$ \mathbf{H} = -\frac{\hbar^2}{2m}\nabla^2 + E_p$$</p>
<p>with</p>
<p>$$E_p = -\frac{e^2}{4\pi \epsilon_0}\left(\frac{1}{r_1}+\frac{1}{r_2}\ri... | <p>The $\frac{e^2}{4\pi\epsilon_0}\frac{1}{r_0}$ term appears in the potential for the electron motion, as Luboš and Vijay point out, to keep the whole energy accounting in place so that the nuclear motion can be properly quantized. The key point is that this potential does not involve the electron coordinates, so that... | 170 |
quantum mechanics | Pertinence of the wave function of the universe, or complete description of system with massive number of dof | https://physics.stackexchange.com/questions/24032/pertinence-of-the-wave-function-of-the-universe-or-complete-description-of-syst | <p>I have heard couple of times about the concept of wave function of the universe, an object that would capture every degrees of freedom inside it (every particle, me, even you dear reader, etc...) and it always sounded fallacious or at least non pertinent, what would be the point of using that gigantic object to desc... | <p>Yes, it is a logically positivistically meaningless notion, so in an absolute sense it is complete bullshit--- you can't measure the wavefunction of the universe, nor give a sense to the idea that it is A and not B when the overlap of A and B is nonzero. But it is <em>useful</em> bullshit, as a figure of speech, use... | 171 |
quantum mechanics | what is difference between these two expectation values? | https://physics.stackexchange.com/questions/30155/what-is-difference-between-these-two-expectation-values | <p>what is difference between these two expectation values $\langle \hat A \hat B\rangle$ and $\langle \hat B \hat A\rangle$?
where the $\hat B$ and $\hat A$ are two operators.</p>
| <p>$$\langle \hat{A}\hat{B} \rangle -\langle \hat{B}\hat{A} \rangle = \langle \hat{A}\hat{B}-\hat{B}\hat{A} \rangle$$</p>
<p>So it is simply the expectation of the commutator.</p>
| 172 |
quantum mechanics | Feynman's 'diamond jumping out of a box' parody, how would this work? | https://physics.stackexchange.com/questions/30172/feynmans-diamond-jumping-out-of-a-box-parody-how-would-this-work | <p>I have been told that Feynman deduced from a path integral formulation an equation that predicts the amount of time it would take for a diamond to 'jump' out of a box:</p>
<p>$t > \dfrac{x \Delta{x} m}{ h} $</p>
<p>where $x$ is the size of the box, $\Delta x$ is the distance the diamond must travel to leave the... | <p>The diamond must become quantum as a unit, and the wave function of the quantum diamond must then disperse sufficiently to extend outside the box. At that point the diamond as a whole unit has a probability of jumping outside the box.</p>
<p>The first criterion is by far the most difficult, because it can only be ac... | 173 |
quantum mechanics | Partial decoherence from interaction between two qubits | https://physics.stackexchange.com/questions/409753/partial-decoherence-from-interaction-between-two-qubits | <p>$\renewcommand{\ket}[1]{\left\lvert #1 \right \rangle}$</p>
<p>If a quantum system $A$ becomes entangled with another quantum system $B$, then $A$ can no longer be described by a pure quantum state.
For example, given a Bell state
$$
\ket{00} + \ket{11} \, ,
$$
the state of either qubit by itself is a classical... | <p><span class="math-container">$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$</span>
The propagator for this Hamiltonian is
<span class="math-container">$$
U(t) = \exp(-iHt/\hbar) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \cos(gt) & -i \sin(gt) & 0 \\
0 & ... | 174 |
quantum mechanics | Difference between state function and eigenfunction | https://physics.stackexchange.com/questions/425519/difference-between-state-function-and-eigenfunction | <p>Is there any difference when we say in quantum mechanics the <em>eigenfunction</em> of an operator and the <em>state function</em>? Can we use the two terms as supplementary to one another?</p>
| 175 | |
quantum mechanics | Does color filter change the frequency of light? | https://physics.stackexchange.com/questions/426844/does-color-filter-change-the-frequency-of-light | <p>If I glow a white LED bulb and then put a color filter around it (for example, a red color filter or a violet color filter) , then will it change the frequency of light.</p>
| <p>The frequency will not be changed, but rather the color that is reflected/transmitted (depending on the nature of the filter). If you observe the white light THROUGH the filter and it appears to be red, then the filter either reflects or absorbs the other wavelengths while it lets the red ones through. The frequency... | 176 |
quantum mechanics | How light source effects the nature of electron in double slit experiment | https://physics.stackexchange.com/questions/430914/how-light-source-effects-the-nature-of-electron-in-double-slit-experiment | <p>I know that electrons behaves as a particle instead of waves once observed under a light source.
But what I really want to know is that how the photon particles are forcing the electrons to deviate from their wave character?</p>
| 177 | |
quantum mechanics | Error in paper showing internal inconsistency in QM? | https://physics.stackexchange.com/questions/431053/error-in-paper-showing-internal-inconsistency-in-qm | <p>In a recent <a href="https://www.nature.com/articles/s41467-018-05739-8.pdf" rel="nofollow noreferrer">paper</a>, “Quantum theory cannot consistently describe the use of itself”, D. Frauchiger and R. Renner describe a modified Schroedinger's Cat <em>gedankenexperiment</em>, in which there are two boxes (A and B), ... | 178 | |
quantum mechanics | How do you calculate the probabilities associated with eigenfunctions of a wave function? | https://physics.stackexchange.com/questions/433443/how-do-you-calculate-the-probabilities-associated-with-eigenfunctions-of-a-wave | <p>I'm watching <a href="https://www.youtube.com/watch?v=EftlcEdaO_8&list=PLy73XOgPrzuYW-RBSJz8IbdanXgIUuwsR&index=17" rel="nofollow noreferrer">Lecture 03-05 of the MIT 3.024 lecture series on Electronic, Optical and Magnetic Properties of Materials</a> by Polina Anikeeva, specifically the discussion from the ... | <p>Yes, the wavefunction does change when you make a measurement, and hence the probability of the system ending up in a given eigenstate. You would need to reset the experiment each time as you made these repeated measurements. If you just repeatedly measured the same system without resetting it, then you would alwa... | 179 |
quantum mechanics | Distinguishability and energy of a system | https://physics.stackexchange.com/questions/440262/distinguishability-and-energy-of-a-system | <p>I'm studying distinguishability in quantum mechanics but I'm confused with the calculation of energies.</p>
<p>Suppose we are given a hamiltonian for 1 particle with two possible sites<br>
<span class="math-container">$$ H = \begin{bmatrix}
0 & t \\
t & 0
\end{bmatrix}$$</span></p>
<p>With this h... | <p>The Hamiltonian of composite system is not a tensor product, but a sum</p>
<p><span class="math-container">$$
H_{comp} = H'_1 + H'_2
$$</span></p>
<p>where <span class="math-container">$H_1'$</span> acts on the part of state vector that describes the first system, and <span class="math-container">$H_2'$</span> act... | 180 |
quantum mechanics | Seeming energy paradox in quantum system? | https://physics.stackexchange.com/questions/441452/seeming-energy-paradox-in-quantum-system | <p>Imagine an electron in atomic or ascillator potential - any bound electron state. The WF has bell like shape fading at infinity - where classical energy of the electron by all means greatly exceeds the state energy eigenvalue. Now, if we place a trap for the electron at this distance and finally get the electron tra... | <p>I guess you need first to offer a description of the trap. For example, if this trap is just a deep potential well at a significant distance from the minimum of the initial potential, then the electron can indeed tunnel into the trap, but if the minimum of the trap well is below the minimum of the initial potential,... | 181 |
quantum mechanics | Planetary-sized pure quantum states | https://physics.stackexchange.com/questions/442432/planetary-sized-pure-quantum-states | <p>Picture a planet wandering intergalactic space. Such a planet would only couple to vacuum flucuations and the cosmic microwave background. (Ignore stray Hydrogen atoms.)</p>
<p>If this planet started as a pure quantum state, how fast would that state lose its coherence?</p>
<p>In such a system, clearly there are... | <p>I am just going to quote Schlosshauer as being pertinent to this question and discussion in comments.</p>
<p>Reference: Decoherence and the Quantum-to-Classical transition (page 84):</p>
<p>To summarize, we have distinguished three different cases for the type of preferred pointer states emerging from interactions... | 182 |
quantum mechanics | Bose Einstein Condensate | https://physics.stackexchange.com/questions/447059/bose-einstein-condensate | <p>In Bose Einstein condensate photons stop moving, but with reference to what frame of reference? Photons move at constant speed in all reference frames so what happens to Maxwell equations at zero degree kelvin? </p>
| <p>In a Bose-Einstein condensate the electrical field of the light interacts with the condensate and the photons and the condensate become entangled. Once this happens the light forms a quasiparticle called a <a href="https://en.wikipedia.org/wiki/Polariton" rel="nofollow noreferrer">polariton</a>. The velocity is no l... | 183 |
quantum mechanics | How can we use usual notations in relational quantum mechanics? | https://physics.stackexchange.com/questions/449641/how-can-we-use-usual-notations-in-relational-quantum-mechanics | <p>in this <a href="https://arxiv.org/abs/quant-ph/9609002" rel="nofollow noreferrer">relational quantum mechanics paper</a> Rovelli tries to reconstruct the theory avoiding to use the notion of a state associated to a particle. he says that the only correct thing would be to speak
of the information tbat an observer O... | 184 | |
quantum mechanics | Quantum numbers in spherical symmetric potential | https://physics.stackexchange.com/questions/450330/quantum-numbers-in-spherical-symmetric-potential | <p>Can we proof the relation that the principal quantum number <span class="math-container">$n$</span> and azimuthal quantum number <span class="math-container">$l$</span> have the relation <span class="math-container">$l=0,1,...n-1$</span> in any spherical symmetric potential <span class="math-container">$V(r)$</spa... | <p>This relationship does not hold for all spherically-symmetric potentials. For example, for the <a href="https://en.m.wikipedia.org/wiki/Quantum_harmonic_oscillator#Example:_3D_isotropic_harmonic_oscillator" rel="nofollow noreferrer">3D harmonic oscillator</a> the relationship between <span class="math-container">$n$... | 185 |
quantum mechanics | Deriving time-dependent Hamiltonian | https://physics.stackexchange.com/questions/452481/deriving-time-dependent-hamiltonian | <p>Consider a two state system <span class="math-container">$\rho$</span> with some Hamiltonian <span class="math-container">$H$</span>. I am interested in the specifics of the systems time-evolution.</p>
<p>The article I am reading gives me the following time evolution:</p>
<p><span class="math-container">$$\lvert ... | <p>You can directly use Schrödinger's equation:
<span class="math-container">$$ i \hbar \frac{d}{dt} \lvert\psi(t)\rangle = H \lvert\psi(t)\rangle $$</span></p>
<p>Start with the given time evolution:
<span class="math-container">$$
\begin{align}
\lvert\psi(t)\rangle &= \cos(\Omega t) e^{i \omega_1 t} \lvert 1\ran... | 186 |
quantum mechanics | Photoelectric Effect (and Electric Current) | https://physics.stackexchange.com/questions/452798/photoelectric-effect-and-electric-current | <blockquote>
<p><strong>When intuition fails: photons to the rescue!</strong><br>
When experiments were performed to look at the effect of light amplitude and frequency, the following results were observed:</p>
<ul>
<li>The kinetic energy of photoelectrons increases with light frequency.</li>
<li>Electric current remai... | <p><a href="https://physics.stackexchange.com/questions/333813/are-number-of-photons-in-an-incident-radiation-proportional-to-its-intensity">This question and its answer</a> may help explain the relationship between intensity and number of photons. As Thomas Fritsch pointed out, electric current is the number of electr... | 187 |
quantum mechanics | Why the available parities of the infinite well solutions change as we change the boundaries positions? | https://physics.stackexchange.com/questions/453228/why-the-available-parities-of-the-infinite-well-solutions-change-as-we-change-th | <p>So, I have noticed something in the solutions of the infinite quantum well and I don't quite understand it. The solutions are of the form</p>
<p><span class="math-container">\begin{equation}
\phi_{n}(x) = A\cos(kx)+B\sin(kx)
\end{equation}</span></p>
<p>If the boundaries of the well are at <span class="math-contai... | <p>The <em>energies</em> will be the same but the solutions - of course - will not. The simplest example is to compare <span class="math-container">$\cos(x)$</span> which is even, with <span class="math-container">$\sin(x)$</span>, which is odd. If you just translate by <span class="math-container">$\pi/2$</span>, th... | 188 |
quantum mechanics | Ehrenfest theorem and the distinction between moment ordinal 1 and moment ordinal 2 of the measured probability distributions | https://physics.stackexchange.com/questions/459903/ehrenfest-theorem-and-the-distinction-between-moment-ordinal-1-and-moment-ordina | <p>The proof of the statement of Ehrenfest theorem in the Schrodinger picture does not depend on the state vector. However, <a href="https://en.wikipedia.org/wiki/Ehrenfest_theorem" rel="nofollow noreferrer">Wikipedia claims</a> that:</p>
<blockquote>
<p>for states that are highly localized in space, the expected po... | 189 | |
quantum mechanics | Does the "particle exchange" operator have any validity? | https://physics.stackexchange.com/questions/464019/does-the-particle-exchange-operator-have-any-validity | <p>In introductory quantum mechanics books the topic of identical particles often introduces a "particle exchange" operator. This operator, when applied to a multi-particle wave-function, exchanges the positions of two identical particles.</p>
<p>However, it seems to me that this is a non-physical thing. Particles c... | <p>If I'm reading your question right, I think you're having a relatively common issue. Feynman himself had the same issue when confronted with a creation operator for an electron. "How can an electron really be created? It violates the conservation of charge!"</p>
<p>The point is that not every operator needs to repr... | 190 |
quantum mechanics | which waves give rise to interference pattern in a double slit diffraction experiment | https://physics.stackexchange.com/questions/467873/which-waves-give-rise-to-interference-pattern-in-a-double-slit-diffraction-exper | <p>In a double slit experiment,the fringes obtained on screen are due to superposition of single slit diffraction from each slit and the double slit interference pattern,</p>
<p>when we talk about single slit diffraction,we say the single slit is divided into many number of slits and the waves originating from them su... | 191 | |
quantum mechanics | As electrons are present in many places at the same time so how can it not violate conservation of energy? | https://physics.stackexchange.com/questions/469062/as-electrons-are-present-in-many-places-at-the-same-time-so-how-can-it-not-viola | <p>I was just wondering that as according to quantum mechanics electrons are present in many places at the same time, now as according to Einstein as <span class="math-container">$$E = mc^2$$</span> doesn't it violate energy conservation ?</p>
<p><code>Edit</code>-</p>
<p>I just meant by energy conservation that a... | <blockquote>
<p>according to quantum mechanics electrons are present in many places at the same time</p>
</blockquote>
<p>This is definitely not what QM says at all. This is usually stated in pop-science articles to explain QM to the layperson, but this is not what the theory says. The electron is actually not at an... | 192 |
quantum mechanics | Electrons Fired One at a Time in Double-slit Experiment | https://physics.stackexchange.com/questions/469718/electrons-fired-one-at-a-time-in-double-slit-experiment | <p>I have read that electrons fired individually through a 2 or more slits still form an interference pattern. I think this may be due to the fact that moving electrons produce electromagnetic waves (like in a transmitter aerial), and EM waves move electrons (like in your TV aerial). While each electron can only pass t... | <p>The wavefunction of an electron is intrinsic and dependent on its mass and momentum; it does not have any important connection to the electron's electromagnetic field, at least in the context of a double-slit interferometer.</p>
<p>The electron's wavefunction, which represents (the square root of) the probability d... | 193 |
quantum mechanics | Can different types of properties be entangled with each other? Say, the spin of particle A with the polarization of particle B? | https://physics.stackexchange.com/questions/476238/can-different-types-of-properties-be-entangled-with-each-other-say-the-spin-of | <p>Does this question make sense? Can measuring the spin of one entangled particle 'determine' the polarization of the other?</p>
| <p>When two particles become entangled, the whole new system will have a common wavefunction that will describe the whole system. This system in your case will have both particles, so this wavefunction will describe both their characteristics, and not just their spins, but all of their characteristics.</p>
<p>So yes, ... | 194 |
quantum mechanics | not separable electron state | https://physics.stackexchange.com/questions/485410/not-separable-electron-state | <p>Consider electron spin operator. It acts on Hilbert space <span class="math-container">$\mathbb{C}^2$</span>. Next, electron position operator acts on space <span class="math-container">$\mathbb{L}^2$</span>. Can we describe all electron features in one, "joint" Hilbert space? Certainly, both spaces can be combined... | <p>It is certainly possible to entangle the position state of a particle with its spin state. This is exactly what a Stern-Gerlach apparatus does, producing quantum correlations between position and spin. If you do not observe the output of the Stern-Gerlach experiment directly (and so do not collapse the state vecto... | 195 |
quantum mechanics | Finding the eigenfunctions of the operator $x$ | https://physics.stackexchange.com/questions/485951/finding-the-eigenfunctions-of-the-operator-x | <p>On pg 104 of "Introduction to Quantum Mechanics" by Griffiths, we are asked to find the eigenfunctions of the <span class="math-container">$x$</span> operator. Hence, we have to find functions such that <span class="math-container">$$x f(x)=\lambda f(x)$$</span> I have used the notation <span class="math-container">... | <p>Here is a formal wisecrack to reassure you: work in momentum space.</p>
<p>Up to normalization constants that do not matter that much for your un-normalizable wave function, consider
<span class="math-container">$$
f_\lambda (x)=\langle x| f_\lambda\rangle= \int dp \langle x|p\rangle \langle p|f_\lambda\rangle = \... | 196 |
quantum mechanics | What is the mechanism when a photon is absorbed causing an electron orbital expansion to a larger orbital? | https://physics.stackexchange.com/questions/489054/what-is-the-mechanism-when-a-photon-is-absorbed-causing-an-electron-orbital-expa | <p>We know a photon is absorbed when an electron's orbit expands, but what is the mechanism?</p>
<p>We know that an electron is not simply a particle, so the photon can't be converted into only kinetic energy.</p>
<p>Is the photon converted into space, thus increasing the orbital size?</p>
| <p>The photon gives the electron the energy it needs to move further away from the nucleus, against the attractive electrostatic force. In doing so, it increases the electron’s electrostatic potential energy, making it <em>less negative</em>. The kinetic energy of the electron actually <em>decreases</em>, because when ... | 197 |
quantum mechanics | How is the mass of an electron distributed? | https://physics.stackexchange.com/questions/502287/how-is-the-mass-of-an-electron-distributed | <p>Since the position of an electron is not determined, how is it’s mass distributed ? In other words, how does an electron curve space time ?</p>
<p>.</p>
| 198 | |
quantum mechanics | Understanding quantum superposition of molecules | https://physics.stackexchange.com/questions/507168/understanding-quantum-superposition-of-molecules | <p>How can the results from <a href="https://www.nature.com/articles/s41567-019-0663-9.epdf?referrer_access_token=P6jczrHmpk_eNzSKdu3YCdRgN0jAjWel9jnR3ZoTv0MQ7n-_yUmyvBsKfkN6FLBu98G0Nrx30Fucun-h3w8WX9IlwWelmQLTVb70WHA4Y1pxsdPmhmq4QvZ3kk0dRycJILQYYHZN26qmO72UylFXKFgM9i70iUUDcZdPwblhMv0RzePEbjpRDvq9ofxAtGf2St2MnNM6St8f1-... | <p>The double slit experiment, regardless of the size of the particles (electrons, neutrons, molecules) does not prove that those particles exist in two places at once, as claimed by the SciAm article. The difficulty of understanding this experiment in classical physics is caused by the use of an unsuitable classical m... | 199 |
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