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Questions tagged [quantum-mechanics]

Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Scan Quantum mechanics: Status of current research

I recently stumbled on a new interpretation of Quantum mechanics, called Scan Quantum Mechanics, given by Beatriz Gato-Rivera. She suggests a quantity called quantum inertia, which divides the ...
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Difference between angular momentum $L$ and $J$ in quantum mechanics

what is the difference between angular momentum L and generalized angular momentum J and their components?
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The order of excited states

The energy level of electron in an infinite square well in three dimensions is given by $E_{n_1 n_2 n_3} =\frac{ \hbar^2 \pi^2}{2mL^2}(n_1^2 + n_2^2 +n_3^2)$. It is understood that $E_{111}$ ...
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If you pass light through a double slit and back through the same slit again is there an interference pattern or particle pattern?

If you pass light through a double slit and back through the same slit again (say by using mirrors to bounce it back around) is there an interference pattern or particle pattern? I'm curious what ...
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Question about a “general” definition for a black hole and his validity outer General Relativity

Consider the following definition: A Black Hole in an asymptotically flat spacetime $\mathfrak{M} \equiv (\mathcal{M}, \mathrm{\textbf{g}})$ is the set of events that do not belong to the causal ...
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What is the meaning of this operator?

In interferometry with coherent light, the final output is differenced detectors. That is, $\left<N\right> = \left<N_1\right> - \left<N_2\right>$ where $N_i$ is the number operator ...
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Quantum energy levels of a point mass rotating about a fixed point

The question is: A particle of mass m is attached to a fixed point in space by a massless rigid rod of length a and can freely rotate about this point. Find the quantum energy levels of the system. ...
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Symmetry in quantum mechanics

My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$ Can you give me an easy explanation for this definition?
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Relation between standard and Kubo-transformed quantum correlations

Via path integral molecular dynamics it is possible to measure the Kubo transformed correlation function between two operators $\hat A$ and $\hat B$ \begin{equation*} K_{\hat A\hat B} = \frac 1 {Z_\...
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Virtual terms in the Dyson series (time dependent perturbation theory)

Let the interaction evolution operator in the interaction picture be $$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$ where $T$ is the time order operator and $H_I=H-H_0$ is the ...
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Understanding spin & spatial components of wave-functions

I would like to understand the following statement: The $S$ and $P$ states can be expressed as products of the spin wavefunctions, $|+\rangle$ and $|-\rangle$, and the spatial wavefunctions, $|...
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Discrete time evolution in a non-Euclidean space?

The time independent schrödinger equation can be written as $$i\frac{\partial \psi}{ \partial t}=H\psi$$ if we consider the case of a 1D particle we can evolve it in time by discretising the ...
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Relation between the propagator and probability for the infinite well

This may be an easy question, but I am really confused about it. For the infinite square well, the (time-dependent) energy eigenfunctions are (inside the well):$$\psi_n(x,t) = \sqrt{2/L}\:e^{-iE_nt/\...
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The Heisenberg uncertainty principle (“Quantum Computation and Quantum Information” by Nielsen and Chuang)

I'm currently reading Quantum Computation and Quantum Information by Nielsen. I'm struggling to understand The Heisenberg uncertainty principle (Box 2.4) Suppose $C$ and $D$ are two observables. ...
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Interpreting wave-particle duality due to wave crests [closed]

I have been thinking a lot about the double slit experiment and am wondering whether any theorist has ever considered the following interpretation for wave-particle duality: Could the reason we ...
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If measurement of state of a photon yields some result R1 out of some mixed states then R1 is likely be the result just after the measurement?

In a popular introductory QM textbook, I came across this experiment which measures the state of polarization of the photon (light) by passing it to some experimental setup. Suppose the incident ...
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transition dipole moment for periodic systems?

The transition dipole moment $\mu=\langle\varphi_a|q\hat r|\varphi_b\rangle$ can be viewed as the off-diagonal matrix element of the position operator $\hat r$,multiplied by the charge $q$. As we ...
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Microcausality when quantizing the real scalar field with anticommutators

We know by the spin-statistics theorem that the real scalar field has to be canonically quantized by commutators. But if we try to use anticommutators, we would expand the field $$\phi(x)=\int\frac{d^...
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Reference for canonical quantization of E.M field (for quantum optics purposes in the end)

I am looking for a nice book or online course treating in a rigorous way the quantization of the E.M field. My needs in the end are to do quantum optics, but I would like to see the proof in a nice ...
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What does the Heisenberg principle actually mean? [duplicate]

As far as I can understand, the Heisenberg principle limits the possibility of calculating the exact position and momentum of electrons, as the light we use to observe it changes it's velocity. But ...
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How did Goudsmit and Uhlenbeck figure out the electron has spin $\frac{\hbar}{2}$?

Most stuff I read online says that to explain the Anomalous Zeeman Effect they had to assume the electron's gyromagnetic ratio is $\frac{-e}{m}$ instead of the classical $\frac{-e}{2m}$. But, since ...
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1answer
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Two-level system Hamiltonian from electric-dipole approx

After making the electric-dipole approx., I can express the interaction of a monochromatic field with angular frequency $\omega$ and a dipole moment ${\bf \mu(x)}$ as $V({\bf x},t) = - {\bf \mu(x)} \...
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Why don't electron-positron collisions release infinite energy?

Questions of the form: An electron and a positron collide with E MeV of energy, what is the frequency of the photons released. quite often come up in my A Level course (for often fairly arbitrary ...
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Measurement of a State Not in the Eigenbasis of the Operator

Suppose I have a two dimensional Hilbert space $\{ |0 \rangle,|1\rangle \}$ with these states being orthonormal. Now suppose I have the Hamiltonian $H=|1\rangle \langle 0|+|0\rangle \langle 1| .$ It ...
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A particle on a ring: orthogonality of eigenstates

Let us consider the quantum-mechanical problem---a particle on a ring of a circumference as $2\pi$ with a magnetic flux $A$ inserted through it: \begin{eqnarray} H=(-i\partial_\phi-A)^2, \end{...
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2 in the Fermi’s Golden Rule

In the derivation of the Fermi's golden rule many authors expand periodic perturbation in this form $$\hat{V}=\hat{F} e^{-i \omega t}+\hat G e^{i \omega t}$$ However I do not understand the reason. ...
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Does helium have any accidental degeneracies?

Does helium have any accidental degeneracies, i.e. are there reducible eigenspaces of $$-\Delta_1 - \Delta_2 - \frac{2}{r_1} - \frac{2}{r_2} + \frac{1}{r_{12}}~?$$
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Time-independent and time-dependent perturbation theory yield different results

First, here's the problem statement. Suppose you have an infinite square well of length $a$, where the box extend from $x=0$ to $x=a$. At $t=0$, you add a perturbation $H'$ of the form: \begin{...
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2answers
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What qualifies a set of operators as a “vector operator” in QM?

In quantum mechanics, there are many vector operators like position, momentum, all the types of angular momentum, etc. In Binney's QM book, he often references vector operators and scalar operators. ...
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Critical parameter for 1D quantum system corresponding to $T_c$ of 2D Classical model

Utilizing the fact that there is a correspondence between a $d$ dimensional quantum system and a $d+1$ dimensional classical system (c.f. Trotter Decomposition), my question regards what the critical ...
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Low energy electrons occupy holes

In Tipler's Physics it is said that for a temperature $T>0$ the only electrons that can gain energy from collisions are the ones with initial energy greater than $E_F-K_BT$. I understand that, ...
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Gauge transformation in quantum mechanics

For the Schrodinger equation $\left( i \hbar \frac { \partial } { \partial t } - q V \right) \Psi = \frac { ( i \hbar \overrightarrow { \nabla } + q \vec { A } ) ^ { 2 } } { 2 m } \Psi$ we can do ...
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Schwinger oscillator model

How do we reach the equation 3.8 .10 I didnt understand how its second last step changes into equation 3.8.10
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Calculate phonon density of states

I need to calculate phonon density of states for a cuprate superconductor. I know there is a general formula for the calculation of phonon density of states by Einstein models like this $$D(\omega)=(\...
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In the Rayleigh–Jeans Law, why polarization is two?

When we count mode of wave in cavity radiator, why the "2" is multiplied by polarization? Polarization has two base, but the number of polarization state maybe infinity and continues. Linear ...
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How does quantum tunneling work in a bigger mass?

im trying to figure out the probability of a 9 y/o kid going right through an object. im trying to do this with a compacted form of schrodingers equation: e^(-2*(m*v/H/2pi)*l) where l = 22,34mm, h = ...
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Tight binding hopping

The definition of the tight-binding model is that the electrons are sited on certain points(the Wannier centers) and that they can hop in linear paths, when we refer to sites that belong to the ...
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Calculate the matrix size for ground states BEC using difference with Gross–Pitaevskii equation

The Ground states BEC at $0$ temperature can be described by Gross–Pitaevskii equation as $(-\frac{\hbar^2}{2m}\nabla^2+V(r)+g|\psi|^2)\psi(r)=\mu\psi(r)$ We limit the BEC in 2D where $V(r)=\frac{1}{...
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Do infinite dimensional systems make sense? [closed]

I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective. I would like to understand if infinite-dimensional systems make sense in ...
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1answer
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Is the Quantum Singleton Bound Compatible with the Toric Code?

The Quantum Singleton Bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qudits, and some subsystem $R$ of $m$ qudits that can 'access the entire quantum code', it ...
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Interpreting the expectation values of $\langle p \rangle$ and $\langle p^2 \rangle$

Why the expectation value of momentum $\langle p \rangle$ is zero for the one dimensional ground-state wave function of an infinite square well? And why $\langle p^2 \rangle = \frac{\hbar^2 \pi^2}{L^2}...
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Classical period of Morse potential [closed]

A particle of mass $m$ and energy $E<0$ moves in a one-dimensional Morse potential: $$V(x)=V_0(e^{-2ax}-2e^{-ax}),\qquad V_0,a>0,\qquad E>-V_0.$$ From the only other question I ...
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When do the eigenkets of an operator form a(n) (orthogonal) basis for the Hilbert space?

When do the eigenkets of an operator form a(n) (orthogonal) basis for the Hilbert space? Is it always the case when the operator is Hermitian? Does the operator need to be Hermitian?
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Is it true that $\int_{\mathbb{R}} \! dx|x\rangle \langle x|x= \hat{x}$?

Does $\int_{\mathbb{R}} \!dx|x\rangle \langle x|x= \hat{x}$, since $\int_{\mathbb{R}} \! dx |x\rangle\langle x| = \mathbb{1}$? If not, why not?
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Quantization on (2,1)-signature hyperplane

QFT states, roughly speaking, belong to a certain subset of functionals over the field configuration on the space-like hyperplane, usually chosen as $t = 0$. What would happen if we chose a mixed-...
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Is the “probabilistic nature of quantum mechanics” and quantum randomness the same?

Digital Physics are a branch of hypotheses about the fundamental physics of our universe. They basically describe the universe as an analogy to a computer and defend that everything in the universe is ...
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Leaking potential [on hold]

I'm trying to model the leaking of a probability from a semi-infinite well into free space. I'm not sure exactly what potential to start with are what form of the wave function to go with. I'm making ...
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How can theories about 1D or 2D systems be generalized for 3D systems?

I was watching a lecture video from MITx $^\dagger$ by professor Barton Zwiebach. He proved a pretty cool theorem "every attractive 1-dimensional potential has a bound state"; however, that only holds ...
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Identical particle

If the wave function of a system of identical fermions is totally antisymmetric then can you understand me that how can be electron,proton,neutron all fermions are distinguishable particle?
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Confusion about average value of $v^2$

I know that expected value of velocity means $$\langle v\rangle=\frac{d\langle x\rangle}{dt}$$ and since I know how to calculate $\langle x\rangle$ namely $$\langle x\rangle=\int_{-\infty}^{\infty}x|\...