Questions tagged [semiclassical]

Semiclassical descriptions involve a base/background part described classically, and quantum parts representing an effective development in powers of Planck's constant, ħ. They cover systematic approximations such as the WKB, intuitive approaches to the correspondence limit, and a broad class of interstitial physical phenomena.

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Contradiction in my understanding of wavefunction in finite potential well

Most things like to occupy regions of lower potential. So the probability amplitude should be higher in a region of lower potential. I denote the potential by V. However, we also know that the kinetic ...
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What is the best criterion to discern between classical and quantum physics? [duplicate]

I ask this question here knowing there are similar questions on this site, but not having found a satisfactory answer for myself below those. Or at least, one in which a comparison between different ...
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Phase-amplitude decomposition of a wave

Consider the stationary Schrodinger equation in 1D: $$ \psi''(x) + Q(x) \psi(x) = 0 \quad x \in [0, L] $$ I am specifically interested in the case where $Q(x)$ is monotonic and gives a single turning ...
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How to derive Landau level with semiclassical approach?

I'm trying to derive the Landau level by applying semiclassical dynamics and the time-dependent Schrodinger equation. From that, I success to derive $E = \hbar\omega_c n$, but I fail to derive the ...
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Possibility of the quantum phase transition with the linear equations of motion in the classical limit

Let us say we have a Lindblad equation for the density matrix $$ \dot{\rho}=-\frac{i}{\hbar}[H, \rho]+\sum_{i=1}^{N^{2}-1} \gamma_{i}\left(L_{i} \rho L_{i}^{\dagger}-\frac{1}{2}\left\{L_{i}^{\dagger} ...
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In relation to the correspondence principle, what happens when the orbital magnetic quantum number $m_\ell$ is very large?

If for each value of the orbital quantum number $\ell$ there are $2\ell+1$ possible associated magnetic quantum numbers $m_\ell$, and they are interpreted as the only allowed orientations that the $...
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What is the meaning/justification for the matching conditions in the WKB approximation?

Consider the time-independent Schrodinger equation in one dimension with a potential $V(x)$ at fixed energy $E$. In the WKB approximation, we obtain solutions in the classically allowed region (i.e. ...
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What is the difference between Bohr-Sommerfeld and Wilson-Sommerfeld quantisation rule?

Is Bohr-Sommerfeld and Wilson-Sommerfeld quantisation rule the same thing? If not, what is the difference?
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Why is the time evolution of occupied and unoccupied levels in a band under the influence of applied fields the same?

I am studying the semiclassical model for solid state physics as described in Chapter 12 of Ashcroft & Mermin. While I am familiar with the concept of holes, a formal argument is made in this ...
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Help to evaluate an integral given in appendix of Quantum Field Theory in a Nutshell [duplicate]

On p. 16 in appendix 3 in section I.2 of Quantum Field Theory in a Nutshell by Zee the integral to be evaluated is $$I = \int_{-\infty}^{+\infty}dqe^{-(1/\hbar)f(q)}.$$ Where $f(q)$ is expanded as $$...
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Additional Examples of Quantum and Classical Analogs?

I was wondering if there are any other important classical/quantum analogs, along the lines of these examples: Schrödinger Equation $\leftrightarrow$ Hamilton-Jacobi Formalism Path Integrals $\...
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Classical limit of quantum mechanics in terms of center of mass

When we say that quantum mechanics reproduces classical mechanics at macroscopic scales, is it a statement about the center of mass of a macroscopic system? More specifically, let $\psi (x)$ be the ...
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Second linearly independent solution of Airy Differential equation

The Airy differential equation is $$ \frac{d^2y}{dx^2}=xy. $$ After Fourier transforming the equation, we get $$ y=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\left(kx+\frac{k^3}{3}\right)}dk. $$ Here $k$...
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WKB and Dirac Delta potential

The semi-classical approximation for using WKB in simple words says that in a region where the potential doesn't vary sharply compared to the wavelength of the wavefunction the momentum (or the ...
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Old quantum theory and the energy levels of the hydrogen atom bounded states [closed]

I'm trying to obtain the energy levels of the hydrogen atom using the old quantum theory. In particular using the Bohr-Sommerfeld quantization rules \begin{equation} \displaystyle\int p_i dq_i = n_ih \...
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1 answer
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Expansion of $\phi (x)$ in the derivation of WKB approximation

In the derivation, we assume the eigenfunctions of $H$ to have the form $$\psi (x)=e^{i \phi(x)/\hbar}$$ where $\phi (x)$ is allowed to take any complex value. But then suddenly we assume this ...
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Why macroscopic bodies should exist as wavepacket?

Based on my understanding, we assume that the electrons, exist as wavepackets in the solids while deriving the transport equations for transistors, we create wavepackets out of momentum eigenstates ...
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6 votes
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Does the stationary phase approximation equal the tree-level term?

Consider the scalar field transition amplitude $$\tag{1} \mathcal{A} = \int_{\phi_i}^{\phi_f} D\phi e^{iS[\phi]/\hbar}. $$ Let $\phi_{cl}$ solve the classical equation $\frac{\delta S}{\delta\phi}=0$. ...
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1 answer
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"Classical" adiabatic approximation: please help me understand better

In this1 paper they use an adiabatic approximation to reduce two differential equations to one. Could you please recommend some alternative reading for this (semi) classical adiabatic approximation, ...
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17 votes
8 answers
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How can Planck's constant take different values?

I have seen books and papers mentioning "In the semiclassical limit, $\hbar$ tends to zero", "the scaled Planck's constant goes as $1/N$ where $N$ is the Hilbert space dimension" ...
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3 answers
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How is classical mechanics recovered when the commutator is zero?

If $X$ and $P$ commute, then the rate of change of expectation value of $X$ becomes zero, assuming $$\frac{d}{dt} \langle X \rangle= \langle [X, P^2+V(x)] \rangle=0.$$ This is not what classical ...
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Spin coherent state for periodic orbit

Usually the spin coherent state $|\theta,\phi\rangle$ represents a localized density centered around a point ($\theta,\phi$) in phase space, which is visible in the Husimi distribution of the ...
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4 votes
3 answers
195 views

Energy conservation in semiclassical gravity

I'd like to know whether semiclassical gravity models contain an energy conservation law with the following (heuristic) form: $$``\text{Energy of classical spacetime + expected energy of quantum ...
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5 votes
1 answer
104 views

Is there a (semiclassical) electric field operator?

So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator $$ \hat{\rho}(r) = q \...
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3 votes
4 answers
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Would a high energy Hydrogen atom start emanating electromagnetic radiation?

We know that the total energy of the hydrogen atom is proportional to the inverse of the square of the principal quantum number $n$: $$E_n \propto -\frac{1}{n^2}$$ So at high quantum numbers the ...
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2 votes
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Why are classical equations of energy used to derive equations in Quantum Mechanics? [duplicate]

I read before that one can't derive a more fundamental theory, and Quantum Mechanics is a more fundamental theory than classical physics. I understand the equation $$E=\frac{1}{2}mv^2+U$$ to be a ...
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Overlap of quasiclassical and patching regions in WKB

I'm trying to clarify in my mind the use of Airy functions as matching functions across a turning point in the WKB approach. Quoting from Section 3.1 of Migdal and Krainov, Approximation methods in ...
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Understanding the formula $2\pi r=n\lambda$ [duplicate]

In my book this formula: $$2\pi r=n\lambda$$ is mentioned only. However, its derivation or explanation has not been given. They only said, "So, the circumference of the circular path with radius $...
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When is it necessary to quantize the electromagnetic field?

I've seen both classical and quantized electromagnetic fields in quantum mechanics problems - for example, classical in linear response and the Coulomb interaction, and quantum in photon absorption - ...
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7 votes
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Does Feynman's path integral include complex trajectories?

The WKB approximation provides the correct exponential decay of eigenstates inside classically forbidden regions if one allows classical momenta to be imaginary. The typical example is a double well ...
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2 votes
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Single particle density of states for non-free particle

I am trying to find the single particle density of states in terms of the energy, for a system with the single particle 2D Hamiltonian: $$H=\frac{p^2}{2 m}+\alpha x \text { with } 0<y<L, x>0$$...
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2 votes
1 answer
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How to understand that drifted-diffusion equation in semi-conductors is semi-classical?

I am studying the mathematical models in drifted-diffusion equations and find that drift-diffusion equation belongs to semi-classical models. However, it seems that compared to the Boltzmann equations,...
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4 votes
0 answers
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Why does metaplectic correction fix the vacuum energy?

In geometric quantization we want to go from a symplectic manifold $\left( M, \omega \right)$ to a Hilbert space $H$. If $M$ is prequantizable, we find a prequantum bundle $L \rightarrow M$ with ...
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3 votes
2 answers
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Quantum properties of long wavelength electromagnetic radiation

How could we have known that Electromagnetic radiation is quantized if we only knew about long wavelength radiation? What are the 'quantum' properties shown by long wavelength electromagnetic ...
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2 votes
0 answers
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Why is the conductance of a metal-insulator-metal tunnel junction parabolic?

For bias voltages below the tunnel junction barrier heights (and below the Fowler-Nordheim limit), tunnel junctions have a parabolic conductance as a function of bias. Is this due to the metallic ...
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0 answers
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WKB approximation problem [closed]

I have some issues on solving this problem: use the WKB method to estimate the ground state energy of a particle of mass $m$ that moves in a three dimensional potential $V(r)=kr$, where $k$ is a ...
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0 answers
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How do you adapt the saddle-point integration where the amplitude function has a phase shift?

If anyone can help me with this, I'd be very appreciative. I have tried researching online, but I haven't been able to find many articles/textbooks which are helpful. I am trying to do an integral of ...
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0 votes
4 answers
121 views

Momentum operator generator of translation classical limit

Classical limit in quantum mechanics proof this question is based on my previous closed question but it is a more specific part and hopefully I will get help. The classical limit of quantum mechanics ...
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4 votes
1 answer
287 views

Classical limit in quantum mechanics proof

Several questions are about the limit $\hbar\rightarrow 0$, e.g. When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail? Classical ...
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1 vote
2 answers
251 views

Does the integral of a Wigner function over a finite region mean anything?

I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
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4 votes
0 answers
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Has anyone derived a classical radation reaction term directly from QED?

As far as I know, pretty much the only aspect of classical EM that's still actively controversial within the physics community is the best way to treat the radiation reaction force exerted on an ...
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Bohr's Correspondence Principle's definition and its examples

I learned that Bohr's Correspondence Principle comes true when the quantum number $n$ goes to infinite. However, some documents are saying that $h\rightarrow 0$ or high energy/mass/length are also the ...
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1 answer
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Assumption made for the WKB approximation in radial coordinates [duplicate]

I was thinking the other day, if you had the Schrodinger equation in 3-dimensions, and had a spherically symmetrical potential. Ie.: $$-\frac{ℏ^{2}}{2m}∇^{2}ψ+V(r)ψ=Eψ$$ Then you could simplify the ...
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What are some of the practical system to which sudden approximation can be applied?

I've been trying to find the practical applications of sudden approximation but on one side, adiabatic approximation has a lot of practical applications but sudden approximation seems not to have any ...
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1 answer
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Response Functions in Field Theory - Subtleties?

The definitions I saw of response functions, e.g. in Landau & Lifschitz (SP Sec. 125), or in Altland & Simons (Ch.7), are given in terms of expectation values of some physical quantity $\...
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Feynman Lectures on Physics Vol-I 32-3 Radiation damping. How does this classical result relate to QM?

The following is from https://www.feynmanlectures.caltech.edu/I_32.html#Ch32-S3 Now let us actually calculate the Q of an atom that is emitting light—let us say a sodium atom. For a sodium atom, the ...
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1 vote
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What is the proof of Bohr's quantisation rule? [duplicate]

In his atomic model, Niels Bohr proposed that electrons can be present only in those orbits where their angular momenta is an integral multiple of $\frac{h}{2π}$. That is $mvr=\frac{nh}{2π}$, where $n=...
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1 answer
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Why do we rule out orbits with non-constructive interference for the atom? [duplicate]

It is said that de Broglie explained the quantization of Bohr's orbitals with the idea of the "matter wave" of the electron being forced to have orbits where it can interfere constructively ...
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3 votes
2 answers
117 views

Quantum tunnelling in space vs. time

In the Gamov model of alpha decay they use the WKB approximation to find the magnitude of the stationary state wavefunction of an alpha particle with a given fixed energy $Q$ that has tunnelled ...
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What are the applications of WKB approximation? [closed]

This method is generally used for: WKB approximation for bound states. WKB approximation for tunneling. What are the examples of these two topics other than the well?
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