All Questions
Tagged with wkb-approximation or semiclassical
370 questions
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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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134
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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 - ...
- 583
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1
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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 ...
- 4,113
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124
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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:
$$H=\frac{p^2}{2 m}+\alpha x \text { with } 0<y<L, x>0$$...
- 21
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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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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 ...
- 569
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2
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498
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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 ...
user300251
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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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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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4
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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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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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771
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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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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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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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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 ...
- 2,052
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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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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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154
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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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WKB application on symmetric potential well
I am a little confused how one can find a wave function by using WKB approximation? I do know the oscillation frequency $$\Omega ~=~ {2E\over h}{\rm Re} \langle L|R \rangle~=~ {E\over \pi\hbar}{\rm Re}...
user288166
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Why the dipole interaction term in the Hamiltonian has all diagonal elements to be zero in the energy eigenbasis?
I have been studying the semi-classical light matter interaction from the book, "Light matter interaction" by Weiner and Ho. They have defined the total Hamiltonian of a two level atom ...
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Approximately sinusoidally varying Fourier transforms
Given a function $f(x)$ and its Fourier transform $\tilde{f}(k) = \int_{-\infty}^{\infty}f(x) e^{ikx}dx$, if I decompose the Fourier transform as
$$\tilde{F}(k) = A(k)e^{ikl(k)}$$
Under what ...
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Wavepacket for hydrogen atom?
We normally observe classical behaviour due to the time dependent schrodinger equation in simple quantum systems when we introduce 'Gaussian wavepackets' which have bell shaped uncertainty in energy, ...
user86425
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Problems deriving the Quantum Hamilton-Jacobi equation
This is my first question at Physics SE so please be kind. I am well versed in the etiquette over at Math SE, but not so much here. Anyway, I thought this question was better suited to this site ...
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337
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How does the linearity of the Schroedinger equation reflect the interactions?
There is a common lore that linear equations describe non-interacting systems, why non-linearities correspond to non-trivial interactions. My (loose) question is how is that compatible with the ...
- 1,594
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Superextremal galactic Reissner-Noström blackholes
To celebrate 10 upvotes but no attempt at answers of this 8-year old question regarding extremality of Reissner-Noström blackholes, Let's formulate a physically feasible scenario for driving a ...
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WKB and Virial Theorem contradiction in determining Bound States
Consider a potential that on the left of some point $x=x^*>0$ is infinite and on the right of that point it is of the form $$V(x)=-\alpha x^{-3}.$$
I tried to use the WKB method to determine the ...
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How does the energy splitting in a double well potential scale?
It is well known in quantum mechanics that in an 1D double well potential there is an energy splitting between the ground state $\psi_0(x)$ and the first state $\psi_1(x)$ despite them being nearly ...
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What is the meaning of "screening-out a perturbation" in a nanosurfaces context?
When reading my lecturer's notes on the Thomas-Fermi screening length, I am told that:
"The Thomas-Fermi screening length $\lambda_{TF}$ is a
rough guide to how rapidly an electron gas can
screen ...
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When is a logarithm of the wavefunction well-defined?
It is sometimes convenient to write the wavefunction as
$$
\Psi(x,t)~=~ e^{\Phi(x,t)}
$$
and then work with $\Phi$ instead. This is particularly sensible in the context of the WKB approximation, where ...
- 1,141
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57
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Semiclassical quantization of chaotic classical system
So far in the introduction of quantum chaos, I have read that in the early day's physicists quantized classically chaotic systems but could not find any signature of chaos in quantized systems. My ...
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Correct indices on coupling matrix elements
Lets say i have an initial state $|i\rangle$ and a final state $|f\rangle$. A transition from $|i\rangle \rightarrow |f\rangle$ is coupled by an operator $\hat O$. Is the relevant coupling matrix ...
- 1,616
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How is Electromagnetically-Induced Transparency a result of "destructive quantum interference between two pathways"?
It has been described that Electromagnetically-Induced Transparency (EIT) is a result of "destructive quantum interference between two pathways." To quote from this source:
In these simple ...
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Compute probability current from WKB approximation
I struggle to reproduce a calculation from the Appendix of the paper "Anharmonic Oscillator: A Study of Perturbation Theory in Large Order", Physical Review D, 7 (6) 1973, link to abstract.
...
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Does spin really have no classical analogue?
It is often stated that the property of spin is purely quantum mechanical and that there is no classical analog. To my mind, I would assume that this means that the classical $\hbar\rightarrow 0$ ...
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Solving the energy spectrum for a $\cot^2$ potential well using the WKB approximation
I am taking graduate quantum mechanics, and we are now discussing the WKB/semiclassical approximation. I am trying to solve a problem where I would need to find the energy spectrum of the 1D motion in ...
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Semiclassical approximation self-study
Can someone give a review of good books to learn semiclassical physics? Somehow, I would like to know if there is a text at the level of Lanczos or Gelfand. By this, I mean that I am interested in the ...
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On the computation of functionals in QFT
Using the Gaussian (path)-integral
$$
\int \mathcal{D}\eta e^{i\int_{t_i}^{t_f} dt \eta(t) O(t) \eta(t)} = N [\operatorname{det} O(t)]^{-1/2}
$$
my book claims that we can compute the following ...
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The connection between classical phase space and quantum multiplicity
I am aware of the relationship $N = V/h^n$ where $N$ is the quantum multiplicity, $n$ is the number of position (or momenta) degrees of freedom, $V$ is the volume of classical phase space and $h$ is ...
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WKB approximation difficulty - deciding what term to neglect
Consider the following quantum well:
Region 1 is a classically forbidden region, and hence the WKB wave-function will take the form of equation
$$\psi(x) = \frac{C}{\sqrt{q(x)}}e^{+\int_b^a q(x')dx'/\...
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Wigner Function and Spin in the Classical Limit?
This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system:
$$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n) $$
Now, let's say this system obeys has spin. As far ...
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When can I use Gaussian integration to compute a path integral?
In reading 14.4 of Gregory Moore's notes on abstract group theory, I was left with some questions on the computation he did of the path integral that may be general features.
Let consider a spacetime $...
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How to get Bohr model from Schroedinger equation? [duplicate]
New theories must in some approximation must reduce to older approximately correct theories.
In what approximation exactly does the angular momentum quantization condition from Schroedinger's equation ...
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378
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Can the WKB approximation be used for this double well potential?
For a positive constant $C$, the double well potential $V(x) = -C|x| $ exists between two infinitely high potential walls at $x=a$ and $x=-a$.
I wish to use the WKB approximation to obtain an equation ...
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497
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Direction of Unruh radiation
In case of a black hole, the direction of the Hawking radiation is from the horizon to the observer. The corresponding effect in the Rindler spacetime is the Unruh radiation.
Intuitively, a rapidly ...
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Approximating the energy levels of the anharmonic oscillator using WKB [closed]
I got stuck trying to solve this problem:
Given the potential $$V(x) = \frac{m\omega^2x^2}{2}-\beta x^4,\ \beta>0$$ I need to evaluate the deviation of the energy levels from the harmonic ...
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On the Application of the Einstein-Brillouin-Keller (EBK) Method
Consider the Born-Sommerfeld quantization condition (modified) [see Einstein–Brillouin–Keller (EBK)]
$$I_{i} = \frac{1}{2\pi}S_{i} = \frac{1}{2\pi}\oint p_{i} dq_{i} = \hbar \left(n_{i} + \frac{\mu_{...
- 1,359
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Why the classical configuration always static when applying saddle point (semi-classical) approximation?
For an Green function/partition function:
$$\int D[\phi]e^{\frac{i S[\phi]}{\hbar}}$$
We can make saddle point approximation and gives classical configuration:
$$\delta \mathcal{S}=0\Longrightarrow \...
- 1,652
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Confusions on expectation value for $\hbar$ going to zero
In Matthew D. Schwartz's QFT book, Chapter 28, the author claims when $\hbar \rightarrow 0$, the following equality (eq 28.4) holds:
So how can I see the second "$=$" holds? It seems the ...
- 1,035
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Stationary Phase approximation with multiple coordinates?
The stationary phase approximation can be used to find an approximate value for the path integral
\begin{equation}\int Dx e^{-S[x]} \approx e^{-S[\bar{x}]} \left(\det{\frac{\hat{A}}{2 \pi}}\right)^{-1/...
