All Questions
Tagged with wkb-approximation or semiclassical
370 questions
3
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Deriving classical trajectories from quantum mechanics
A paper [1] by David Wallace contains a brief description of how classical trajectories emerge from quantum mechanics. I've summarised the relevant parts below:
Wallace says that decoherence lets us ...
0
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0
answers
17
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Is this derivation of the WKB approximation ruined by both principal and non-principal roots of $-i$ appearing in different steps? [closed]
In the first screenshot of pg 203 of my textbook below, discussing the turning point conditions in the WKB approximation in quantum mechanics, you can see they state
$i(-i)^{1/3} = -1$
which implies ...
- 11
0
votes
0
answers
34
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WKB tunneling probability calculations
I'd like to calculate the WKB tunnelling probabilities from different vibrational levels. I hope the question is not too chemical.
I have a compound which undergoes a chemical reaction. For the ...
- 1
2
votes
1
answer
67
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Question about the bulk theory in the 't Hooft limit of the AdS/CFT correspondence
In the AdS/CFT correspondence we can take the 't Hooft limit in which the theory in the bulk becomes classical supergravity, $g_s → 0$ and $α'/L^2 → 0$.
My question is, how classical is this bulk ...
- 1,809
2
votes
0
answers
49
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Is there a Bohr-Sommerfeld quantization condition for fields?
is there a Bohr-Sommerfeld quantization condition avaliable for Fields and not for particles?
for example for particles we have that $ n_i h = \oint p_i.dq_i .$
but the same for Fields with a field $ \...
- 65
0
votes
2
answers
65
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Action as the complex phase of the wave function of quantum mechanics
I heard that the action of classical mechanics can be seen as the complex phase of the wave function of quantum mechanics
$$\psi=\rho \exp\left(\frac{iS}{\hbar}\right)\tag1$$
I am more familiar with ...
- 904
2
votes
2
answers
161
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Is there a difference between saddle-point and steepest descent methods?
In this paper there is a clear distinction between steepest descent method and saddle-point approximation method. Eg. in the first page it says:
Indeed, Picard-Lefschetz theory establishes that a ...
- 814
0
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0
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36
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What is the thermal radius of the universe's "horizon"?
I have repeatedly come across the statement that every time there's a horizon (could be an event horizon of a black hole, or a Rindler horizon associated with acceleration), the vacuum state differs ...
- 7,455
2
votes
1
answer
109
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Should the quantum and classical parts of the Maxwell-Liouville equation be solved self-consistently?
Should the Maxwell-Liouville equation be solved self-consistently or is that unnecessary?
In semi-classical theory for nonlinear optics it is common to treat the electromagnetic field $E(r,t)$ ...
- 1,616
2
votes
2
answers
159
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WKB Approximation of the Quasinormal Mode Spectrum of the Poschl-Teller (PT) Potential
In Black Hole Spectroscopy, it is well known that the Pöschl-Teller (PT) potential behaves approximately, or similarly to the more complicated Regge-Wheeler (RW) Potential.
The WKB Approximation has ...
- 481
18
votes
3
answers
3k
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Is there a second-order non-linear addition to Maxwell's equations?
Maxwell's equations are famously linear and are the classical limit of QED. The thing is QED even without charged particles is pretty non-linear with photon-photon interaction terms. Can these photon-...
2
votes
0
answers
62
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How can we calculate simple quantum tunneling processes from the path integral?
I've been reading through Altland and Simons' Condensed Matter Field Theory, and am confused a bit by their discussion on tunneling and instantons. However I don't quite understand how this relates to ...
- 21
0
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0
answers
28
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dipole-radiation in semiclassical dynamics solid state
Using the semiclassical dynamics in solid state physics (electrons on a lattice with periodic potential, constrained to a band structure), we usually obtain that in the presence of external fields (...
- 51
1
vote
1
answer
97
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"Deriving" Poisson bracket from commutator
This Q/A shows that deriving P.B.s from commutators is subtle. Without going into deep deformation quantization stuff, Yaffe manages to show that $$\lim_{\hbar \to 0}\frac{i}{\hbar}[A,B](p,q)=\{a(p,q),...
- 822
1
vote
2
answers
118
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Why Normalise by $h$ in the Partition Function for Classical Harmonic Oscillator?
I was wondering if anyone could explain the reasoning behind the $h$ normalization constant when calculating the partition function for a classical harmonic oscillator.
I know that the partition ...
- 31
6
votes
0
answers
89
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Classical limits of Quantum Electrodynamics?
Quantum Electrodynamics is the theory that studies the interactions between matter and radiation (somewhat).
How would one explain for example the movement of an electron in a constant electric field ...
- 330
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2
answers
77
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Resource for WKB approximation formula
Is there any source that explicitly writes down the WKB "function" (to be defined soon) in orders of time derivative of the frequency over the frequency? Of course only to some finite order.
...
7
votes
1
answer
676
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What happens to branching in the Many-Worlds Interpretation of quantum mechanics in the limit when Planck's constant goes to 0?
We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why ...
0
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0
answers
27
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Stimulated emission semiclassical model for atom recoil
In the context of Saturated absorption spectroscopy, I'm having trouble modeling stimulated emission, and getting the result that is written in articles, such as this article. I tried to use a non-...
- 143
4
votes
2
answers
553
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Making sense of stationary phase method for the path integral
I am trying to understand this paper/set of notes. I have already seen the following related question: Does the stationary phase approximation equal the tree-level term? but had some trouble following ...
- 3,992
-1
votes
2
answers
190
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The question about commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ seemingly can't match with Poisson bracket $\{x,\,p\}=1$ [duplicate]
At the limit $\hbar\rightarrow 0$, all "quantum" should tend to "classical", but why is the quantum commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ equal to $0$, but ...
3
votes
1
answer
315
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Classical formulation of mechanics applied to Quantum Mechanics
According to Ehrenfest's theorem, the expectation values of observables such as position ($x$), momentum ($p$), etc. behave not only in a deterministic way but in fact in a classical way. Therefore, ...
- 1,880
4
votes
2
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129
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Semi-classical Quantum Ping-Pong in an infinite well potential
The general one particle state in a simple infinite well of size $L$ is a superposition of all the Hamiltonian eigen-states:
$$\tag{1}
\psi(x, t) = \sqrt{\frac{2}{L}} \sum_{n = 1}^{\infty} c_n \, e^{-\...
- 7,687
2
votes
1
answer
110
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Relation between the wavelength and the particle-wave duality
I will go straight into an example. Let's take the case of an electron of mass $m$ confined in an infinite 1D box of width $a$. Solving the Schrödinger equation and pay attention to the boundary ...
- 123
6
votes
2
answers
1k
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Change of variable in Schrödinger's first paper
On the first page of his first paper in series "Quantization as an Eigenvalue Problem", Schrödinger begins with
$$H(q, \frac{\partial S}{\partial q})=E$$
and then takes a change of ...
- 1,419
2
votes
2
answers
179
views
Classical limit of quantum harmonic oscillator
I have read that if in the quantum harmonic oscillator, $n$ is very large, then the probability density is similar to the classical one.
In the case of a simple harmonic oscillator:
$$P_{clas}=\frac{1}...
0
votes
2
answers
150
views
How Quantum Mechanics reconciles with Classical Mechanics?
Imagine we have to charged particles. The kinetic energy of the system is:
$$
T = \frac{1}{2}(m_1 + m_2) \mathbf{\dot{R}}_{cm}^2 + \frac{1}{2} \mu \dot{R}^2 + \frac{L^2}{2 \mu R^2}
$$
and its ...
- 1,573
3
votes
1
answer
211
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$ \hbar^2$ Correction to the Bohr-Sommerfeld Quantization Condition
We can get the Bohr-Sommerfeld quantization from the WKB method as answered. Since we use approximation, there should be an error in the system, I know this is not right all the time; in some ...
- 185
3
votes
1
answer
168
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The classical limit of quantum mechanics through Ehrenfest's theorem
Consider Ehrenfest's theorem:
\begin{align}
m\frac{d\langle x\rangle}{dt}=\langle p\rangle \\
\frac{d\langle p\rangle}{dt}=-\langle V'(x)\rangle.
\end{align}
Suppose $V(x)=x^2+x^{n+1}$ where $n>1$. ...
- 786
3
votes
0
answers
130
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Double-well potential and non-perturbative energy splitting
(A reference for the topic is a QFT note (chapter 2
Instantons in Quantum Mechanics) here by Yoichi Kazama at University of Tokyo, see page 30)
Consider the double well potential in quantum mechanics,
...
- 345
2
votes
2
answers
243
views
WKB method as a Semiclassical Approach
A naive question about WKB approach. It is dubbed to be a "semiclassical" method. What is precisely mean in quantum mechanical context to be "semiclassical"? Wikipedia states that ...
- 1,555
1
vote
0
answers
35
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Question about semiclassical approach to QCD
I'm struggling to understand the usefulness of the semiclassical approach to QCD. In particular, by using this approach, we can analyze the vacuum structure of QCD, including theta-vacua, $n$-vacua, ...
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0
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28
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In which cases does the action obey $\frac{\partial S}{\partial t}=-E$? [duplicate]
I'm reading https://web.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf and the article states that there are cases where the action obeys $\frac{\partial S}{\partial t}=-E$. Is ...
- 1
0
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2
answers
128
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Motion/momentum of a wave packet
I'm reading Dirac's "Principles of quantum mechanics" right now, being a little confused about the following part: (Chapter V$\S$31: "The Motion of Wave Packets").
He's making the ...
- 255
3
votes
1
answer
106
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Asking for explanation of Einstein's critique of the non-invariance of Bohr-Sommerfeld quantization
I am looking to understand better what problem might come from the claimed non-invariance of the Bohr-Sommerfeld quantization, which Einstein criticizes in his article On the Quantum Theorem of ...
- 149
4
votes
0
answers
104
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What do the authors of the paper mean here exactly by path integral?
First of all, please forgive me if i am asking a dumb question. I don't have a physics background. I was reading this paper by Hawking & Hertog on populating string theory landscape and came ...
- 67
0
votes
2
answers
145
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Confusion with Bohr-Sommerfeld quantization condition
The Bohr-Sommerfeld quantization condition reads $$\oint p(x)dx=(n+1/2)h,$$
where $n=0,1,2,\ldots$ and in general, the RHS is nonzero. Now,
$$\oint p(x)dx=\int_{x_1}^{x_2}p(x)dx+\int_{x_2}^{x_1}p(x)dx=...
- 12.3k
2
votes
1
answer
62
views
Intution for the physical meaning of high energy limit of a quantum states and uniform distribution in phase spacehow of a particle
Zeev Rudnick state in his talk Quantum Ergodicity for the Uninitiated (around 12 minute 40 second mark at the last text section of the slide) that a "a possible interpretation of the statement ...
- 125
2
votes
2
answers
125
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Resource on quantum to classical
I am looking for a book/paper which derives classical mechanics starting from quantum mechanics, to better understand the transition. Expected level of mathematical rigour is equivalent to graduate ...
1
vote
0
answers
45
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Mesons as a two-body problem is semiclassical QCD?
In particle physics and quantum field theory, mesons are interpreted as a system composed of a quark and an anti-quark, and the color charge of both must be at each opposite moment (green/anti-green, ...
- 1,670
3
votes
1
answer
258
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The WKB approximation and boundary conditions
To estimate a quantization rule using the WKB approximation, one is usually working in the 'classically allowed region'. You apply the WKB approx in the middle of the region and use the Airy ...
- 472
1
vote
1
answer
86
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Semiclassic limit of a QFT in Zinn-Justin
I am reading the Zinn-Justin book "Quantum Field Theory and Critical Phenomena" and i have come across a perplexing point.
Given the partition functional, in Euclidean QFT:
$$Z[J, \hbar] = \...
- 1,853
2
votes
1
answer
375
views
How to use saddle point approximation with path integrals?
i would like to evaluate $$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$
and it is my understanding that the way to do so is using the saddle point ...
- 31
3
votes
0
answers
107
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Meaning of equations associated with Legendre transform
In the famous paper about semiclassical Bloch theory https://arxiv.org/abs/cond-mat/9511014, the Lagrangian
\begin{eqnarray}
L (\mathbf{k},\dot{\mathbf{k}}) = -e \delta \mathbf{A}(r,t)\cdot\dot{\...
1
vote
1
answer
259
views
Classical limit of Moyal bracket in integral representation
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$
...
3
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0
answers
95
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Can the WKB approximation be used to get eigenenergies for negative potential 'barriers'?
I recently took a course that discussed the WKB approximation for linear potential. In class and in the exercises we only looked at pretty simple potentials that are just a constant times |x|. What I ...
- 31
1
vote
1
answer
117
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How limiting a potential affects the energy eigenvalues?
I am asking myself, how constraining a potential changes the energy eigenvalues.
With the WKB-Approximation-Method one can derive that the dependence of the eigenenergies regarding a potential $V(x) \...
- 57
2
votes
0
answers
59
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All closed orbits in semiclassical model
I'm studying from "Solid State Physics" by Ashcroft-Mermin. In particular, in chapter 12 it talks about the semiclassical model and tries to reason about the Hall effect in the limiting case ...
- 693
2
votes
0
answers
92
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Semi-classical limit of Feynman path integral
I am reading Blau's note on The Path Integral Approach to Quantum Mechanics. I am troubled for the derivations of semi-classical limit of Feynman path integral, which is located on Page.50 of Blau's ...
- 1,461
0
votes
1
answer
273
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How Feynman's path integral lead to least action principle? Math proof needed [duplicate]
I have read about Feynman path integral which leads to classical limit.
It said that because $\hbar \rightarrow 0$ in classical view. The function of path integral $\int e^{\frac{1}{\hbar}f(x)} dx$ ...
