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.
302
questions
7
votes
2
answers
528
views
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 ...
- 481
0
votes
2
answers
39
views
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 ...
0
votes
0
answers
37
views
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 ...
- 906
2
votes
0
answers
44
views
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 ...
- 21
0
votes
0
answers
31
views
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
$$
\dot{\rho}=-\frac{i}{\hbar}[H, \rho]+\sum_{i=1}^{N^{2}-1} \gamma_{i}\left(L_{i} \rho L_{i}^{\dagger}-\frac{1}{2}\left\{L_{i}^{\dagger} ...
- 135
0
votes
1
answer
42
views
In relation to the correspondence principle, what happens when the orbital magnetic quantum number $m_\ell$ is very large?
If for each value of the orbital quantum number $\ell$ there are $2\ell+1$ possible associated magnetic quantum numbers $m_\ell$, and they are interpreted as the only allowed orientations that the $...
- 113
1
vote
0
answers
61
views
What is the meaning/justification for the matching conditions in the WKB approximation?
Consider the time-independent Schrodinger equation in one dimension with a potential $V(x)$ at fixed energy $E$. In the WKB approximation, we obtain solutions in the classically allowed region (i.e. ...
- 31
0
votes
0
answers
26
views
What is the difference between Bohr-Sommerfeld and Wilson-Sommerfeld quantisation rule?
Is Bohr-Sommerfeld and Wilson-Sommerfeld quantisation rule the same thing? If not, what is the difference?
- 1
1
vote
0
answers
38
views
Why is the time evolution of occupied and unoccupied levels in a band under the influence of applied fields the same?
I am studying the semiclassical model for solid state physics as described in Chapter 12 of Ashcroft & Mermin. While I am familiar with the concept of holes, a formal argument is made in this ...
1
vote
0
answers
51
views
Help to evaluate an integral given in appendix of Quantum Field Theory in a Nutshell [duplicate]
On p. 16 in appendix 3 in section I.2 of Quantum Field Theory in a Nutshell by Zee the integral to be evaluated is $$I = \int_{-\infty}^{+\infty}dqe^{-(1/\hbar)f(q)}.$$
Where $f(q)$ is expanded as
$$...
- 35
2
votes
0
answers
66
views
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 $\...
- 411
4
votes
1
answer
157
views
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 ...
- 917
2
votes
1
answer
52
views
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 $k$...
- 219
3
votes
1
answer
84
views
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 ...
- 1,273
1
vote
0
answers
66
views
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
\...
- 25
1
vote
1
answer
61
views
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 ...
- 2,853
1
vote
1
answer
45
views
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 ...
- 450
6
votes
1
answer
142
views
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 $\frac{\delta S}{\delta\phi}=0$. ...
- 549
1
vote
1
answer
39
views
"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, ...
- 911
17
votes
8
answers
4k
views
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" ...
- 189
2
votes
3
answers
125
views
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 ...
- 2,853
0
votes
0
answers
11
views
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 ...
- 701
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:
$$``\text{Energy of classical spacetime + expected energy of quantum ...
- 549
5
votes
1
answer
104
views
Is there a (semiclassical) electric field operator?
So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator
$$ \hat{\rho}(r) = q \...
user312272
3
votes
4
answers
218
views
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 ...
- 34.4k
2
votes
0
answers
63
views
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 ...
- 1,826
1
vote
0
answers
59
views
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 ...
- 38.3k
0
votes
0
answers
28
views
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 $...
- 754
1
vote
0
answers
80
views
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 - ...
- 521
7
votes
1
answer
123
views
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 ...
- 3,731
2
votes
0
answers
75
views
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
2
votes
1
answer
29
views
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,...
- 21
4
votes
0
answers
96
views
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 ...
- 457
3
votes
2
answers
449
views
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
2
votes
0
answers
20
views
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 ...
- 135
1
vote
0
answers
46
views
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 ...
- 11
1
vote
0
answers
51
views
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 ...
- 47
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 ...
- 51
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 ...
- 51
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 ...
- 17.1k
4
votes
0
answers
60
views
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 ...
- 41.3k
0
votes
0
answers
31
views
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 ...
0
votes
1
answer
38
views
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 ...
- 177
0
votes
0
answers
28
views
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 ...
- 1
0
votes
1
answer
74
views
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 $\...
- 1,529
1
vote
0
answers
21
views
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 ...
- 1,667
1
vote
0
answers
56
views
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=...
- 679
0
votes
1
answer
73
views
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 ...
- 633
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 ...
- 2,282
0
votes
1
answer
147
views
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?
- 19