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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WKB solution in QFT: classical action and particle vs antiparticle case
Consider the theory of a complex scalar field
$$S[\psi, \psi^\dagger] = -\int d^4x \left(\hbar \partial_\mu \psi^\dagger \partial^\mu\psi + \hbar^{-1} m^2 |\psi|^2\right)$$
giving the Klein-Gordon ...
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1answer
94 views
Noether's Theorem and the Measurement? (In Copenhagen)
Background
I'm trying to recently understand the Copenhagen interpretation. We all know due to Noether's theorem energy is translational invariance in time.
However, in quantum mechanics the ...
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1answer
79 views
Degenerate ground state of Hamiltonian from analytical perspective
Suppose I have a Hamiltonian that depends on the continuous vector parameter $\boldsymbol{\theta}$, and the ground state corresponds to line/plane or some other $1$ to $p-1$ dimensional subspace of ...
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29 views
Einstein–Brillouin–Keller quantization rule, what does it really mean?
The Einstein–Brillouin–Keller method is a quantization rule going from classical mechanics to quantum mechanics, according to wikipedia:
I have several question regarding the above description:
what ...
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29 views
Transmission coefficient of second-order WKB approximation
In WKB approximation we expand the action in powers of $\hbar$ $$S(x)=S_0(x)+\dfrac{\hbar}{i}S_1(x)+\left(\dfrac{\hbar}{i}\right)^2S_2(x)+...$$ In standard treatment only terms up to and including $...
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2answers
82 views
The link between discrete energy level in quantum mechanics and harmonic series in Acoustics
Consider a quantum square potential well with infinite depth:
$$
V(x)=\begin{cases}
0, &|x|<a \\
+\infty, &\text{otherwise}.
\end{cases}$$
Solving the Schodinger equation of a particle with ...
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1answer
98 views
Why are Grassmann variables the classical limit of fermions?
In many texts the anti-commutation relations for fermions are given as
$$\{ \bar{\psi}^\alpha (\vec{x}), \psi^\beta(\vec{y}) \} = \delta^{\alpha\beta} \delta(\vec{x} - \vec{y})$$
$$\{ \psi^\alpha (\...
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Bohr’s Quantisation Condition [duplicate]
I am a grade 12 student from India and my physics textbook does not delve deep in the bohrs quantisation condition but has given us a paragraph to figure out what it is:
“Consider Motion of an ...
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1answer
30 views
Show that angular momentum is independent of the angle
I was thinking about the Bohr atomic model which states, that the angular momentum (L) must be an integral multiple of the reduced Planck constant, this implies that $L=mvr$ must be constant for a ...
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1answer
122 views
Difference between QFT In curved spacetime, semiclassical, and quantum gravity?
Could someone describe the difference, qualitatively, between QFT in curved spacetime, semiclassical gravity, and quantum gravity? I know that each is an approximation to the next and the end goal is ...
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1answer
66 views
What was the origin of Bohr-Sommerfeld's quantization rules?
What made Bohr and Sommerfeld think momentum and angular momentum is quantized? What is the meaning of momentum quantization in harmonic oscillator? Can we imagine it (by some classical example)?
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Semiclassical limit $S \to\infty$ in spin model
In many literature, the limit $S \to \infty$ is considered as a semiclassical limit. My question is that when this approximation is valid? Since paticles, say electrons, have the fixed spin number $S=...
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2answers
94 views
Quantum corrections in path integral
I am working the following exercise:
Calculate the generating functional $$Z[j]=\int \mathcal{D}\Phi \exp\left(\frac{i}{\hbar}S[\Phi,j]\right),\quad S[\Phi,j]=\int d^4x(\mathcal{L}(\Phi)+j\Phi),$$ $...
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1answer
49 views
Validity of kinetic theory
I am reading this paper, which uses (chiral) kinetic theory.
The authors write:
The Boltzmann method is valid only in the semiclassical limit, where $\omega_B \tau \ll 1$, $\omega_B \ll \mu$ and $\...
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24 views
Domain of applicability of Quantum Correspondence Principle
I know Correspondence Principle states that (as wiki put it) that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in ...
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1answer
49 views
Connection formulas: Why do we assume asymptotic behavior of the Airy functions?
The derivation is obtained from Introduction to Quantum Mechanics by Griffiths
Let assume the turning point occurs at $x=0$, then the WKB solutions right and left to the turning point are:
$$
\psi=\...
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1answer
32 views
Relation of Corresponding principle and law of large numbers
Is it possible that Corresponding principle can be derived from the law of large numbers? Also is the principle a postulate of Quantum Mechanics?
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1answer
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What is wavelength at classical turning points using WKB Approximation? [closed]
According to what I know is that a classical turning point in Newtonian Mechanics is a point where a particle has a zero kinetic energy (Total energy is equal to potential energy) and must be ...
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38 views
Incompatibility between the Bohr-Sommerfeld quantization condition and the Dunham expansion
I'm trying to apply the RKR (Rydberg-Klein-Rees) method which computes the classical turning points, $a(E,J)$ and $b(E,J)$ of a diatomic molecule for a rotational-vibrational energy value E, and ...
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1answer
80 views
Volume of state in phase space free particle
I have to how a quantum state of a free particle between 0 and a occupies an area of $h$ in the phase space.
What I did was to calculate $\Delta x \Delta p$ and show that it was of order $h$, but I ...
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1answer
162 views
The WKB approximation and the Cotangent bundle
When we say (see pag. 9 of Lectures on the Geometry of Quantization) that
the image of the differential of the phase function lies in the level set of the classical Hamiltonian
is it simply ...
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WKB near turning point by means of complex integration (Landau & Lifshitz, Quantum Mechanics) [duplicate]
The question is basically about section 47. in Landau's Quantum Mechanics (non-relativistic theory)
everything is fine until the sentence (at the beginning of the second page) where he says: it is ...
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21 views
Semi-classical Hydrogen Atom Angular Momentum under Magnetic Field
Suppose we have a semiclassical hydrogen atom in its ground state at the x-y plane with the proton being at the origin. Let there be a magnetic field $\vec{B}=B\hat{z}$. By deriving the orbital ...
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Does Graviton bend light?
In the weak gravitational regime where the low-energy effective action holds,(linearized) gravity can be quantized, and we can treat graviton as quantum fields on a Minkowski background. One can also ...
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1answer
48 views
Why does a stable orbit of $H$ atom contain an integer number of de Broglie wavelengths? [duplicate]
I was trying to understand why a stable orbit of a hydrogen atom has to satisfy that the orbit length must be a multiple of de Broglie wavelength.
I have seen some related questions like This.
But ...
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WKB connection formulae from the path integral
The semiclassical, or WKB, approximation is one that is far more natural in the path integral formalism than it is when derived from the Schrodinger equation directly.
Furthermore, the connection ...
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2answers
142 views
Wilson-Sommerfeld Quantization of Dirac delta in Infinite Square Well (ISW)
I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$)
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2)
$$
using Wilson-Sommerfeld ...
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0answers
60 views
Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]
I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form
$$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$
where the interest is in the ...
3
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0answers
56 views
Difference between different approximations in QM and other definition of the integral
I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question:
It sounds maybe a bit strange ...
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26 views
Hannay's angle for a spin half interacting with a magnetic field that varies adiabatically over time
Does it make sense to talk about the Hannay's angle for a magnetic moment with spin 1/2?
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1answer
148 views
How to obtain the quantization of a simple pendulum using Bohr-Wilson-Sommerfeld rule? [closed]
How to obtain the quantization of a simple pendulum using Bohr-Wilson-Sommerfeld rule?
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Why do the matrix elements of an operator correspond to the Fourier components of the observable in Heisenberg's Matrix Mechanics?
It is well-known that Heisenberg $a$ began developing his Matrix Mechanics by creating matrix components
$$A(n,n-a,t)=A(n,n-a)e^{i\omega(n,n-a)t}$$
or
$$A_{nm}(t)=A_{nm}e^{\frac{i}{\hbar}\omega(nm)t}$$...
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1answer
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Are powers of the harmonic oscillator semiclassically exact?
The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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2answers
78 views
$\hbar \approx 0$ and the spread of QM wave function
Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\hbar \approx 0$? In that case the Ehrenfest ...
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173 views
The classical limit of QM as a Hamilton-Jacobi equation?
I'am having difficulties to understand the so-called classical limit in quantum mechanics. There is a popular method to transform the Schrödinger equation into two coupled equations that are the ...
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2answers
156 views
Negative probabilities with Wigner quasi-probability distributions
I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is:
$$P(x)= \text{e}^{-\beta V(x)}\left(...
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2answers
158 views
Klein Gordon equation in the nonrelativistic and semiclassical limit in a Wigner approach
I would like to analyse the semiclassical and nonrelativistic limit of the Klein-Gordon equation,
\begin{equation}
\frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \frac{M^2 c^2}{\hbar^2} \phi =0.
...
3
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2answers
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Stability of the Hawking-Hartle vacuum in semiclassical gravity
Consider a free quantum field theory defined upon a static Lorentzian spacetime possessing a bifurcate Killing horizon, such as Schwarzschild spacetime.
These assumptions are sufficient to define a ...
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Maslov's method of deriving the WKB approximation
For a generic one-dimensional potential, the WKB approximation yields the quantization condition
$$ \oint p dq = (n + 1/2)\hbar . $$
Here, the correction factor $1/2 $ was obtained by Kramers by ...
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1answer
111 views
Isn't Bohr's correspondence principle obvious?
I was taking an introductory course in quantum mechanics when I came across the Bohr's correspondence principle. According to Wikipedia, the correspondence principle states that the behavior of ...
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0answers
48 views
why is a Lagrangian submanifold a semi-classical state and not a classical state?
I read that the Lagrangian submanifold can be regarded as a semi-classical state when classical mechanics is formulated using symplectic geometry. Does anyone know why it would be a semi-classical ...
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2answers
106 views
Bohr quantization hypothesis [closed]
To explain Rydberg formula, Bohr have assumed somewhat general hypothesis which is applicable to various classical system. As far as I know he assumed that for any classical system with periodic ...
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3answers
601 views
Derivation question of WKB method
Quantum Mechanics (2nd Edition) by Bransden and Joachain contains the following passage:
Substituting (8.176) into (8.171), we obtain for $S(x)$ the equation
$$-\frac{i\hbar}{2m}\frac{\mathrm{d}^...
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Why is the semiclassical approximation of the abelian Chern-Simons theory exact?
I was told that in abelian Chern-Simons theory (say, with a general level matrix $K$), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the ...
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117 views
Energy values for particle in a box with strange potential
I am trying to analyze a particle in a box with a rather strange potential inside the box: $V(x) \propto x^{3/2}$
I've tried using the WKB approximation, but I get some strange results and I don't ...
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1answer
64 views
How to deal with negative potential in the WKB approximation?
I'm trying to model a system as being inside an infinite potential well with $V(x)=-ax^v$ where $a$ and $v$ are some positive real numbers.
However I'm a bit confused: if I take the - sign inside ...
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31 views
periodic orbits in Gutzwiller's trace formula
It is said that in the Gutzwiller trace formula, one sums over the periodic orbits.
I do not know how to derive the formula, but a simple question arises for me.
That is, for some classical ...
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111 views
High-field Hall effect and magnetoresistance (Ashcroft & Mermin: Solid State Physics)
I'm reading Mermin's Solid State Physics, chapter 12: The semiclasssical model of electron dynamics. I know the current density from the $n$ band is
$$ \mathbf{j}=(-e)\int_{\text{occupied}}f(\...
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3answers
527 views
How was Bohr's third postulate later found to be wrong?
Why does the Bohr's third postulate was later found to be wrong? I read it in a note but don't know why is it?
The third postulate is:
The orbits of electronic motion are circular and well ...
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297 views
WKB Connection Formula Clarification
From Griffiths, if we have some potential $V(x)$ and energy $E$ such that $E=V(0)$ where $V(x)<E$ for all $x<0$ and $V(x)>E$ for all $x>0$. In the patching region, Griffiths uses only one ...
