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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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
34 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
28 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
50 views
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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32 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
53 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
148 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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26 views
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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16 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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38 views
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
44 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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63 views
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
128 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)
$$
using Wilson-Sommerfeld ...
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57 views
Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]
I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form
$$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$
where the interest is in the ...
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52 views
Difference between different approximations in QM and other definition of the integral
I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question:
It sounds maybe a bit strange ...
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23 views
Hannay's angle for a spin half interacting with a magnetic field that varies adiabatically over time
Does it make sense to talk about the Hannay's angle for a magnetic moment with spin 1/2?
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1answer
111 views
How to obtain the quantization of a simple pendulum using Bohr-Wilson-Sommerfeld rule? [closed]
How to obtain the quantization of a simple pendulum using Bohr-Wilson-Sommerfeld rule?
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32 views
Why do the matrix elements of an operator correspond to the Fourier components of the observable in Heisenberg's Matrix Mechanics?
It is well-known that Heisenberg $a$ began developing his Matrix Mechanics by creating matrix components
$$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}$$...
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1answer
81 views
Are powers of the harmonic oscillator semiclassically exact?
The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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2answers
72 views
$\hbar \approx 0$ and the spread of QM wave function
Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\hbar \approx 0$? In that case the Ehrenfest ...
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154 views
The classical limit of QM as a Hamilton-Jacobi equation?
I'am having difficulties to understand the so-called classical limit in quantum mechanics. There is a popular method to transform the Schrödinger equation into two coupled equations that are the ...
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2answers
114 views
Negative probabilities with Wigner quasi-probability distributions
I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is:
$$P(x)= \text{e}^{-\beta V(x)}\left(...
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2answers
105 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.
...
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2answers
75 views
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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43 views
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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52 views
Feynman Path Integrals and Bohr-Sommerfeld Quantisation Condition [duplicate]
I'm currently learning about Feynman Path Integrals, and I came across the following paragraph:
"Periodic classical orbits will carry a complex phase which will in general average to zero over many ...
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1answer
91 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
43 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
96 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
571 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
$$-\frac{i\hbar}{2m}\frac{\mathrm{d}^...
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0answers
60 views
Why is the semiclassical approximation of the abelian Chern-Simons theory exact?
I was told that in abelian Chern-Simons theory (say, with a general level matrix $K$), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the ...
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111 views
Energy values for particle in a box with strange potential
I am trying to analyze a particle in a box with a rather strange potential inside the box: $V(x) \propto x^{3/2}$
I've tried using the WKB approximation, but I get some strange results and I don't ...
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1answer
56 views
How to deal with negative potential in the WKB approximation?
I'm trying to model a system as being inside an infinite potential well with $V(x)=-ax^v$ where $a$ and $v$ are some positive real numbers.
However I'm a bit confused: if I take the - sign inside ...
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0answers
28 views
periodic orbits in Gutzwiller's trace formula
It is said that in the Gutzwiller trace formula, one sums over the periodic orbits.
I do not know how to derive the formula, but a simple question arises for me.
That is, for some classical ...
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0answers
100 views
High-field Hall effect and magnetoresistance (Ashcroft & Mermin: Solid State Physics)
I'm reading Mermin's Solid State Physics, chapter 12: The semiclasssical model of electron dynamics. I know the current density from the $n$ band is
$$ \mathbf{j}=(-e)\int_{\text{occupied}}f(\...
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3answers
388 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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0answers
240 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 ...
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45 views
Multiple Classical Limits of a Quantum Theory [duplicate]
I recently learned that one of the many lessons that one can learn from the AdS/CFT correspondence is that there could be two classical limits (the bulk with gravity in $D+1$ spatial dimensions, and ...
2
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1answer
164 views
Time-reversal operator in Phase Space Representation
Consider the simplest possible case in which the time reversal operator $\hat{\mathrm{T}}$ is given by the operation of complex conjugation $\hat{\mathrm{K}}$.
We can view $\mathrm{T}$ is an anti-...
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65 views
Algebraic solution of problems in classical mechanics
We know from the theory of the quantum harmonic oscillator that the energy spectrum can be determined nearly effortlessly once we are aware of the simple algebraic structure.
In a certain sense, we ...
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1answer
203 views
Energy quantization in the path integral and the Fourier spectrum of the action
I offered a bounty on this question for a simple way to see that the Feynman path integral yields discrete energy levels for bound states, in one dimensional quantum mechanics. As shown there, there's ...
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0answers
42 views
Maslov correction N-dimensional harmonic oscillator with different frequencies
I am working on the semiclassical dynamics of a N-dimensional harmonic oscillator based on an autocorrelation function approach and I like to treat a system with independent oscillators, which have ...
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0answers
156 views
Using the correspondence principle, how does one show that in the classical limit, the expectation value of $H$ is the classical energy?
I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on:
So far, $H$ is only an abstract Hermitian operator in the equation $H\Psi = i\hbar\dfrac{\partial\Psi}{\...
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0answers
49 views
Correspondence principle for macroscopic orbits
According to the correspondence principle, quantum laws ought to reduce to classical ones in the limit of macroscopic bodies, right?
But I don't see how the probability clouds of electrons in ...
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1answer
390 views
On Groenewold's Theorem and Classical and Quantum Hamiltonians
I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping
$$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$
...
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3answers
428 views
If Bohr model is outdated and we know that there is no such thing as an “electron orbital circumference” then how is $2\pi r=n\lambda$ still valid?
We know that Bohr model is outdated and we know that there is no such thing as an "electron orbital circumference" then how is $2\pi r=n\lambda$ still valid?
Edit :
If the electrons for higher ...
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1answer
185 views
Quantum Mechanical Hamiltonians without Classical Analogues [duplicate]
Recently I found myself in a state similar to that which @senator found himself here. I too have been reading Dirac's Lectures on Physics and am particularly confused by the notion of Hamiltonians ...
2
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1answer
85 views
How I can see that everyday life systems behave classical (from QFT path integrals)?
If I would try to treat macroscopic systems consisting of a super-large number of particles (also when environment is included), I have to compute $2N$-point correlation functions with very large ...
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1answer
192 views
Classical Limit of Quantum Mechanics recovered from the Path Integral Formalism
From Zee's Quantum Theory in a Nutshell he explains how the classical limit of quantum mechanics can be recovered from the path integral formalism.
It can be shown that the path integral formalism is:...
5
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2answers
222 views
Deriving the Old Quantum Condition ($\oint p_i dq_i=nh$)
A body undergoing periodic motion in an orbit of quantum number $n$ will have a period $T$, determined by
$$T=\oint \frac{ds}{v}=\oint \frac{ds}{\sqrt{\frac{2}{m}(E-V)}}$$
Where $ds$ is an ...
