All Questions
Tagged with semiclassical classical-mechanics
58 questions
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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 ...
1
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2
answers
119
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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
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 ...
3
votes
1
answer
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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, ...
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2
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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
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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:
$$P_{clas}=\frac{1}...
0
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2
answers
150
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How Quantum Mechanics reconciles with Classical Mechanics?
Imagine we have to charged particles. The kinetic energy of the system is:
$$
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
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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$. ...
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2
votes
2
answers
243
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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 ...
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2
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2
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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 ...
3
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0
answers
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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{\...
0
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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$ ...
3
votes
2
answers
259
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Does The Classical Limit of Quantum Computing Exists?
In any standard college book on quantum mechanics and field theory, one for sure has encountered some expressions like " the classical limit corresponds to setting hbar to zero" or quantum ...
4
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1
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In what sense a path integral can be approximated by the classical contribution $\exp{[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}}]$?
People often say that the amplitude $K(b,a)$ to go from $a$ to $b$ can be approximated by $$K(b,a) \sim \exp{\left[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}(b,a)\right]},\tag{1}$$
where $S_{\text{cl}}(b,a)...
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3
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4
answers
255
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Does quantum mechanics require classical mechanics for its own formulation?
Is true that quantum mechanics requires classical mechanics (as a limiting case) for its own formulation?
user343397
2
votes
1
answer
158
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Is the correspondence principle intentionally a physical statement regarding perturbation theory?
As a new physics reader I accidentally stumbled across perturbation theory as an idea to relate the modeling tactics I see repeatedly employed in
Statistical mechanics to approximate molecular motion ...
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3
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Why does the expectation value in quantum mechanics correspond to the classically measured value?
I understand that we can use the Heisenberg picture to show, for a Hamiltonian of the form
$$
\hat{H}=\frac{\hat{P}^{2}}{2m}+\hat{V}(\hat{X})
$$
the Ehrenfest theorem:
$$
m\partial_{t}\langle \hat{X}\...
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1
vote
0
answers
84
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Understanding the phrase "Classical mechanics corresponds to the high frequency limit of quantum mechanics"
Recently I have taken an interest in mathematical physics and as my background is mostly in math itself, I have quite a lot of catching up to do regarding my knowledge of physics. One phrase that I ...
- 149
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2
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600
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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 ...
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1
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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 $...
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2
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0
answers
85
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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 $\...
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5
votes
1
answer
305
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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 ...
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3
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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 ...
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2
votes
0
answers
70
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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 ...
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votes
4
answers
246
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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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5
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1
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773
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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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38
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7
answers
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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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2
votes
1
answer
150
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Why the action is taking phase in considering Huygens principle in matter waves?
From Dirac's remarks
$$\langle x_2,t_2|x_1,t_1\rangle=\exp\left[ \frac{i\int_{t_1}^{t_2}\mathrm dt\, L_{\text{classical}}{\left(\dot{x},x\right)}}{\hbar}\right].$$
How can I conclude from Huygens ...
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How can the Hamilton-Jacobi equation represent the motion of a particle as a wave? [duplicate]
While it’s often said that the Schrödinger Equation cannot be derived and “came from the mind of Schrödinger”, a quick google search led me to Schrödinger’s original paper where he introduced the ...
- 2,173
1
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0
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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 ...
- 6,187
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1
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93
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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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0
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1
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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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3
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1
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167
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When do you use Quantum Mechanics? [duplicate]
Given a problem, how does one know whether to use quantum mechanics or classical mechanics?
Take for example electron scattering from a nucleus. The electrons are given a wavefunction in this case. ...
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0
votes
0
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610
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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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1
vote
1
answer
299
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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 ...
- 147
0
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0
answers
49
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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 ...
user87745
1
vote
0
answers
84
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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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0
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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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0
answers
71
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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 ...
- 2,984
0
votes
1
answer
484
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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 ...
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2
votes
1
answer
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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 ...
- 3,518
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1
answer
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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:...
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1
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1
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Quantum Anomalies: Is there a way to show that we recover a classical symmetry that does not exist quantum mechanical in the classical limit?
Quantum Anomalies: Is there a way to show that we recover a classical symmetry that does not exist quantum mechanical in the classical limit?
From undergraduate quantum mechanics, I know that we ...
- 7,860
4
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1
answer
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Example of a quantum-mechanical theory with nontrivial classical limit
I am looking for a toy model example of a well defined quantum-mechanical theory with the following properties:
It can be constructed via canonical quantization starting from some classical theory ...
- 16.3k
6
votes
2
answers
2k
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Why does the classical path give the dominant contribution in the path integral?
Why is it that the classical path gives the dominant contribution in the quantum mechanical path integral? How do we understand this?
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3
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3
answers
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Transition from quantum to classical mechanics
As I understand it, if $S \gg h$ then we are in the classical realm, whereas if $S \leq h$ we are in the quantum realm. My question is what happens somewhere in between those 2 limits? Are we quantum ...
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1
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1
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Sommerfeld's modifications to Bohr's atomic model
While I was reading the derivation of Sommerfeld's model it was stated in the book that an electron moving in ellipse has Two degrees of freedom namely the radial distance $r$ and the azimuthal angle.
...
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How do I know how many classical limits (if any) a given quantum theory is going to have?
I was reading this, where it is mentioned that some quantum theories can have no classical limit or even more than one classical limit.
A possible example might be quantum spin, which doesn't have a ...
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5
answers
807
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Can I swap quantum mechanical ground state for some classical trajectory distribution and have it sit still after the swap?
Suppose that I have a single massive quantum mechanical particle in $d$ dimensions ($1\leq d\leq3$), under the action of a well-behaved potential $V(\mathbf r)$, and that I let it settle on the ground ...
- 135k
3
votes
4
answers
270
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When to use Quantum Mech.? [closed]
Is there any parameter (in terms of physical quantities such as mass, length, charge...) which can be used to decide when to treat a system quantum mechanically and not classically?
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