All Questions
Tagged with semiclassical classical-mechanics
58 questions
58
votes
6
answers
12k
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Tree-level QFT and classical fields/particles
It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree-level cross-section for electron-electron scattering ...
Squark
38
votes
7
answers
7k
views
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 $\hbar\rightarrow 0$ ...
- 2,494
19
votes
3
answers
4k
views
How does one quantize the phase-space semiclassically?
Often, when people give talks about semiclassical theories they are very shady about how quantization actually works.
Usually they start with talking about a partition of $\hbar$-cells then end up ...
- 970
18
votes
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
13
votes
3
answers
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views
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:
$$
m\partial_{t}\langle \hat{X}\...
- 941
13
votes
2
answers
5k
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Classical Limit of the Feynman Path Integral
I understand that in the limit that $\hbar$ goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...
- 951
12
votes
1
answer
512
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How do I know how many classical limits (if any) a given quantum theory is going to have?
I was reading this, where it is mentioned that some quantum theories can have no classical limit or even more than one classical limit.
A possible example might be quantum spin, which doesn't have a ...
- 5,287
7
votes
1
answer
676
views
What happens to branching in the Many-Worlds Interpretation of quantum mechanics in the limit when Planck's constant goes to 0?
We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why ...
6
votes
2
answers
1k
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Change of variable in Schrödinger's first paper
On the first page of his first paper in series "Quantization as an Eigenvalue Problem", Schrödinger begins with
$$H(q, \frac{\partial S}{\partial q})=E$$
and then takes a change of ...
- 1,419
6
votes
2
answers
2k
views
Why does the classical path give the dominant contribution in the path integral?
Why is it that the classical path gives the dominant contribution in the quantum mechanical path integral? How do we understand this?
- 27.2k
5
votes
1
answer
305
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Classical limit of quantum mechanics in terms of center of mass
When we say that quantum mechanics reproduces classical mechanics at macroscopic scales, is it a statement about the center of mass of a macroscopic system?
More specifically, let $\psi (x)$ be the ...
- 959
5
votes
1
answer
773
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Classical limit in quantum mechanics proof
Several questions are about the limit $\hbar\rightarrow 0$, e.g.
When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?
Classical ...
- 61
4
votes
4
answers
1k
views
Classical Limit in Quantum Mechanics
Suppose I have a wave function $\Psi$ (which is not an eigenfunction) and a time independent Hamiltonian $\hat{\mathcal{H}}$. Now, If I take the classical limit by taking $\hbar \to 0$ what will ...
- 399
4
votes
1
answer
379
views
In what sense a path integral can be approximated by the classical contribution $\exp{[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}}]$?
People often say that the amplitude $K(b,a)$ to go from $a$ to $b$ can be approximated by $$K(b,a) \sim \exp{\left[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}(b,a)\right]},\tag{1}$$
where $S_{\text{cl}}(b,a)...
- 349
4
votes
1
answer
801
views
Classical mechanics from Quantum mechanics
I'm looking at a way to prove that one recovers, under ad hoc assumptions, classical mechanics from quantum theory. Usually, we can find in textbooks that the propagator
$$K(x,x_0;t)=\langle x|e^{-i ...
- 12.1k
4
votes
1
answer
214
views
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
3
votes
3
answers
1k
views
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 ...
- 31
3
votes
4
answers
3k
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How can I use a magnet to lift a paperclip?
This question has been asked already, but no satisfying answer came up. All the answers seem to push the problem further, but do not explain clearly what is going on.
I will reformulate it for a ...
- 5,270
3
votes
1
answer
315
views
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
3
votes
4
answers
270
views
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?
- 592
3
votes
1
answer
167
views
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. ...
- 4,276
3
votes
1
answer
2k
views
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
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
3
answers
269
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 ...
- 6,666
3
votes
2
answers
259
views
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 ...
3
votes
1
answer
58
views
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 ...
3
votes
0
answers
107
views
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{\...
3
votes
4
answers
255
views
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
views
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 ...
- 131
2
votes
1
answer
155
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 ...
- 3,518
2
votes
1
answer
150
views
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 ...
- 519
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
2
votes
2
answers
125
views
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 ...
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:
$$P_{clas}=\frac{1}...
2
votes
0
answers
85
views
Additional Examples of Quantum and Classical Analogs?
I was wondering if there are any other important classical/quantum analogs, along the lines of these examples:
Schrödinger Equation $\leftrightarrow$ Hamilton-Jacobi Formalism
Path Integrals $\...
- 429
2
votes
0
answers
70
views
Why are classical equations of energy used to derive equations in Quantum Mechanics? [duplicate]
I read before that one can't derive a more fundamental theory, and Quantum Mechanics is a more fundamental theory than classical physics.
I understand the equation
$$E=\frac{1}{2}mv^2+U$$
to be a ...
- 2,021
1
vote
3
answers
4k
views
Why do we obtain classical physics by taking the limit of Planck's constant to zero?
Why if we specifically set Planck's constant equal to zero (the limit of it) do we sometimes get classical physics? I mean, what does it mean physically to set the constant equal to zero? Or to say it ...
- 7,860
1
vote
1
answer
163
views
Is Gravitational Red shift equal to $mgh$
Is gravitational shift - $\frac{gh}{c^2}$ (according to pound-rebka experiment) always equal to $PE=mgh$? because assume the gravitational pull, $g$, is equal to $1$ then we can say $g = 1$ similarly ...
user43495
1
vote
1
answer
403
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:...
- 895
1
vote
1
answer
299
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 ...
- 147
1
vote
1
answer
146
views
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
1
vote
1
answer
3k
views
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.
...
- 281
1
vote
2
answers
119
views
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
1
vote
0
answers
84
views
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
1
vote
0
answers
90
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 ...
- 6,187
1
vote
0
answers
84
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 ...
- 1,080
1
vote
0
answers
350
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}{\...
- 2,984
0
votes
1
answer
484
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 ...
- 435
0
votes
4
answers
246
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 ...
- 61
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