Questions tagged [deformation-quantization]
A description of quantum mechanics in phase space a common ambit with classical mechanics, through the Wigner map from Hilbert space. May be used to address Quantum Mechanics in phase space, the star product binary operation controlling composition of observables, and Wigner, Husimi, and other distribution functions in phase space.
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"Deriving" Poisson bracket from commutator
This Q/A shows that deriving P.B.s from commutators is subtle. Without going into deep deformation quantization stuff, Yaffe manages to show that $$\lim_{\hbar \to 0}\frac{i}{\hbar}[A,B](p,q)=\{a(p,q),...
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Poisson Bracket and commutators in quantum mechancs [duplicate]
how did they reach the conclusion that quantization of the Poisson brackets
$ (A,B) $ was equal to the commutator $ \frac{1}{i\hbar}[A,B] $
in quantum mechanics?
so the quantum equations of motion ...
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On the Born-Jordan quantization being an equally weighted average of all operator orderings
On my way studying quantization schemes, I came across the expression saying that the Born-Jordan quantization rule is the equally weighted average of all the operator orderings and that the Weyl's ...
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$ \hbar^2$ Correction to the Bohr-Sommerfeld Quantization Condition
We can get the Bohr-Sommerfeld quantization from the WKB method as answered. Since we use approximation, there should be an error in the system, I know this is not right all the time; in some ...
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Operating with the Weyl transform on a wave function
I'm very new to studying quantum mechanics in phase space, so I'm trying to demonstrate some results that I see in books to get used to the formalism. I recently got stuck when i applied the Weyl map $...
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Why is $[ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}$?
If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by
$$ [ \hat{A},\hat{B} ] \...
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Quantization of $x$ and $p$ through the Weyl transformation
I have a question about the development of the integral for calculating the quantization of the classical variables $x$ and $p$ using the Weyl transformation method. The notation that the textbook I'm ...
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Weyl Quantization Integral
I have some doubts when calculating the integral for Weyl Quantization symbol.
If I understand correctly, quantization using the Weyl symbol takes a function in phase space and takes it to an operator ...
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What does it mean for an operator to depend on position or momentum?
While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
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Lie algebra with $\sim \!N^3$ generators [closed]
Is there a Lie algebra whose number of generators scales as $N^3$, or in general $N^p$ with $p$ an arbitrary positive integer?
All the familiar examples, such as $\mathrm{U}(N)$ or $\mathrm{SU}(N)$ or ...
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Elastic potential energy formula
From the Wikipedia page on elastic energy, we can find a bunch of formulas to describe it. For example, in the continuum section it talks about energy per unit of volume (density?):
$U=\dfrac{1}{2}C_{...
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Classical limit of Moyal bracket in integral representation
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$
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Does geometric quantization work for arbitrary "particle with constraint + potential" systems?
I was struck by the following line in Hall's Quantum Theory for Mathematicians (Ch. 23, p. 484):
In the case $N = T^*M$, for example, with the natural “vertical” polarization, geometric quantization ...
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Is there a relationship between the phase space path integral and phase space quantum mechanics?
I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space?
I also ...
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Commutator Constant
I have seen a lot of commutators in quantum mechanics having a constant factor $i\hbar$. I have read about Dirac supplanting Poisson Brackets with commutators having a constant $i\hbar$.
I want to ...
