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Reconciling the expression for the Wigner function involving $\langle x+\xi/2|\rho|x-\xi/2\rangle$ with the one using the characteristic function

Let me slightly rephrase the question as follows: how to switch between the definition of Wigner function as $$W(x,p) = \int_{\mathbb R}\frac{d\xi}{2\pi} e^{-ip\xi} \langle x+\frac\xi2|\rho|x-\frac\...
glS's user avatar
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Punchline of Liouville's Theorem

As I write and prove in [1] and rewrite in [2], the punchline has more to do with the total derivative than the partial derivative.... [Liouville's theorem] Consider a Hamiltonian system ...
Michael Levy's user avatar
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What does Liouville's Theorem actually mean?

In my estimation, Liouville's theorem is generally relevant to a Hamiltonian system characterised by the Hamiltonian $ H\!\left(t,\boldsymbol {q},\boldsymbol {p} \right)$, where the state of the ...
Michael Levy's user avatar
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A proof of Liouville’s theorem

Preface The title of the OP's question is "A proof of Liouville’s theorem". Yet, the precise question asked therein by the OP is very specific as it relates to a specific proof of Liouville'...
Michael Levy's user avatar
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Is there a Schrödinger equation for phase space?

Yes, there can be found Schrödinger equation in phase-space, allowing to consider Feynman/Boltzmann ensembles of smooth paths (in contrast to standard infinite velocity diffusion-like), see e.g. https:...
Jarek Duda's user avatar
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Describe the characteristics of a Hamiltonian System to a non-scientist

Hamiltonian systems are an abstract formulation of dynamical systems, that have two interdependent degrees of freedom per dimension. In its easiest to describe in an one dimensional case, the two ...
Roland F's user avatar
1 vote

Relation of entropy given in terms of phase space volume vs. multiplicity

In classical mechanics, the states of a system live on a smooth manifold: $\vec{p}$ and $\vec{q}$ are continuous variables for each particle that can take an infinite number of values, even after you ...
Rokas Veitas's user avatar
1 vote

How to Find the Phase Space Integral Bounds with a Regulating Photon Mass?

The integral over $z = \cos \theta$ is of the form $$ \int_{-1}^1 dz \delta \left( 2 - x_1 - x_2 - \sqrt{ x_1^2 + x_2^2 + 2 x_1 x_2 z + 4 \beta } \right) $$ The argument of the delta function vanishes ...
stringynonsense's user avatar

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