# Questions tagged [phase-space]

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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### Are the Wigner and Husimi transforms injective?

I am wondering if the Wigner function is injective. By injective I mean, that, for every density matrix $\rho$, there is a different Wigner distribution. The same question applies to the Husimi ...
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### Bopp operators and Wigner-Weyl representation

I am learning about the Wigner-Weyl transformations to move a $c$-number Lindblad operator $A(x,p)$ back into operator form. As far as I know, to move back and forth normally requires a four variable ...
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### Hodge dual and the Moyal bracket? Any link? [closed]

I have already asked this on the mathematics Stack exchange but I thought I'd try it here too! The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an ...
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### Confusion about the use of the term “Phase Space” in Strogatz text

I've just started learning about Hamiltonian mechanics, and from the definition given in Taylor's classical mechanics, phase space must always have an even dimension. However, I recall from reading ...
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### What are resonant tori?

What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?
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### Physical meaning of phase-plane representation

If I have a dynamic equation I know I can represent it in the phase- plane ($x$,$\dot{x}$) and there I can study the stability of the system just looking at the shape of the curve: if it is an ellipse ...
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### Can rotational motion can be reduced to translational motion?

Susskind says that all physics (dynamics of the world) can be described in terms of particle positions and their first order derivatives. Does such translational motion gives rise to rotation?
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### Equilibrium in Stat Mech and Phase space density

I was wondering if there is any relationship between equilibrium in Stat Mechanics and the phase space density of a system? This does not seem to be completely independent, as Entropy is maximized in ...
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### Meaning of phase space density

I am trying to understand Liouville's theorem physically. It says that $\frac{\partial \rho}{\partial t} + \{\rho,H\} = 0$. Thus, we have $\frac{d \rho(q(t),p(t),t)}{dt}=0$. I would like to ...
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### Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all ...
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### Why aren't classical phase space distribution functions always delta functions?

The phase space distribution function (or phase space density) is supposed to be the probability density of finding a particle around a given phase space point. But, classically, through Hamilton's ...
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### Poincaré maps and interpretation

What are Poincaré maps and how to understand them? Wikipedia says: In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the ...
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### Damped simple harmonic oscillator problem

I'm supposed to calculate and draw the phase space trajectory for this: for the two different cases when and . I've never done this sort of question before, how are they done? I've tried googling ...
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### Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
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### Liouville's theorem and the preservation of topology

What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume? By Liouville's theorem, the total phase space volume is ...
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### How to check whether a given $W(x,p)$ represents a Wigner function of a physical state? [duplicate]

For simplicity let us consider one-dimensional quantum-mechanical systems only. Given any state $\rho\in\mathcal{B}(\mathcal{H})$ and its Wigner function $W_\rho(x,p)$, there are several properties it ...
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### Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
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### How to prove that a Hamiltonian system is *not* Liouville integrable?

To show that a system is Liouville integrable, we just need to find $n$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For ...
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### Canonical ensemble, energy, heat bath

I am studying through the book Thermodynamics and Statistical Mechanics by Walter Greiner and I’ve got a couple of doubts when I was reading about the classical ensembles, specially the Canonical ...
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### Is there a physical system whose phase space is the torus?

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of ...
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### Can statistical mechanics be formulated generally in terms of phase space?

In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of ...
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### Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
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### Deriving probability distributions from the Wigner distribution

I know that I can calculate the probability distributions of $x$ and $p$ from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by ...
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### Phase space appellation

Does anyone know why they called the momentum-position space the phase space in the first place? To clarify what I mean a bit more, I'll give you an example: The name configuration space for the ...
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The conjugate momentum of a charged particle moving in a uniform magnetic field is given by $$\textbf{p}=m\textbf{v}+q \textbf{A}$$ This expression is not unique because $\textbf{A}$ is not unique. $... 3answers 973 views ### Probability distribution in phase space and Liouville's theorem? We can define a probability distribution over phase space (say 1D)$\rho(x,p)$such that, for example, $$\langle x\rangle = \int x \rho(x,p) dxdp$$ etc. It can be shown here that such a distribution ... 1answer 70 views ### All angle dependence in$\mathrm{d}LIPS_2$? Recall that$\mathrm{d}LIPS_2$(one particle decaying into two particles of the same mass) is given by $$\mathrm{d}LIPS_2 = \frac{\vert{\bf k_1'}\vert}{16\pi^2\sqrt{s}}\mathrm{d}\Omega_{cm}.$$ In a ... 1answer 2k views ### Which transformations are canonical? Which transformations are canonical? 1answer 832 views ### Phase Plot for Harmonic Oscillator This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the$sin(\omega t)\$: ...
Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$G = p_1\delta x_1 +p_2 \delta x_2$$ Now, we know clearly that this group is an Abelian ...