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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1answer
303 views

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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Volume of highdimensional Sphere vs volume of spheres shell

When calculating the phase space volume $\Omega$ in the microcanoncial ensemble with fixed energy $E$, one integrates over a shell that includes all energies in between $E$ and $E+\delta E$: $\Omega=\...
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A question on Lagrangian dynamics an the velocity phase space

I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
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Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?

I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13: What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space ...
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When can phase trajectories cross?

It's said in elementary classical mechanics texts that the phase trajectories of an isolated system can't cross. But clearly they can, for example for the pendulum, the trajectories look like this: ...
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Is Liouville's theorem valid for dimensionally restricted systems?

Liouville's theorem states that the phase space volume of a system is conserved over time. Intuitively, this seems to imply that if a system is at some time constrained to, say, a curve in phase space,...
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Is it possible for the phase of electric charge to change over large general relativistic distances?

Jackson provides examples of how magnetic charge and electric charge form together to create complex charge, \begin{align} \rho = \rho_e+i\rho_m \end{align} which gives rise to the complex faraday ...
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264 views

conservation of volume in phase space

I was reading through a proof of Liouville's theorem on conservation of volume in phase space from David Tong's lecture notes (Chapter 4: "Hamiltonian formalism") and on page 89 it says that $\det(J)=...
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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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1answer
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Conjugate momentum is not gauge invariant

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. $...
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3answers
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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 ...
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1answer
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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 ...
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Which transformations are canonical?

Which transformations are canonical?
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1answer
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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 $sin(\omega t)$: ...
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Group of translations in two dimensions - A weird treatment

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 ...
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Constraints of relativistic point particle in Hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: $$S=-m\...
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Question about canonical transformation

I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from $(q, p)$ to $(Q, P)$ is one that if which the original coordinates obey ...
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Phase Space Flow

Phase space flow shares characteristics with fluid flow such as incompressibility by Liouville's theorem. Extending the similarities one might be curious, does phase space flow have a characteristic ...
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Why are we living in the $q$ part of the phase space?

In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person ...
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Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
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Doubts regarding dimension of a system:Definitions and algorithms

I need to do phase reconstruction from time series data. In doing so, I encountered Takens' embedding theorem and Cao's minimum embedding dimension $d$ by nearest neighbor method. In paper "Optimal ...
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1answer
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Ensemble of harmonic oscillators

I have some problems with problem 2.3 from Reif's Fundamentals of statistical and thermal physics: Consider an ensemble of classical one-dimensional harmonic oscillators. a) If we assume ...
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1answer
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Current density in phase space

$\newcommand{\dd}{{\rm d}}$ I have a question which arises from looking at the impact free Boltzmann equation. Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The ...
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1answer
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Canonical transformation generated by hamiltonian?

Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any ...
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1answer
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Probablity density function $f(\mathbf{x},\mathbf{v},t)$

The phase density function is usually denoted as $f(\mathbf{x},\mathbf{v},t)$ which gives probable number of particles moving with velocity $ \mathbf{v}$ at position $\mathbf{x}$ at time t. Also we ...
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Peculiar Hamiltonian Phase space

I was solving an exercise of classical mechanics : Consider the following hamiltonian $H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$ Where $\lambda,m,\...
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What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)

I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
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Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...