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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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Why exactly is the Husimi-Q distribution not a real probability distribution?

From this question I understood that the uncertainty principle is causing a problem because two points $x,p$ and $x',p'$ in phase space can be confused. Why exactly is this a problem? I don't grasp ...
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Mixed canonical transformation

Wikipedia and most authors denote four types of canonical transformations: $F_1(\mathbf{q},\mathbf{Q})$ , $F_2(\mathbf{q},\mathbf{P})$, $F_3(\mathbf{p},\mathbf{Q})$ and $F_4(\mathbf{p},\mathbf{P})$. ...
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Time-independent canonical transformations

Lie's criterion tells us that $(q,p) \to (Q,P)$ is a canonical transformation, for a system with Hamiltonian $H$ and "Kamiltonian" $K$, if and only if the identity $$\sum_k p_k dq_k -Hdt = \sum_k P_k ...
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Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis. There was an important equation called Cauchy-Riemann equation. When I saw it at first, I recalled a book's sentence stating ...
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When can action-angle variables be defined?

According to Goldstein, "We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (...
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Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
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Do compact symplectic manifolds play a role in physics?

In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact. Do compact symplectic manifolds have physical meaning? ...
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Translating between classical treatment of non-autonomous systems and time evolution in quantum mechanics

When I read an introduction to (classical) dynamical systems, the system was considered in a phase space, and the state of the system evolving in phase space. For a non-autonomous system, an ...
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Solving Hamilton-Jacobi via canonical transformations

Given a solution to a Hamilton-Jacobi equation in $(X, P)$ variables and a canonical transformation from $(x, p)$ to $(X, P)$, how does one write down the solution to the Hamilton-Jacobi in terms of ...
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Phase space meaning [duplicate]

In the field of medical physics, specifically in monte carlo simulation of radiation beams produced by electron accelerators, people call ‘phase space’ to a file that contains the data of a large ...
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Going from quantum master equation to Fokker-Planck equation for the Wigner function

I am trying to understand how to go from a quantum master equation to a Fokker-Planck equation for the Wigner function. For instance, in this article https://arxiv.org/pdf/quant-ph/0605166.pdf , they ...
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Arithmetic of Hamiltonian in canonical transformation

I have the following Hamiltonian: $$ \mathcal{H} = \frac{p^2}{2m} + V(q-X(t)) + \dot{X}(t)p, $$ and I make the usual canonical transformation for the momentum: $$ p \rightarrow p' = p + m\dot{X},$$ ...
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A question over Liouville’s Theorem

I have some doubts about Liouville theorem, probably its just something conceptual. So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved. But ...
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Why any operator is specified by this characteristic function?

On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2: Any operator referring to a harmonic oscillator — position operator ...
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Why is dynamics first order in phase space?

I have watched some lectures in which the lecturer said that system dynamics are (generally?) first order in phase space, forming a system of coupled differential equations. At a basic level I see ...
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Dirac bracket for the Madelung (polar) form of the Schrodinger field

I'm having an issue with obtaining the Dirac bracket in the Madelung (polar) representation of the Schrödinger field: \begin{equation} \Psi=\sqrt{\rho}e^{i\theta/\hbar}. \label{eq:...
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anistropic or asymetric particles distribution

As a preamble, I have several questions and I know that is preferable to ask one question per post, but they are really linked together. This is why I chose to ask them together. I am considering a ...
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Microstates - Why Position and Momentum?

Why is it that when we discuss the microstate of a system of particles, we use Position and Momentum? How does Position and Momentum tell us everything we need to know about a single particle? I've ...
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A confusing point of the Hamiltonian for a particle interacting with electromagnetic fields

In non-relativistic quantum theory the Hamiltonian for a particle interacting with electromagnetic fields is $$H=\frac{(\mathbf{p}-\mathbf{A}*e/c)^2}{2m}+e\phi+\int\,d^3x \frac{\mathbf{E^2}+\mathbf{B^...
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How can a pendulum have amplitude greater than $\pi$?

How can a pendulum have amplitude angle greater than $\pi$? I've been reading about phase plots, which are graphs of the $\frac{d\theta}{dt}$ on the $y$ axis and $\theta$ on the $x$ axis, shown below. ...
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Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
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Wigner Function for an Entangled Composite System

How is it possible to compute the Wigner function for a composite system that is prepared in an entangled state? In particular, consider the state $|ψ_{AB}\rangle=\frac{1}{\sqrt{2}}(|0_A\rangle|1_B\...
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Showing time-dependent transformations are canonical

In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$ Q = Q(q,p,t) \tag{9.4a} $$ and $$ P = P(q,p,t) \tag{9.4b} $$ such that if $H$ ...
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Wigner flow for open quantum systems

In the paper by Friedman and Blencowe the Wigner flow for an open quantum system is derived. On page 3 the Wigner flow for the harmonic oscillator is derived. Substituting the potential $V = \frac{1}{...
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Is a phase space a function? [closed]

I saw a graph of a phase space of a pendulum and it looks like an $x-y$ plane with a spiral representing the speed and position (I assume from the origin). Are all phase spaces two dimensional, or is ...
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What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid. However, in the phase space formulation ...
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Comparing the Madelung and Groenewold-Moyal pictures of quantum mechanics

We can consider a dynamical theory to be a "transport theory" if it can be described entirely by a series of continuity equations of the form: $$\frac{\partial \rho}{\partial t} + \nabla \cdot \left({...
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How do contact transformations differ from canonical transformations?

From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated ...
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Symplectic Standard map [closed]

I have come across this map, which the notes call standard symplectic map. Why is it symplectic? How do I show it? Are those action-angle variables? $I(t+1)=I(t)+K\sinθ(t)$ $θ(t+1)=θ(t)+I(t+1) \quad ...
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Physical interpretation of differences between classical and quantum ensemble dynamics

The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. ...
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Lagrangian and Hamiltonian dynamics, momentum and canonical transformations

I am relatively new to Lagrangian and Hamiltonian dynamics. I am aware of how to form the equations of motion using the Legendre Transformation. I, however, have one fundamental question and I was ...
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Linear canonical transformation represented by a unitary operator

I am reading a paper on Squeezed states which mentions the following fact "a linear canonical transformation can be represented by a unitary transformation" and then used a operator $\hat{U}$ for ...
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Wigner 's unreasonable effectiveness of mathematics in natural sciences [closed]

This question is related to Wigner's problem, related to the unreasonable effectiveness of mathematics in natural sciences. Understanding a phenomenon means constructing a mathematical model , and ...
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Density function in phase space

What does density function in phase space physically mean? How does it indicate, the more familiar density that we are accustomed to ( an analogy may be), in phase space?
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Problem with the regularity condition for a constraint

So, I'm considering Lagrangian: $$L=\frac{1} {2}e^{q_1}\dot{q}_2^2. $$ I obtain the primary constraint $\phi=p_1=0$. The canonical Hamiltonian is $H_c=\frac{1} {2}p_2^2e^{-q_1} $, and the total ...
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The formula for the average number of fermions $\langle N \rangle$

In the context of Fermi gases (or fluids in general), one would typically in the grand-canonical formalism use the formula $\langle N \rangle = -\frac{\partial \psi}{\partial \mu}$, where $\psi$ is ...
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Four body decay rate

I would like to calculate a 4-body decay rate and I'm stuck. I have a massive scalar particle (with mass $M$) decaying to four massless particles (2 fermions and 2 scalars). I am not sure how to ...
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Poisson Brackets And Angular Momentum Components

Related: Poisson brackets of angular momentum When Poisson Brackets are taught as part of an Analytical Mechanics courses, examples are commonly shown which anticipate analogue results in QM. One ...
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One dimensional system a Hamiltonian system?

I have the following equation of motion: $$ \dot x = \beta x y $$ with $y=1-x$. I would like to see if it is Hamiltonian or not. Due to it being one dimensional, I think it should be locally ...
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Probability for a phase space flow to return to its original state

A Hamiltonian system of $n$ interacting atoms, each of mass $M$, is confined within a cubical box of sides $V$. The average initial speed of each particle is $v$. How do I estimate the timescale for ...
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Deriving a generator $G(q,p)$ given certain conditions

So let's consider a mechanical system in Lagrangian and Hamiltonian formalism; it has Lagrangian $L(q,q',t)$ and Hamiltonian $H(q,p,t)$. I know that $L$ invariant under infinitesimal changes $q → q + ...
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Phase space volume doesn't change under canonical transform

I have given a set of generalized coordinates $(q_1,..q_n,p_1,..p_n)$. Suppose I had a canonical transform $(q_i,p_i)\rightarrow (Q_i,P_i).$ I am trying to show that the phase space volume element ...
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Statistical mechanics and thermal averages in $\mu-$space and $\Gamma-$space

What is the relation between the thermal averages in $\mu-$space and $\Gamma-$space of a system having $f$ degrees of freedom in statistical mechanics? For a system with $N$ particles (and having $n=...
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Four special cases of canonical transformations

Let $(q,p) \mapsto (Q,P)$ be a diffeomorphism of phase space. Then this is a canonical transformation if $$p\dot{q}-H(q,p,t)=P\dot{Q}-K(Q,P,t) + \frac{dM}{dt}\tag{1}$$ for some $M=M(q,p,Q,P,t)$. The ...
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Invariant Phase space volume under canonical transfromation

I have a volume element in phase space: $$ d\omega = \prod _{i=1}^{N}(dq_{i},dp_{i})$$ Now I should show the invariance of this product under canonical transformations. I think first I would have to ...
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What is the purpose of phase space? [duplicate]

Why is phase space important? As far as I'm concerned, you're just rewriting the dynamic law using momentum instead of velocity and mass. $$m \space \frac {d \ \vec v}{d\ t}=\vec F \\ \frac{d \ \vec r}...
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Hamilton-Jacobi theory: Differentiating wrt. the constant $E$?

Let's say we have a 1D harmonic oscillator, its Hamiltonian is given by $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2$$ we wish to solve it via the Hamilton-Jacobi equation so we have $$\frac{1}{2m}\...
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What is meant by “phase space structure”?

I've heard that non-equilibrium systems have the property that their phase space has a structure, as opposed to 'structure-less' phases spaces of equilibrium systems. What does this precisely mean? I'...
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How to determine whether two variables are canonical or not?

I'm currently reading about Ashtekar's variables, and found out that the new variables, $A$ and $E$, are canonical, so they satisfy the following poisson bracket \begin{eqnarray} \left\{ A^i_a(x), A^...
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Time-reversal operator in Phase Space Representation

Consider the simplest possible case in which the time reversal operator $\hat{\mathrm{T}}$ is given by the operation of complex conjugation $\hat{\mathrm{K}}$. We can view $\mathrm{T}$ is an anti-...