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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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61 views

Question about ergodicity and the evolution of the probability distribution under Liouville's theorem

According to Liouville's theorem, the probability distribution function $\rho$ evolve in phase space with $$ \frac{d \rho}{d t} = \frac{\partial \rho}{\partial t}+\left\{\rho,H\right\}_{P.B} =0 $$ ...
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Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
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1answer
77 views

Quantum micro canonical ensemble

In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (...
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Discrepancy regarding Husimi Probability distribution calculation

I am trying to simulate a system of j qubits and for visualization of the dynamics considering the Husimi distribution of the state. To carry out the projection onto coherent states I have proceeded ...
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62 views

Why does the Boltzmann equation deal with single-particle phase space density?

Why does the Boltzmann equation deal with single-particle phase space density $\rho_{1}(\textbf{r}_1,\textbf{p}_1,t)$ rather than the N-particle phase space density $\rho(\{\textbf{r}_i,\textbf{p}_i,t\...
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86 views

Derivation of Proportionality of Phase Space Volume log(Γ)∝N

In the derivation of extensivity of entropy for the micro-canonical ensemble, we assume an ensemble of two systems with the energies $E_1$ and $E_2$. The total energy is given as $E<E_1+E_2<E+\...
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78 views

Problem on deriving canonical transformation condition

I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with $$\omega=\stackrel{...
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1answer
377 views

Lorentz invariance of volume element from the four-volume element: why on-shell?

In Srednicki's QFT book, in chapter 3 (eqn. 3.16 onwards) he talks about the lorentz invariance of the volume element. For this he writes $d^3k/f(k)$ should be invariant under lorentz transformations. ...
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59 views

Difference between conserved quantities and constants of motion?

In Hamiltonian mechanics, consider extended phase space, the trajectory followed by a particle in that space is formed by an intersection of different 2n dimensional surfaces, all of these surfaces ...
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70 views

Hamilton-Jacobi theory vs Hamiltonian formalism

I'm writing some notes on Hamilton-Jacobi Theory and I'd like to find an example of a system that is quite difficult to integrate in the usual Hamiltonian formalism, but quite easy in the Hamilton-...
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1answer
121 views

Boundary conditions for calculus of variations in phase space and under canonical transformations

This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ ...
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129 views

How can momentum and position be combined into a phase space when they have different units?

Elaboration of the question: What is the geometrical interpretation of units? As in, a unit of length is a choice of scaling of the coordinate systems i.e. it is a choice of diffeomorphism, but then ...
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Mean free path for $3\rightarrow 1$ scattering

I want to calculate the mean free path of an antineutrino in nuclear matter where it can undergo the reaction $p+e^-+\bar{\nu} \rightarrow n$, which I imagine will involve calculating the rate of that ...
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1answer
170 views

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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129 views

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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170 views

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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233 views

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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387 views

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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64 views

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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83 views

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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282 views

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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110 views

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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181 views

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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1answer
47 views

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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126 views

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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45 views

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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121 views

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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3answers
118 views

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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2answers
102 views

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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86 views

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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153 views

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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97 views

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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1answer
106 views

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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171 views

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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1answer
151 views

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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1answer
99 views

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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1answer
215 views

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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56 views

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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1answer
808 views

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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2answers
156 views

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 ...