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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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Hamiltonian transformation and a symplectic transformation

I tried to solve this question but I can not, show that the transformation $ψ(Q, P) = (P, −Q)$ is symplectic.
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Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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Use of generating function in canonical transformation

In the theory of Canonical transformations, initially we use the fact that the new and the old system of $(q_i, p_i)$ with the Hamiltonian $H$ satisfy the modified Hamilton's principle. Now here, the ...
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Poisson Brackets in inhomogenous magnetic field [on hold]

This question came in my classical mechanics paper and I still can’t solve it. A particle of mass $m$ and electric charge $e$ is moving under the influence of an inhomogeneous magnetic field $\vec{...
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Derivation of Hamilton-Jacobi theory using canonical transformations

The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function. Is it possible to use a another type of generating function, ...
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23 views

Generating function in action-angle method and Hamilton-Jacobi theory

I think that in action angle method, generating function which generates such a canonical transformation does not explicitly depend on time, so new and old hamiltonians are equal. But in H-J method, ...
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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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Phase space in Statistical mechanics

I'm currently struggling with the concept of phase space in Thermodynamics/Statistical physics. In particular I have trouble understanding the use of the "one-particle phase space". If we look at a ...
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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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38 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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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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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
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Lorentz invariance of volume element

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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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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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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Expansion to show $g$ is conserved if $H$ is invariant

On Shankar QM page 99 it says that If $H$ is invariant under the following infinitesimal transformation $$q_i\rightarrow\bar{q_i}=q_i +\epsilon\frac{\partial{g}}{\partial{p_i}}$$ $$p_i\...
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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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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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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 ...