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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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1answer
882 views

The Hamilton's equations of a charged particle in electromagnetic field

For the relativistic charged particle in EM field we have the following equation for the hamiltonian $$H\left( {\vec r,\vec P,t} \right) = c\sqrt {{m^2}{c^2} + {p^2}} + e\varphi = c\sqrt {{m^2}{c^2} ...
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The distribution function in statistical mechanics/kinetic theory

From Wikipedia: ... a particle's distribution function is a function of seven variables, $f ( x , y , z , t ; v_ x , v_y , v_ z )$, which gives the number of particles per unit volume in single-...
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2answers
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Hamilton's Formulation and Independent Coordinates

In Lagrange's formulation we know that $q,\dot {q}$ are independent of each other i.e, $$\frac { \partial q }{ \partial \dot { q } } =0.$$ My question is, is this true for $p$, $q$ in Hamilton's ...
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How is Liouville's theorem important to statistical mechanics?

I have come across Liouville's theorem in the first chapter of many statistical mechanics textbooks, still I don't quite get how it is important to statistical mechanics. How is it related to ...
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1answer
258 views

Entropy and classical mechanics

I was trying to understand what entropy means in the context of classical mechanics, but unfortunately I'm now more confused than I started. Reading, for example, the Wikipedia article on the Second ...
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1answer
63 views

How to find the right constraints on a canonical transformation?

I have a 1-dof Hamiltonian in the variables $p,q$. I am not sure on how to find the constraints on a canonical tansformation $\psi(P,Q)=(p,q)$ such that the Hamiltonian in the new variables $P$ and $Q$...
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1answer
297 views

Poisson brackets' property as necessary and sufficient condition for transformation to be canonical?

I read (Landau, Lifshitz: Mechanics) and then I want to know if conditions (45.10) are sufficient for transformation $p,q \to P,Q$ to be canonical (obviously, they are necessary).
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1answer
172 views

Ambiguity of Coarse Graining in Classical Stat Mech?

In classical statistical mechanics, the partition function is usually defined by: $$ Z = \int \prod dx_i \int \prod dp_i e^{-\beta H(x_i,p_i)} $$ The standard justification for this definition is that ...
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2answers
73 views

Maximal number of conserved quantities (classical integrability)

In these notes on page 4 the author says that if a $2d$-dimensional phase space has $d$ conserved quantities $F_{\mu}$ that Poisson commute, then $H$ can be written as a function of the $F_{\mu}$. Why ...
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1answer
118 views

Evaluating limits of action angle problems

I am really troubled with finding the limits in "action-angle integral" problems. It is said that the limit is taken over generalised coordinate $q$ such that we have a complete liberation or rotation ...
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1answer
205 views

Deriving the Husimi Function of Harmonic Oscillator Eigenstates by Convolution

In phase space formulation of quantum mechanics, the so-called Husimi function can be defined as the convolution of the Wigner function by an appropriate Gaussian. There are apparently alternative ...
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1answer
164 views

Example of a quantum-mechanical theory with nontrivial classical limit

I am looking for a toy model example of a well defined quantum-mechanical theory with the following properties: It can be constructed via canonical quantization starting from some classical theory ...
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64 views

Why are Hamiltonian's used with non-Hamiltonian brackets to define equations of motion?

I am reading Non-Hamiltonian equilibrium statistical mechanics and I am confused about how they are using Hamiltonians. In Section II, they introduce the "non-Hamiltonian bracket", \begin{align} \{a, ...
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3answers
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Phase space in classical mechanics

I am new in classical physics and I frequently come across the terms phase space and phase trajectory. Can anyone please explain to me what they are in a simple language?
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2answers
106 views

Basic confusion with quantum mechanical operators

Given a classical observable, $a(x,p),$ Weyl quantization gives the correspondent QM observable as: $$\langle x | \hat{A} | \phi \rangle=\hbar^{-3}\int \int a \left(\frac{x+y}{2},p\right)\phi(y) e^{2\...
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133 views

Part 2 - - Terminology — phase space embedding

I am facing difficulties in understanding the definition of Takens' delay embedding from a non-physicist point of view....too much technical jargons. Can somebody please provide a simpler way to ...
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Is every observable a function of position and momentum?

In the first answer of this question it is said that every quantum observable, let's say $\hat{A}$, can be represented as a function of position and momentum observables. In other words, as I ...
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2answers
182 views

Part1 — Beginner level confusion regarding terminologies — symbolic dynamics, trajectory, phase space

I came across the topic of symbolic dynamics when studying about time series analysis. Since I have not formally taken any course on chaotic dynamics, I have some difficulties in understanding some ...
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Does an stereographical projection of a physical plane have any physical meaning?

Mathematically, an arbitrary 2D plane can be mapped onto a sphere by stereographical projection. Each line on the plane is equivalent to each line on the sphere. If the sphere rotates under the $SO(3)$...
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1answer
91 views

Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space?

Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space (or the tangent space thereof)? If not, why is this result obtainable only in the Hamiltonian ...
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1answer
575 views

Proof of a property of the Poisson bracket

I have seen written in many courses of statistical mechanics that, for two functions of the general coordinates and momenta $f(q,p)$ and $g(q,p)$ to satisfy $$ \{f,g\}=0 \tag{1} $$ in a 2D phase ...
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1answer
92 views

Independent Quantities in Canonical Transformations

I was looking through some lecture slides and I came across this page: I understand that the equation highlighted blue (top right corner) is obtained from the Principle of Least Action. Given a ...
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1answer
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In which dimension should nearest neighbor distances be calculated? Confusion regarding some concepts

In this paper, Abstract—The intrinsic dimensionality of a set of patterns is important in determining an appropriate number of features for representing the data and whether a reasonable two- or ...
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1answer
322 views

How does a Legendre transformation $H\to L$ work in non-canonical coordinates?

Let $H(z)$ be a Hamiltonian and $\omega_{ij}$ the symplectic form on the phase-space and $\omega^{ij}$ its inverse $\omega_{ij} \omega^{jk} = \delta^k_i$. We know that the Hamilton's equations are ...
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648 views

Is closed phase trajectory a necessary feature of any one-dimensional periodic motion?

The phase trajectory of a one-dimensional simple harmonic oscillator is a closed one (In particular, it's an ellipse). Is closed phase trajectory a generic feature of any periodic motion at least in ...
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333 views

Physical Interpretation of Phase Space Volume

Perhaps one of the most important results of the whole of Classical Mechanics is that the volume occupied by an ensemble in the phase space remains constant in time. Another very interesting result is ...
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1answer
247 views

Can the Wigner function be described using coherent states?

The Wigner function for a wave function $\Psi(\vec{r})$ is $$ W(\vec{r},\vec{k}) = \frac{1}{2\pi} \int dy e^{-i \vec{k} \cdot \vec{y}} \Psi^{*}(\vec{r}-\vec{y}/2) \Psi(\vec{r}+\vec{y}/2) . \tag{1} $$ ...
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1answer
168 views

Can states with negative Wigner function exhibit no quantumness?

I always thought that the negativity of the Wigner function is a direct manifestation of nonclassicality, but the accepted answer to the question "Are negativity of the Wigner function and quantum ...
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1answer
151 views

Gram determinant for the $2\to 3$ scattering cross-section

I'll repeat the part of my previous question regarding the topic: There is a book of Byckling and Kajantie, "Particle kinematics", discussing in particular (Chapter V) the kinematics of the $2\to 3$ ...
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1answer
130 views

Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics) [closed]

$\newcommand{\dd}{{\rm d}}$ In the paper "On the principles of elementary Quantum Mechanics" am trying to get from equation EQN 4.25 to EQN 4.27. I need help on exponential identities and integration ...
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4answers
450 views

Definition of symmetrically ordered operator for multi-mode case?

As I know, Wigner function is useful for evaluating the expectation value of an operator. But first you have to write it in a symmetrically ordered form. For example: $$a^\dagger a = \frac{a^\dagger ...
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1answer
258 views

A question regarding 3-particle phase space for cross-section $2\to 3$

There is a book of Byckling and Kajantie, "Particle kinematics", discussing in particular (Chapter V) the kinematics of the $2\to 3$ cross-sections. It can be easily found in internet. I just want to ...
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3answers
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The geometrical interpretation of the Poisson bracket

"Hamiltonian mechanics is geometry in phase spase." The Poisson bracket arises naturally in Hamiltonian mechanics, and since this theory has an elegant geometric interpretation, I'm interested in ...
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1answer
144 views

Non-autonomous Hamiltonian flow in phase space is volume preserving

How does one prove that for a system whose Hamiltonian is dependent explicitly on time ($H (q,p,t)$), the volume of an element in phase space is conserved i.e. $\frac{d V}{dt} = 0$ ? In what follows ...
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1answer
140 views

$2\to 3$ cross-section phase space simplification

Suppose the $2\to 3$ cross-section: $$ \sigma = (2\pi)^4\int \frac{d^{3}\mathbf p_{3}}{(2\pi)2E_{3}}\frac{d^{3}\mathbf p_{4}}{(2\pi)^{3}2E_{4}}\frac{d^{3}\mathbf p_{5}}{(2\pi)^{3}2E_{5}}|M(\mathbf p_{...
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2answers
740 views

Conjugate variables in thermodynamics vs. Hamiltonian mechanics

According to Wikipedia, the canonical coordinates $p, q$ of analytical mechanics form a conjugate variables' pair - not just a canonically conjugate one. However, the "conjugate variables" I directly ...
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0answers
222 views

Generating function of point transformation

I am asked to show that the generating function corresponding to a point transformation in Lagrangian mechanics can be taken as null. The point transformation consists of $$ Q_i=Q_i(q,t), $$ and ...
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1answer
131 views

Alternate interpretation of number of degenerate fermions formula in phase space

I'm writing personnal notes on statistical mechanics and I'm tempted to write something that may turn out to be false. So I need a confirmation/infirmation and opinions on the following (I suspect it'...
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1answer
109 views

The phase space volume $g(E)$ of constant energy surface

Consider a system with two particles described by a Hamiltonian of the form $$H(P,Q,p,r)=\frac{P^2}{2M}+\frac{p^2}{2\mu}-\frac{Gm^2}{r}$$ where $(Q,P)$ are coordinates and momenta of the center of ...
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1answer
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Is it possible to derive Liouville's Theorem purely from maximum differential entropy?

Typically in physics (at least the way I learned mechanics), this is derived using the multi-dimensional divergence theorem on the $2N$-dimensional phase space i.e. $0=\partial_t \rho + \sum\limits_{...
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1answer
155 views

Why is an ellipse not a self-intersecting curve?

For a Hamiltonian which is time-independent, the phase trajectories don't intersect. But the Hamiltonian of a one-dimensional harmonic oscillator with constant energy, for example, has an elliptical ...
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1answer
75 views

Does $\{\rho(q,p),H(q,p)\}=0$ necessarily imply $\rho(q,p)=\rho(H)$?

In classical statistical mechanics, the Liouville's theorem tells that for a system in equilibrium the Poisson bracket $$\{\rho(q,p),H(q,p)\}=0.\tag{1}$$ 1. Does it necessarily imply $\rho(q,p)=\rho(...
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Why the Galileo transformation are written like this in Quantum Mechanics?

In Quantum Mechanics it is said that the Galileo transformation $$\mathbf{r}\mapsto \mathbf{r}-\mathbf{v}t\quad \text{and}\quad \mathbf{p}\mapsto \mathbf{p}-m\mathbf{v}\tag{1}$$ is given by the ...
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1answer
784 views

What is the generator for scaling transformation in one dimension?

consider the hamiltonian of 1D harmonic oscillator H = Px^2/2m + 1/2 kx^2 and let H' = Px'^2/2m + 1/2 kx'^2 such that x' = Ax and Px' = (1/A)Px then the wavefunction of the two hamiltonians ( H ...
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0answers
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Area and Entropy in Liuoville’s theorem [duplicate]

If Liouville’s theorem states that an area in phase space doesn't change, then can the area go scattered, but with total area conserved? If not, wouldn't it be contradicted to the entropy law?
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2answers
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Meanings of the word “phase”

I have been confused at points due to multiple uses of the word "phase". Mainly, when I think of a phase diagram, I think of a graph relating temperature to pressure, and segments the possible ...
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1answer
528 views

Heisenberg's equations of motion dynamical system

A quick question about Heisenberg's equations of motion: $$i\hbar\dot{x} = [x,H],\qquad i\hbar\dot{p} = [p,H],$$ where $H$ is the Hamiltonian. I realize that strictly, $x$ and $p$ are operators, but ...
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2answers
582 views

Is Liouville's equation an axiom of classical statistical mechanics?

Suppose we have a classical statistical problem with canonical coordinates $\vec{q} = (q_1, q_2, \dots, q_n)$ and $\vec{p} = (p_1, p_2, \dots, p_n)$ such that they fulfill the usual Poisson brackets: \...
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2answers
1k views

Phase of a vibrating particle

I came across a paragraph that, as I interpreted, conveyed that: 1). The phase of a vibrating particle is the ratio of the displacement of the vibrating particle to its amplitude, at any instant. 2)....
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1answer
216 views

Phase Locking vs. Synchrony

Consider two cosinusoidal signals given by \begin{gather*} z_1(t) = A_1\cos\phi_1(t)\\ z_2(t) = A_2\cos\phi_2(t) \end{gather*} with \begin{gather*} \phi_1(t) = (\omega_1 t + \theta_1)\\\phi_2(t) = (...