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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
70 views

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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1answer
479 views

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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1answer
107 views

Four special types of canonical transformations

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

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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1answer
116 views

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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1answer
283 views

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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55 views

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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1answer
195 views

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-...
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3answers
420 views

Meaning of phase relationship for a superposition of states

I have studied an introductory course in quantum mechanics, and yet I still do not understand the significance of a phase difference between quantum states that a system is in a superposition of. In ...
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253 views

Hamiltonian equation in Cartesian coordinates

My Lagrangian equation is $$L = \dfrac{1}{2}m\dot{q}^{2} \tag{1},$$ where $q=(x,y)$. Performing the Legendre transformation I get the Hamiltonian equation, \begin{equation} H(p,q) =p\dot{q}-\dfrac{1}{...
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1answer
57 views

When intersections of trajectories in Poincare sections are possible?

If we get intersections of some "trajectories" in non-standard 2D Poincare sections, that have been obtained from numerical integration of Hamilton equations for autonomous system in 2D coordinate ...
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1answer
108 views

Constants of Integration In Hamilton-Jacobi theory

I have had this confusion for a while now. We solve the Hamilton Jacobi equation, $$H+\frac{\partial S}{\partial t}=0$$ Say we get a solution $S(q,\alpha,t)$ where $\alpha$ is a constant of ...
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141 views

Formulation of an Action-Principle in a general Phase Space (No Cotangent-Bundle)

If we come from the physics side, the hamiltonian formalism usually is introduced via generalized coordinates (which are just a collection of numbers stuffed into a vector ($\vec{q}$), and the ...
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203 views

Density of states for a system of $N$ particles (both interacting and non-interacting case)

The density of states $\rho(\epsilon)$ for a single particle (with $g$ number of internal states such as spin) confined in a volume $V$ can be calculated as $$\rho(\epsilon)=\frac{4\pi gV}{h^3}p^2\...
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Conceptual Misunderstanding of Hamilton-Jacobi Theory

I will use the example of one dimensional harmonic oscillator to explain my doubt. $$H=\frac{p^2}{2}+\frac{q^2}{2}.$$ We allow, $S=W-\alpha t.$ We get, $$\frac{1}{2}\left(\frac{\partial W}{\partial ...
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1answer
243 views

Phase space in classical field theory

In Classical Mechanics, a system has configurations described by points of a configuration manifold $Q$. In that setting we define the phase space of the theory to be the cotangent bundle $M = T^\ast ...
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2answers
281 views

Phase space with torus topology

Consider a particular compact 2D symplectic manifold $\mathcal{M}$ defined as follows: The topology of $\mathcal{M}$ is a 2-torus. Let $\theta$ and $\varphi$ be the coordinate patch on $\mathcal{M}$ ...
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How do you set the phase space area for the simulation of a quantum system?

I am simulating a quantum system where I am evolving a Wigner function so i have a grid where I have the X-vaues and P-values. Now, I want to run the simulation for a phase space area of $\hbar$ or ...
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Can any symplectomorphism (1 Definition of canonical transformation) be represented by the flow of a vectorfield?

For this question I will use the definition that a canonical transformation is a map $T(q,p)$ from the phase space onto itself, which leaves the symplectic 2-form invariant (which is the definition of ...
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167 views

Intuition For Canonical Transformations

Mathematically I can understand Canonical Transformations but I don't have an intuitive understanding of them. Why do we need to a canonical transformation? Is it to simplify the form of Hamilton's ...
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reversing time of thermally averaged cross section for 3->2 processes

I am considering the process $a,b\to 1,2,3$ where all particles have the same (or approximately the same) mass. I have found a program micrOMEGAs which can calculate the thermally averaged cross ...
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1answer
103 views

Identify a Hamiltonian system consistent or not?

I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints ...
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1answer
122 views

Hamiltonian under Canonical transformation

We know that 'Hamilton's equations' preserve in canonical transformation. But does this mean than 'Hamiltonian' itself doesn't change under canonical transformation?
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452 views

Two Independent Harmonic Oscillators is NOT Ergodic!

I read on a book that the system of two or more independent harmonic oscillators in classical mechanics is not ergodic. I want to know why a harmonic oscillator is actually ergodic but two or more ...
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505 views

On Groenewold's Theorem and Classical and Quantum Hamiltonians

I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping $$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
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1answer
113 views

Star Product and Poisson Brackets

I have the following definition of star product, \begin{equation} \star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\...
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1answer
117 views

Tensor operator analogy in classical physics

In quantum mechanics, tensor operators are defined through their commutation with the operators of spherical angular momentum components, $$\begin{aligned} \ [L_3,T(k,q)] &= \hbar q\ T(k,q), \\ [...
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1answer
202 views

What is the time average over entire phase space

If this equation is an ensemble average over phase space $$\langle A\rangle=∫_Γ ∏^{3N} \ {\rm d}q_i \ {\rm d}p_i\ A({q_i},\,{p_i})ρ({q_i},\,{p_i})$$ what is the time average of $A$ and how that can ...
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1answer
71 views

Phase of a Wave and Phase Space

What relation does the phase of a wave have with the phase space? Namely, how are they related historically and/or physically? P.S. if it helps, I came across this question while thinking about the ...
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1answer
100 views

Classification of fixed points in 4D phase space

The usual classification of fixed points as used in linear stability analysis is based on planar systems (un-/stable node, un-/stable spiral point, saddle). I need to extend this classification to a ...
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1answer
309 views

Generating function of infinitesimal translation - classical mechanics

I am reading Sakurai's Modern quantum mechanics and at some point it's trying to draw a parallel between classical and quantum mechanics. It says An infinitesimal translation in classical ...
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1answer
69 views

Can a first-order autonomous system not at a fixed point, transition to a fixed point?

The following is from Introduction to Dynamics, by Percival and Richards: At each zero $x_{k}$ of the velocity field $v\left[x\right],$ $$v\left[x_{k}\right]=0,$$ so that a system ...
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0answers
115 views

Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
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1answer
163 views

Measurements in the phase space picture of quantum mechanics

Suppose we are dealing with non relativistic quantum mechanics of point particles, that is we are in the realm of 'classic' quantum mechanics (no quantum fields ect.). In the Heisenberg picture, ...
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1answer
379 views

Phase space and phase line

The simplest system that I can think of, from classical point of view is a single particle moving in one dimension. Even for this system one needs two coordinates to describe the state, its position ...
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1answer
87 views

How to obtain Hamiltonian formalism and phase space for Lagrangian with second-derivatives?

This is a special case of this question of mine, which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case. Take a ...
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2answers
263 views

Symplectic form on covariant phase space

Usually, the phase space of a physical system is defined as the cotangent bundle of the configuration space at some fixed time slice $t = t_0$, conveniently co-ordinatized by $\{q^a, p_a\}$ where $$ ...
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1answer
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Why is $E=\frac{nh} {2\pi}$ equal to the energy in the citation below, if h has the dimension of an action?

In this article on matrix mechanics in quantum theory you can read, in the subsection of the harmonic osscilator, that $$E=\frac{nh}{2\pi},$$ Where $E$ stands for the possible energies of the ...
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259 views

Deriving the Old Quantum Condition ($\oint p_i dq_i=nh$)

A body undergoing periodic motion in an orbit of quantum number $n$ will have a period $T$, determined by $$T=\oint \frac{ds}{v}=\oint \frac{ds}{\sqrt{\frac{2}{m}(E-V)}}$$ Where $ds$ is an ...
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1answer
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Pegg-Barnett phase implementation does not seem to work

I attempt to monitor the phase of a wavevector $|\psi\rangle$. As I found (e.g. here ), a matrix representation for the Pegg-Barnett phase operator in Fock base can be obtained as $$\Phi=\sum_{m,n,...
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0answers
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Is the reversal of the movement of all particles in phase-space, the expansion of space and the development of quantumfields equal to time reversal?

In an earlier question, which was called a duplicate, I already asked a similar question. But I state my question differently now and add something to it. Suppose "we" would reverse the direction of ...
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1answer
201 views

Quantum states in position and momentum phase space

While studying introduction to statistical mechanics ,I came across a new idea phase space where we use both position and momentum coordinates to denote a system .In my book the author calculates the ...
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1answer
293 views

Jacobian in Hamiltonian dynamics

I was trying to show that for an infinitesimal time evolution in classical Hamiltonian dynamics preserves volume by showing that the following Jacobian $$ \left|\frac{\partial(q(t),p(t))}{\partial(q(...
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2answers
136 views

Liouville Theorem analogue in generalized velocities?

The Liouville Theorem concerns dynamics in phase space: does an analogue exist in configuration space, and, if not, could you give a motivation / proof why?
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1answer
107 views

$h^3$ term in probability density function of ideal gas

Other than that we need the $h^3$ to 1) make the units correct and to 2) account for the normalization of the probability distribution, what interpretation can we give to this $h^3$ term in \begin{...
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1answer
137 views

Proof of “non-existence” of marginals of the Husimi $Q$-function

There are many ways to consider the Husimi ($Q$) quasi-probability distribution function, e.g. as the expectation of the density operator in a coherent state or as the Weirstrass transform of the ...
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1answer
237 views

Can any symplectomorphism be called a canonical transformation?

I just want to make sure I am thinking clearly about canonical coordinates and transformations in Hamiltonian mechanics. Suppose we have a Hamiltonian system $(M, \omega, H)$ — where $M$ is the ...
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1answer
43 views

Two first integrals of an hamiltonian field $X_{H}$ are independent $\det \left[ \frac{\partial F_{i}}{\partial p_{k}} \right] \neq 0$

I want to understand how it is established if two first integrals of an hamiltonian field $X_{H}$ are independent. One hypothesis is: Considering two first integrals $F(q^i,p_k)$ $$\det \left[ \...
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5answers
591 views

Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?

This answer by Qmechanic shows that the classical Liouville equation can be extended to quantum mechanics by the use of Moyal star products, where it takes the form $$ \frac{\partial \rho}{\partial t}~...