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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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4
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2answers
99 views

How does one “invert” such infinite-dimensional sympletic form?

In the book "Lectures on the IR structure of gravity and gauge theories" by Strominger the author considers the sympletic form for free electrodynamics: $$\Omega_\Sigma[A;\delta_1 A,\delta_2A]=-\frac{...
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1answer
168 views

Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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1answer
125 views

Free particle 1D phase space

I'm trying to draw the phase space for a particle moving freely between 0 and $L$. I guess $H=E$(total energy, constant)$=\frac{p_{x}^2}{2m}$ so $p_{x}=\pm\sqrt{2mE}$ for every x between 0 and L, and ...
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1answer
206 views

Quantum versus classical computation of the density of states

If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy ...
2
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1answer
193 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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0answers
16 views

Lorentz-invariant phase space element

We're following R.K. Ellis et al.: QCD and Collider Physics (p.99 ff). I want to calculate the Lorentz-invariant phase space element (LIPS) $\mathrm{d}\Pi$ for the processes $p,q\rightarrow l$, $p,q\...
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1answer
44 views

On- & off-shell conserved charges/constants of motion

I am trying to understand how conserved charges generate symmetry transformations via the Poisson bracket, but I am missing something in one part of the derivation. The part I am struggling with is ...
4
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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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7answers
1k views

Why isn't momentum a function of position in quantum mechanics?

In quantum mechanics, the unitary time translation operator $\hat{U}(t_1,t_2)$ is defined by $\hat{U}(t_1,t_2)|ψ(t_1)\rangle = |ψ(t_2)\rangle$, and the Hamiltonian operator $\hat{H}(t)$ is defined as ...
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1answer
26 views

On the derivative of the Hamiltonian of a particle in quantum mechanics

The Hamiltonian function for a particle of mass $m$ moving along the $x$-axis, subject to a potential energy $V(x)$, the Hamiltonian function is: $$H = \frac{p_x ^2}{2m} + V(x).$$ My book states ...
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1answer
188 views

Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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0answers
28 views

Drawing the orbits in the phase space of a particle with the following properties

Considering a 1D space, I'm given the graph where $U(\pmb{q})$ is the potential energy as a function of position $\pmb{q}$, and each $E_0, E_1,E_2$ is a different amount of total energy in the system....
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1answer
61 views

Find the density of states of a bouncing ball

Imagine a ball falling from a maximum height of $h$ and colliding with the ground at $z=0$. The ball only moves in the z-axis and the collisions are elastic. My job is to show that the density of ...
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0answers
31 views

Transformation between a dynamical system to a Hamiltonian system [duplicate]

Consider a dynamical system characterized by these equations $$\dot{x}=x-xy \\ \dot{y}=-y+xy$$ If we transform $\ln(y)=q$ and $\ln(x)=p$, the system can be changed into a Hamiltonian system with $q$ ...
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1answer
91 views

Canonical Transformations that are Complex

I'm self studying through a book that has the following question. The book gives the answer, but I'm trying to understand why: Under what condition is the following transformation NOT canonical? $$Q =...
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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
224 views

Liouville's Theorem in Spherical Coordinates

I'm attempting to verify the invariance of $$d\omega=\prod_{i=1}^{3N}dq_idp_i$$ in the case $N=1,$ and under a canonical transformation into spherical coordinates. I know to let $$q_1=r\sin\theta\cos\...
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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
62 views

An “upper ceiling” for thermodynamics?

Roger Penrose said in "A Road to Reality" (p.701): “There is a common view that the entropy increase in the second law is somehow just a necessary consequence of the expansion of the universe. ...
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2answers
260 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
475 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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3answers
737 views

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
200 views

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
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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1answer
427 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
61 views

Wigner function of thermal state

I am wondering how you would compute the Wigner Function of a Thermal State with average phonon number $\bar{n}_{\mathrm{th}}$. I know the result should be a Gaussian with variance in position $\...
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0answers
25 views

Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times. $\langle x\rangle $, $\langle p\rangle $,$\langle x^2\rangle $, $\langle p^2\rangle $, $\langle xp\rangle $, $\...
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2answers
11k views

Commutator of $X$ and $P$ in quantum mechanics [closed]

So, this is my first contact with Quantum Mechanics and I'm having trouble with this exercise. One of the steps involves calculating $[X,P]$, and I stuck there. Can anyone give me some help?
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1answer
86 views

Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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1answer
156 views

How to check if a generating function produces an identity transformation without substituting the CT equations in the Hamiltonian?

In chapter 9, Goldstein ($3^{rd}$ ed.) includes a discussion and a few "trivial special cases" of Canonical Transformation which keeps the form of the Hamiltonian unchanged and named it Identity ...
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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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1answer
82 views

Question about using Liouville's theorem to calculate time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\...
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1answer
103 views

A proof of Liouville’s theorem

I have found a proof of Liouville's theorem on the internet, which fits me very well except one step I don't understand, the derivation is as follows: In the derivative, it must have used the ...
4
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2answers
395 views

Canonical Transformation of Hamilton Equation

I have a problem understanding the criteria of an canonical transformation. I am preparing for my exam and came across this question: For which $A$, $B$, $C$, $D$ is: $Q = A q^2+B p^2, \quad P = ...
2
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1answer
100 views

The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
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1answer
125 views

Wigner-Weyl ordering in exponential

If the particle number is $\hat{a}^\dagger\hat{a}\leftrightarrow|\alpha_w|^2-1/2 $, it can be mapped on the Wigner fields by assuming symmetric ordering:$|\alpha_w|^2\leftrightarrow\hat{a}^\dagger\hat{...
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1answer
112 views

Hamilton's equation of motion with other momentum

I wrote here a problem couple days ago. I figured out what was the problem there, but now it made another problem. Sorry for similiar question. I'm trying to draw phase portrait for my ODE and for my ...
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0answers
26 views

Relation between Wigner flows and entropy

After reading this question What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics? and some papers of Steuernagel’s group,...
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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
61 views

How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
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0answers
22 views

Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
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1answer
172 views

Differential cross section $d\sigma/dp^{\gamma}_{T}$?

Why we care about $d\sigma/dp^{\gamma}_{T}$? What the physical meaning of it? Why not plot $\sigma$ follow $p^{\gamma}_{T}$?. As in this picture.
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2answers
97 views

Relationship between $\star$-products in phase-space QM and NC geometry

What exactly is the relationship between $\star$-products in phase-space quantum mechanics, i.e. $$ (f \star g) (x,p) = f(x,p) e^{\frac{i \hbar}{2} ( \overleftarrow{\partial_x} \cdot \overrightarrow{\...
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2answers
69 views

Density Of states derivation

In the aspect of density of state derivation or simply assuming the frequency of a solid as a continuous distribution we have to come up with an equation expressing the density of states. Its derived ...
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0answers
87 views

Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
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1answer
83 views

Prove that a transformation is canonical by using $\mathbb{M}^T\cdot \mathbb{J}\cdot \mathbb{M}$ [closed]

So, I was given the following problem to solve: A system with two degrees of freedom is described by the following hamiltonian \begin{equation} H=p_1^2+p_2^2+\frac{1}{2}(q_1-q_2)^2+\frac{1}{8}(...
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1answer
74 views

Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?
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1answer
37 views

Asymmetry in Hamilton Equations

I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's ...
2
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1answer
41 views

Why are marginal eigenvalues of Jacobian of a periodic orbit related to the symmetry?

In ChaosBook, at page 61 of the unstable version of the book, it is stated that $$J_p (x) \mu (x) = \mu (x,)$$ i.e the velocity vector is an eigenvector of the Jacobian along periodic orbit $p$ ...
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2answers
979 views

How to prove that a Hamiltonian system is *not* Liouville integrable?

To show that a system is Liouville integrable, we just need to find $n$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For ...