Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.

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.

Filter by
Sorted by
Tagged with
4
votes
2answers
101 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{...
0
votes
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\...
0
votes
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 ...
1
vote
1answer
46 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 ...
0
votes
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....
0
votes
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 ...
1
vote
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 $q$ ...
2
votes
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. ...
1
vote
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 $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
1
vote
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 $, $\...
1
vote
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 $\...
1
vote
1answer
87 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 ...
1
vote
1answer
158 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 ...
1
vote
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{{\...
1
vote
1answer
95 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 =...
0
votes
1answer
105 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 ...
2
votes
1answer
101 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 ...
-1
votes
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 ...
2
votes
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,...
0
votes
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 ...
1
vote
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 ...
0
votes
2answers
72 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 ...
1
vote
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 $\...
1
vote
2answers
99 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{\...
0
votes
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}(...
0
votes
1answer
127 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{...
1
vote
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
votes
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 $p$ ...
2
votes
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 $K(x',p')=K(p')$?
0
votes
0answers
44 views

Phase space density function and Probability density function

I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of ...
1
vote
0answers
50 views

Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
3
votes
1answer
219 views

Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
1
vote
1answer
53 views

A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
1
vote
1answer
347 views

The phase space trajectory of a single particle falling freely from height is? [closed]

The phase space trajectory of a single particle falling freely from height is? Phase space is a plot between momentum and position, and since kinetic energy increases the momentum must increase with ...
1
vote
1answer
93 views

Volume of state in phase space free particle

I have to how a quantum state of a free particle between 0 and a occupies an area of $h$ in the phase space. What I did was to calculate $\Delta x \Delta p$ and show that it was of order $h$, but I ...
1
vote
0answers
58 views

Relation between the canonical partition function and the phase space volume

In Kerson Huang's Statistical Mechanics (2nd ed.), it is claimed that the phase space volume occupied by the canonical ensemble is the partition function: $$ Q_N (V, T) \equiv \int \frac{dp dq}{N! h^{...
0
votes
1answer
55 views

Phase space harmonic oscillator area and probability

I want to find the probability of finding an oscillator between $x$ and $x+dx$. I calculated the volume $\frac{8\pi EdE}{\omega^2}$ enclosed in the phase space for the oscillator with energy between $...
0
votes
1answer
139 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 ...
1
vote
1answer
46 views

Wigner phase space operator correspondence: how to order?

According to Gardiner-Zoller (Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q) $$a\rho\...
1
vote
1answer
43 views

Finding period from action-angle variable in one dimensional potential [closed]

I want to calculate the period from the action-angle variable for a particle in a one dimensional potential $V = V_0 \tan^2(q \pi/2a)$. After doing some algebra I get $$I = \frac{\sqrt{2mE}}{2\pi} \...
1
vote
0answers
99 views

Why are 2 dimensions needed for every 1 dimension of space in order to determine the motion of a physical system?

In classical mechanics, the phase space of a mechanical system has twice the number of dimensions of "actual" space (i.e. position space). That is, in phase space, each particle has both a position ...
2
votes
1answer
55 views

Thermodynamical conjugate variables

In thermodynamics the potentials are typically only a function of 2 variables, say $$U=U(S,V)$$ with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
3
votes
1answer
70 views

A question about $\delta(x) \star \delta(p)$

Using the Moyal product between two delta functions in $(x,p)$-space one gets $$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$ However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
0
votes
1answer
31 views

Is it possible to have discontinuities in the phase portrait of a dynamical system? If yes what does it really mean?

I've been using Mathematica to draw the phase portrait of a system and I got some jumps along the trajectory. I have a deviation term which might be the reason of this but is it possible to have them ...
0
votes
0answers
16 views

Can all phase space conserving dynamics be described by a Lagrangian system? [duplicate]

Given a system described by a set of ODE's that can be shown to conserve phase space, does there necessarily exist a Lagrangian (or Action) formulation that describes my system? I'm comfortable ...
1
vote
1answer
56 views

Generating function depending on $q$, $p$, $Q$ and $P$

If I have a generating function, say, $$G(q,p,P,Q)= qp - e^Q e^P\tag{1}$$ what are the equations that give me the transformations $Q=Q(p,q)$ and $P=P(q,p)$? I have only seen generating functions ...
1
vote
2answers
122 views

Making sense of phase portrait of simple mass-spring oscillator

I'm new to physics, and I'm having trouble making sense of phase portrait of the following system, $$ m \ddot{x} + k x = 0 $$ whose phase portrait is in here. Since $$ x(t) = \sqrt{\frac{2E}{k}} \...
1
vote
1answer
68 views

Wigner map of the product of two operators

Does anyone know how to prove that for the product of two operators $\hat{A}\hat{B}$ the Weyl-Wigner correspondence reads $$ (AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
1
vote
1answer
53 views

Perelomov coherent states for an arbitrary Hamiltonian

I'm reading about Perelomov coherent states, but I'm not sure if I'm getting it right. From this question and some Perelomov papers I understand the following: The Perelomov coherent states are ...
2
votes
1answer
89 views

Non-commutative Fourier transform of an operator

Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative. $$ \hat{\rho} \xrightarrow[non-comm]{...