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
0
votes
2answers
43 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
58 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
91 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
75 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
72 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
36 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
37 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
71 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
16 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
44 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
210 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
45 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
111 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
57 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
36 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
43 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
61 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
38 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
39 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
54 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
47 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
65 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
28 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
15 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
44 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
61 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
60 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
45 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
70 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]{...
1
vote
1answer
101 views

Fourier transform of cross-spectral density space matrix elements

In order to derive phase space like equation of motion (e.g. the equation of motion for the Wigner function of a single particle in one-dimension), it is an advantage to work with the Fourier ...
2
votes
1answer
78 views

Intuition between this construction of the sympletic form for classical fields

In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the ...
0
votes
2answers
95 views

Integration of Poisson brackets by integration by parts [closed]

In the context of Statistical Mechanics I have to show that the following integral is zero: $$\int \sum_{i=1}^{3N}(\frac{\partial O}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial O}{\...
3
votes
1answer
56 views

What does conservation of probability mean in Classical Mechanics and why is it true?

In the context of the Liouville equation, regularly the conservation of probability is invoked. (Of course, the overall probability is always conserved but this is a truism and not what is meant here. ...
3
votes
2answers
111 views

What does $\frac{d\rho}{dt} =0$ but $\frac{\partial \rho}{\partial t} \neq 0 $ mean intuitively?

For concreteness, let's say that $\rho(q,p,t)$ describes the probability density in phase space. On a superficial level both $\frac{d\rho}{dt}$ and $\frac{\partial \rho}{\partial t} $ tells us how $\...
3
votes
1answer
30 views

Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
3
votes
0answers
43 views

Star-shaped phase space

I am asked to classify the following phase spaces. The phase spaces 2 and 3 are fairly simple (harmonic oscillator and a elastically reflected particle). However, I fail to classify the phase space ...
1
vote
1answer
100 views

Can all canonical transformations be generated using a generating function?

In Classical Mechanics, a gauge transformation is of the form \begin{equation} L \to L' = L + \frac{dF(q,t)}{dt} \, . \end{equation} Any transformation of this kind leaves the Euler-Lagrange equation ...
0
votes
0answers
36 views

Symplectic and Euclidean structure invariance

Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$. Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...
0
votes
1answer
64 views

Are symmetries necessarily canonical transformations?

A canonical transformation is defined as a transformation such that afterwards Hamilton's equations still hold. It can then be shown that this requirement implies that canonical transformations are ...
0
votes
0answers
45 views

Derivation of density of states for a gas with $N$ states

I am trying to find any information on the derivation of the density of states for a system with periodic boundary conditions in 3D. I know how it works with 1 particle since I have seen the ...
0
votes
0answers
13 views

Biconformal space and curvature

I've found very few contributions about the so called Biconformal Space, "a curved phase space". I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
0
votes
1answer
72 views

Plotting quadrature uncertainties in phase space

In most books like in the picture given below, the uncertainties regarding quantum states like coherent and squeezed states are represented in phase space plot by some area enclosed within a circle or ...
0
votes
1answer
68 views

Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
2
votes
1answer
200 views

Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
1
vote
1answer
54 views

Integral limits in phase space

If I am calculating the partition function for $H=cp$, ultrarelativistic gas in three dimensions. And by breaking down $d \Gamma$ into $dq$ and $dq$ and further using spherical coordinates I will get $...
2
votes
1answer
71 views

Liouville equation with Dirac delta as probability density

I would lke to know if the probability distribution given by $$\rho(q,p,t)=\delta(q-q(t),p-p(t)) $$ with the initial condition $\rho(t=0)=\delta(q,p), $ where $q(t)$ and $p(t)$ are trajectories ...
0
votes
1answer
47 views

Statistical averages - integration in phase space

I apologize for my horrible math skills. But how exactly is one supposed to get ensemble averages in the 6N dimensional phase space? Suppose I have two particles in 1D. The phase space element is ...
1
vote
1answer
43 views

Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
0
votes
0answers
35 views

Volume within equi-energetic surface of a classical harmonic oscillator in microcanonical ensemble

$$ V(E) = \int_{H\leq E} d\mu = \int_\Gamma d\mu\, \Theta\bigl(E-H(q,p)\bigr) . $$ To compute the volume within the equinergetic surface in the microcanonical ensamble, we use the formula above, ...
4
votes
1answer
75 views

Liouville's volume theorem in differential forms language

I will cast my question mostly in words. I usually find Liouville's volume theorem cast in two forms: Easy way (without differential forms language): Phase space volume remains preserved under ...