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
1answer
19 views

Second consequence of invariance under regular canonical transformation in Shankar's QM book

Near the end of chapter 2 of R. Shankar's book Principles of Quantum Mechanics, he talks about two consequences of invariance of the Hamiltonian under a regular canonical transformation. My problem is ...
2
votes
1answer
49 views

Expected value in usual quantum mechanics vs quantum information

In standard Quantum Mechanics, one computes the expected value of an operator $A$ (arbitrary state $|\Psi\rangle$) as $$ \langle\Psi|A|\Psi\rangle. $$ This has the virtue that we can compute for ...
6
votes
2answers
471 views

Does quantum mechanics halve the dimension of phase space?

In classical mechanics, a particle confined to move along only the $x$-direction can be fully described by a 2-tuple $(x_1,p_1)$ in phase space. In this case, the phase-space is clearly 2-dimensional. ...
0
votes
0answers
16 views

Intersection of Phase space trajectories

(a) We know that, in the phase space the trajectory (path of time evolution) of a closed system does not cut itself. What is the reason for that? Can a trajectory overlay on itself? (b) Is the ‘closed ...
9
votes
2answers
2k views

Why the Galileo transformation are written like this in Quantum Mechanics?

In Quantum Mechanics it is said that the Galileo transformation $$\hat{\mathbf{r}}\mapsto \hat{\mathbf{r}}-\mathbf{v}t\quad \text{and}\quad \hat{\mathbf{p}}\mapsto \hat{\mathbf{p}}-m\mathbf{v}\tag{1}...
1
vote
2answers
72 views

What does it mean for $P$ functions to be “more singular than a delta”?

Consider the Glauber-Sudarshan $P$ representation of a state $\rho$, which is the function $\mathbb C\ni\alpha\mapsto P_\rho(\alpha)\in\mathbb R$ such that $$\rho = \int d^2\alpha \, P_\rho(\alpha) |\...
12
votes
1answer
1k views

Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)

I am moving into a new field and after thorough literature research need help appreciating what is out there. In the continuos variable formulation of optical state space. (Quantum mechanical/Optical) ...
1
vote
1answer
39 views

Wigner Transform in momentum space coefficient

I am currently reading some about the Wigner transform, and I ran into a problem. The Wigner transform (in the literature I am reading) is defined as: $$\tilde{A} = \int \exp{\big[\frac{-ipy}{\hbar}\...
6
votes
1answer
399 views

What are resonant tori?

What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?
0
votes
1answer
43 views

How to integrate the phase space volume for 2 -> 2 scattering processes?

In the QFT book from Schwartz it is stated that $$ d\sigma = \frac{1}{4E_1E_2|\vec{v}_1-\vec{v}_2|}|\mathcal{M}|^2 d\Pi_{\text{LIPS}}\tag{5.22} $$ where $$ d\Pi_{\text{LIPS}} = (2\pi)^4\delta^4(\sum p)...
0
votes
1answer
45 views

Statistical weight for $N$ harmonic oscillators in microcanonical ensemble

I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators. To do that i use the hamiltonian of the harmonic oszillator: $$H(q,p)=\sum\limits_{i=1}^N \...
1
vote
1answer
29 views

Calculating the Wigner transform of operators

Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then ...
2
votes
1answer
110 views

Understanding derivation of Wigner function for the Harmonic oscillator

In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
2
votes
2answers
1k views

What does Liouville's Theorem actually mean?

Basically, the mathematical statement of Liouville's theorem is: $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\...
4
votes
1answer
266 views

What is a time-dependent symmetry in Hamiltonian mechanics?

I've read something from John Baez which I don't understand: If we consider a single nonrelativistic free particle - in one-dimensional space, to keep life simple - and describe its state by its ...
0
votes
0answers
28 views

How did Glauber come up with his definition of non-classical states?

In their paper, Titulaer and Glauber state The results [...] are derived only for fields with positive-definite P functions. Those are, in fact, precisely the quantum fields which may be described in ...
2
votes
1answer
47 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,...
2
votes
1answer
135 views

Constraints are not functional relations! [closed]

I am reading a Wikipedia article on Dirac brackets. At the bottom of the page "illustration on example provided" the article states that for a system with constraints: $$ \phi_1 = p_x + \...
3
votes
1answer
79 views

Bounding derivatives of the Wigner function using phase-space tails

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} ...
0
votes
0answers
34 views

Curvature in phase space

As I understand it, one way to describe a wave function is as a probability density distribution in phase space. The equations of motion for the wave function would describe how that density ...
3
votes
2answers
106 views

Why there is no commutator term in the pre-sympletic density?

In this post I'm considering the Covariant Phase Space (CPS) formalism as presented by Lee & Wald in "Local symmetries and constraints ". In the CPS formalism we take the Lagrangian form ...
1
vote
0answers
38 views

Relation between the canonical symplectic form on phase space and the Hamiltonian in GR

I am working on the Hamiltonian formulation of the Einstein equations of motion in General Relativity, where the aim is to find the Hamiltonian generating the dynamics from the Einstein equations (...
1
vote
1answer
164 views

Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?

Consider a canonical transformation $(p,q) \rightarrow (P,Q)$ under a generating function $F$. The condition for form invariance of Hamiltonian equations of motion looks like : $$\sum_{s}P_s\dot{Q_s} ...
0
votes
1answer
181 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'...
1
vote
1answer
151 views

Why do we have this difference in the multiplicity of Cartesian position space and momentum space for a gas?

For an ideal gas, the multiplicity of an ideal gas with $N$ molecules in Cartesian position space is $$\Omega_{\text{space}}=\Big(\frac{V}{(\Delta x)^3}\Big)^N.$$ This is pretty intuitive, because we ...
0
votes
0answers
108 views

The mathematical model of the Galilean transformation in Hamiltonian mechanics

In my previous question, I asked about the Galilean invariance of the Hamiltonian. I've got already two answers, probably good but I have difficulties interpreting them. Both answers write the ...
0
votes
0answers
31 views

Splitting (in horizontal and vertical subspace) of the thermodynamic phase space in contact geometry

Im currently studying thermodynamics in a contact geometric context and I've stumbled upon a comprehension problem on my side. We have the equilibrium connection $\Gamma_p$, which generates the ...
0
votes
1answer
75 views

Wigner functional for fermionic fields (QFT in phase space)

I'm curently studying the Wigner functional formulation of Quantum Field Theory, which is derived from the Schrödinger picture: the operators which act on the states of the Fock space are functions of ...
5
votes
2answers
320 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 ...
13
votes
1answer
506 views

FLRW cosmology with a scalar field: what are the phase-space variables?

I'm studying a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology model with a simple scalar field as source (no dust-like matter, no radiation, no cosmological constant). For the moment, the field ...
3
votes
0answers
34 views

Overlap of Liouville densities

I am having a little trouble understanding the meaning of the fact that the convolution of two Liouville densities $\rho_{1}(p,q,t)$ and $\rho_{2}(p,q,t)$ (i.e. the classical probability overlap) ...
0
votes
2answers
616 views

Area of a Harmonic Oscillator in phase space

This might be a dumb question, but its art of my assignment and im stuck at the last part. So here it goes. I have a mass-spring system, i am supposed to workout its equation using conservation of ...
11
votes
1answer
868 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 $$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
0
votes
1answer
25 views

A strange consequence of Isotropy in Phase Space

In Goodstein, States of Matter, page 66 the following is stated: For a single particle the number of states in a region $d^3pd^3r$ of its phase space is: $$\frac{d^3pd^3r}{(2 \pi \hbar)^2} \tag{1.3....
1
vote
2answers
59 views

Are Hamilton's equations really form invariant under canonical transformations?

Let under a canonical transformation $(q,p,t)\to(Q,P,t)$, the Hamiltonian is changed from $H(q,p,t)\to \tilde{H}(Q,P,t)$. But in general, the functional forms of $H$ and $\tilde{H}$ are different. ...
0
votes
1answer
30 views

In canonical transformation, how the new coordinate is the old momenta?

I am studying Canonical transformation using Goldstein (3ed), Ch.9. I do understand everything he does in the first section and why we do need a generating function, $F$. The problem I am facing is ...
6
votes
1answer
119 views

Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)

I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase ...
6
votes
0answers
349 views

Upper bounds on phase space momenta

Suppose I wish to calculate the phase space volume for the process $\overline{X}X \to A_1 A_2 A_3 A_4 A_5$ in the CM frame of $\overline{X}, X$ so that $\sqrt{s} = 2m_X$. The volume is given by $$ V \...
1
vote
1answer
285 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\...
2
votes
0answers
74 views

Why the volume of a cell in phase space should be equal to $(2\pi \hbar)^s$?

We want to properly define the concept of entropy using the Boltzmann's Definition of it. But there is a big problem: the coarse graining problem (Id est: How do we count the number of microstates in ...
7
votes
1answer
741 views

Significance of symplectic form in classical field theory

I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions. Given two solutions $\phi_1$, $\phi_2$ of the ...
1
vote
0answers
18 views

$n$-Body Phase Space Recurrence Relation

On slide 23 of these slides, it is stated that an $n$ body phase space element $d\Phi_n(P; p_1, \ldots, p_n)$ may be decomposed according to the recurrence relation \begin{align*} \mathrm{d} \Phi_{n}\...
0
votes
1answer
716 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 ...
0
votes
1answer
63 views

Are Poisson brackets preserved during a canonical transformation?

Fix a Hamiltonian $H(q, p, t)$. Definition: A transformation $(q, p, t)\mapsto (Q(q, p, t), P(q, p, t), t)$ is said to be canonical iff for the Kamiltonian $K$ defined as $H(q, p, t)=K(Q(q, p, t), P(q,...
1
vote
1answer
125 views

Hamilton-Jacobi method with time dependent Hamiltonian

I have the following phase space $$ M = \{ (p, q) \in \mathbb{R}^2 \mid q \geq 0 \} $$ and the Hamiltonian $H = q^2p^2t$. How does one solve for $q(t)$, with $q(0) = q_0 > 0, p(0) = p_0$ using the ...
19
votes
1answer
2k views

Which transformations are canonical?

Which transformations are canonical?
1
vote
0answers
21 views

Clever phase space parametrization of $n$-body processes

Consider the process $X\to A_1 A_2 A_3$ where $X$ in the frame where $X$ is at rest with mass $m$. It is well known from Dalitz that we may choose a frame to evaluate the Lorentz invariant phase space ...
1
vote
1answer
134 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 ...
1
vote
1answer
42 views

How is the relationship between the old and the new canonical variables justified?

In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
1
vote
0answers
51 views

Multiplicative inverse of Weyl symbol and invertibility of operator

If the Weyl symbol $A_W$ of an operator $\hat{A}$ has a multiplicative inverse at every point of the phase-space, can I conclude that $\hat{A}$ is invertible?

1
2 3 4 5
13