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
0 answers
12 views

Husimi $Q$-function of Infinite Square well

Eigen-Wavefunction of infinite square well is $$\psi(x)=\sqrt{2/l}\sin(n\pi x/l).$$ I want to write Husimi $Q$ function for infinite square well. General expression of Q function is $$Q=(1/2)\pi \...
  • 743
2 votes
0 answers
55 views

$ΔqΔp \geq \frac{\hbar}{2}$ while $h$ is the cell in phase space

I can show how: \begin{equation} ΔqΔp \geq \frac{\hbar}{2} \end{equation} In a general way. You just need to consider the commutation properties of 2 generic operators to do so. Both $p$ and $q$ ...
  • 31
2 votes
0 answers
16 views

Phase space volume for multiple particles

How do I calculate the phase space volume for an ensemble of more than one (in this particular case, three) free moving particle with the same mass $m$ moving in one dimension? I do understand how to ...
0 votes
0 answers
65 views

Does the canonical ensemble prove that the assumption of equal probability is true? [duplicate]

First, the system is copied $\mathscr{N}$, and the number of systems on each microstate $j$ is $n_j$, we have: $$ \begin{align} \sum_{j}^{state} n_j =& \mathscr{N} \tag{1} \\ \sum_{j}^{state} n_j ...
2 votes
1 answer
30 views

Partial derivative of momentum with respect to position in Poisson bracket representation

The representation of a Poisson bracket is given by the following equation: $$\tag{1} \{f,g\} = \sum_{s=1}^n \sum_{i=1}^{d=3}\left ( \frac{\partial f}{\partial x_i^{(s)}} \frac{\partial g}{\partial ...
  • 125
2 votes
1 answer
54 views

What is the difference between the state $| \psi \rangle$ of quantum mechanics and the microscopic state $St(q,p)$ of statistical mechanics?

In terms of physical quantities, the $|\psi \rangle$ of quantum mechanics and the microstate $St(q,p)$ of statistical mechanics are both a vector, and the microstate of statistical mechanics can be ...
2 votes
1 answer
106 views

Proof that Hamiltonian is constant if Lagrangian doesn't depend explicitly on time

I know that on solutions of motion we have $\frac{dH}{dt}=\frac{\partial H}{\partial t} $ and i understand the proof for this fact. Then, we have that $$\frac{\partial H}{\partial t}=-\frac{\partial L}...
  • 31
1 vote
1 answer
22 views

Area of Phase Space and Dependence on Energy

The phase curve for a system is made for some configuration, for example - The Harmonic Oscillator. Now as we increase the energy, the phase curve enlarges i.e. area enclosed by the curve increases. ...
1 vote
0 answers
41 views

A problem understanding primary constraints meaning

I have some problems understanding the meaning of a function that vanishes weakly. As far as I can understand, when somebody writes that a function $F$ in the phase space vanishes weakly, that means ...
1 vote
1 answer
242 views

The equivalence of the Stosszahlansatz and the usual Boltzmann entropy arguments?

Question Below I show one can use the Boltzmann Stosszahlansatz to independently arrive at the Maxwell Boltzmann distribution without using the usual phase space arguments (Assuming $*$ equation has a ...
6 votes
3 answers
930 views

A rule for when phase-space orbits may cross

Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (...
  • 1,178
1 vote
0 answers
30 views

Why is the action an adiabatic invariant in a unidimensional oscillator?

I'm reading Rax's "Méchanique Analytique" but I can't understand a particular step. We consider a unidimensional oscillator system with a potential that depends on a parameter $\lambda(t)$ ...
0 votes
0 answers
37 views

Phase space of $\phi$-Meson decay

A $\phi$-Meson can decay into an electron-positron-pair or a pair of Kaons. In which decay is the phase space bigger?
  • 371
0 votes
1 answer
70 views

Classifying canonical transformation and scaling transformation

Lets assume we have a very simple transformation in 1 Dimension from $(x, p_x)\rightarrow (y,p_y)$ given as $$\begin{aligned} y &= cx \\ p_y &= c^{-1} p_x \end{aligned}$$ Is this a strictly ...
  • 1,823
3 votes
1 answer
97 views

Punchline of Liouville's Theorem

Reif's Fundamentals of Statistical and Thermal Physics, pages 627-628, presents Liouville's theorem. I do not understand the punchline. Starting with Hamilton's equations, they derive $$\frac{\partial\...
2 votes
1 answer
70 views

Why is it useful to learn about Hamiltonian Mechanics in the framework of Symplectic Geometry?

...Other than providing a deeper insight into the mathematical background of dynamical systems. Does casting certain classes of problems in terms of symplectic geometry make solving them easier/...
2 votes
1 answer
96 views

Why do we construct Lagrangian submanifolds after symplectic reductions

I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts. I have read that Lagrangian ...
  • 135
4 votes
1 answer
132 views

Are Hamilton's equations reversible?

Say I define a time dependent vector field $\Psi(t):\mathbb{R}^d\to \mathbb{R}^d$ as reversible (also here) if, for $f(x,y)=(x,-y)$, we have: $$ f\circ \Psi \circ f =\Psi(-t)=\Psi^{-1}(t).$$ Just to ...
2 votes
2 answers
58 views

Difference between stable manifold and basin of attraction?

In 'Nonlinear Dynamics and Chaos' by S. Strogatz, a distinction is made between a stable manifold and basin of attraction of a fixed point in phase space: Here, the stable manifold of a saddle point ...
  • 67
5 votes
1 answer
87 views

On-shell Poisson brackets and time derivative

In classical statistical mechanics, the information about a given system is given by a distribution of probability over phase space $\rho(p,q,t)$. Let $H(p,q, t)$ be the hamiltonian of the system and $...
  • 1,187
0 votes
2 answers
161 views

Why are Hamilton's equations sometimes written with a gradient?

I am used to seeing Hamilton's written as: $$\frac{dq_j}{dt} = \frac{\partial H}{\partial p_j}\\ \frac{dp_j}{dt} = - \frac{\partial H}{\partial q_j}.$$ However I have also seen it written as $$\frac{...
  • 494
2 votes
1 answer
82 views

The different generators of canonical transformations

Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus ...
  • 374
0 votes
1 answer
54 views

Integration by Parts in Liouville's Theorem

I am looking at a proof of Liouville's Theorem, which states that for $F, G \in C_0^\infty$ and a Hamiltonian $H$, the operator $$D_H = \sum_{i=1}^n\Big(\frac{\partial H}{\partial p_i} \frac{\partial}{...
  • 494
0 votes
2 answers
63 views

In an $n$ particle system, why is the Hamiltonian summed over $n$?

Suppose I am working in a system consisting of $n$ particles. Thus the phase space will be $\mathbb{R}^{6n}$, and both the momentum and position space will be $\mathbb{R}^{3n}$ each. Then, for some ...
  • 494
1 vote
2 answers
235 views

What is the theoretical value of this phase space invariant?

So I wanted know how to theoretically calculate this phase space invariant (equation $3.31a$ )$R$ in our universe (FLRW metric) during the cosmological nucleosysthesis: $$R = \int_{p} \frac{\mathcal{...
1 vote
0 answers
34 views

Potentials that prevent the phase flow of the system [closed]

I am trying to solve a question that my professor gave. When a particle moves in one dimension $x$ in a potential $U(x)$ , the resulting motion over a very short time interval is specified by Newton’...
  • 11
1 vote
2 answers
73 views

In Hamilton-Jacobi theory, how is the new coordinate $Q$ time-independent when Hamilton's principal function separates?

Following the notation in Goldstein, the solution to the Hamilton-Jacobi equation is the generating function $S$ for a canonical transformation from old variables $(q,p)$ to new variables $(Q,P)$ ...
  • 13
4 votes
1 answer
102 views

Liouville's Theorem & Flows in Phase Space for Particle in a Box

A Hamiltonian system of $100$ interacting oxygen atoms, each of mass $16$ $m_p$, is confined within a cubical box of sides $1 m$. The average initial speed of each particle is $300 ms^{-1}$. Estimate ...
  • 503
1 vote
0 answers
20 views

Liouville's theorem on the tangent bundle [duplicate]

One interpretation of Liouville's theorem is the determinism and reversibility of classical mechanics, i.e. the mechanical states can't converge or diverge. The theorem is often formulated on the ...
  • 223
0 votes
0 answers
41 views

Volume element in system phase space

Consider $N$ particles in $3D$, with coordinates $q_i$ and momenta $p_i$, so $\{q_1,p_1,q_2,p_2,...,q_{3N},p_{3N}\}$ are variables. Construct a phase space of the system, with axes $(q_1,p_1,q_2,p_2,.....
  • 67
2 votes
1 answer
48 views

Confusion about the action variable definition

Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set $$M_f = \{ (p,q)...
  • 1,018
0 votes
0 answers
17 views

Time-dependent canonical transform step in Hamiltonian perturbation theory (Percival problem 8.20)

In Percival and Richards's great book, "Introduction to Dynamics", problem 8.20 asks the following question. Any insight on how to solve this would be appreciated: Consider a system with ...
0 votes
1 answer
108 views

Is this hamiltonian of the form of some well-known physical system?

I'm doing a homework exercise and I'm asked whether some hamiltonian (that is the result of a canonical transformation of some other hamiltonian) is reminiscent of the hamiltonian of some well-known ...
  • 592
0 votes
0 answers
46 views

In canonical transformation, is there any rules or methods for finding the transformation $(q,p)\to(Q,P)$?

If we get two different Hamiltonian by using two methods of canonical formulation of theory and these two Hamiltonian are equivalent. How can I find the canonical transformation from which we can ...
  • 1
2 votes
1 answer
73 views

How to find canonical transformation to achieve desired Hamiltonian?

I am trying to find a way to transformation that will turn a Hamiltonian from one form into another form: $$(1)\;\;\;H=p^2+e^x\rightarrow\bar{H}=p'^2.$$ I don't know of any systematic ways to do this. ...
  • 57
2 votes
0 answers
38 views

Can conservation of phase space volume be viewed as a consequence of some symmetry via Noether's theorem? [duplicate]

Liouville's theorem says that for the Hamiltonian evolution of a system, the flow of points on the phase space with time is like that of an incompressible fluid i.e. the phase space density is ...
0 votes
1 answer
43 views

Existence of a unitary transform $(q,p) \rightarrow (-q, p)$

If $q$ and $p$ are the canonical position and momentum operators of a quantum harmonic oscillator, is there a unitary that transforms $(q,p)$ into $(-q, p)$? For instance, denoting the annihilation ...
  • 377
0 votes
1 answer
107 views

Microstate and Phase space

Pathria, Statistical mechanics, 4ed,pg32-33 "The microstate of a given classical system, at any time, may be defined by specifying the instantaneous positions and momenta of all the particles ...
  • 830
1 vote
1 answer
55 views

Why is this the requirement for invertibility within the context of canonical transformations in mechanics?

I'm reading "Analytical Mechanics", by Hand and Finch. In page 210, there's the following statement, regarding the generating function $F$ for some lagrangian such that $L'=\lambda L-\frac{\...
  • 592
0 votes
1 answer
84 views

Uncertainty principle in deformation quantization

Deformation quantization procedure is a well-known way to quantize a classical phase space (at least formally for Poisson manifolds which is known as formal deformation quantization). Although it is a ...
  • 391
0 votes
0 answers
25 views

Canonical transformation of 2-body hamiltonian into center of mass and translation components. How to gain an expressions for the conjugate momenta?

I am trying to transfom a quantum hamiltonian as detailed in this website. It starts of by using the classical hamiltonian and conjugate momenta: $$ H\equiv \frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V\...
  • 31
2 votes
1 answer
198 views

Construct the Density Matrix of a Gaussian State from its First and Second Moments / Wigner Function

Borrowing some description for the setup from a question I posted earlier here; Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the ...
1 vote
1 answer
125 views

Ability to represent canonical transformation generating functions in different forms?

In Goldstein's Classical Mechanics (3rd edition) section 9.1, we introduce the generating function method of describing a canonical transformation. We then introduce four types of generating functions:...
0 votes
1 answer
45 views

Given the Symplectic Matrix acting on phase space, find the Gaussian Unitary acting on the Hilbert space

In Gaussian Quantum Mechanics, a unitary preserving the Gaussian nature of the state is a called a Gaussian Unitary. In the phase space picture, a Gaussian state is fully characterized by its first ...
0 votes
0 answers
42 views

Phase flow and potential energy

I have a question that our teacher gave us and this is my very first time I see concepts like phase flows. Prove that a positive potential energy always guarantees a phase flow. I should use ...
0 votes
1 answer
90 views

About the curvature of solutions of Hamilton's equations

I am a math major and have recently stumbled on the Hamilton's system of equations in the context of Hamiltonian Monte Carlo Markov chains on a continuous state space, say $\mathbb{R}^d$. I am trying ...
  • 103
3 votes
1 answer
58 views

Why the generating function $F = q + Q$ form Hamiltonian with old coordinate dependence?

In the book Analytical Mechanics by Louis N.Hand, chapter 6, Question 1, It is asked to use the generating function $$F = q + Q\tag{1}$$ to any Hamiltonian (I have used Harmonic oscillator). By doing ...
-3 votes
3 answers
116 views

Does the phase space exist in reality? [closed]

The concept of phase space really bothers me sometimes and the term is used across many branches of physics such as statistical mechanics, classical mechanics as well as in quantum mechanics. Does ...
1 vote
0 answers
35 views

Is the reverse of Liouville's theorem true?

Liouville's theorem states phase space volume is conserved for Hamiltonian systems. Given a general dynamical system, if it is shown to have conserved phase space volume, will it also have a ...
  • 111
1 vote
0 answers
81 views

How Maxwell-Boltzmann distribution is related to phase space?

Maxwell-Boltzmann distribution give us the probability of particle to have a speed between $u$ and $u+du$. Can we interpret this distribution as how many times a specific speed is found in the phase ...

1
2 3 4 5
16