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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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)...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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{\...
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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 ...
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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\...
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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 ...
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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:...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Behavior of Velocity Distribution under Lorentz Boost

I'm trying to relate the velocity distribution function $f(v)$, defined over phase space such that $f(v)d^3v d^3x$ is the number of particles in the phase space volume $d^3vd^3x$, between two frames, ...
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Sources and sinks in phase space

I am studying the concept of phase space in thermodynamics and there is something for which I'd like to have a physical understanding as to what it is or what it represents. If we have sources or ...
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Spherical co-ordinates are not canonical?

One of the conditions for canonical transformations is that all momentum variables should commute. But $(L_x ,L_y)=L_z \neq 0$. Does that mean these are not canonical co ordinates? But aren't point ...
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What do Canonical Transformations actually represent in phase space?

While we read canonical transformations in classical mechanics we learn that Generalized co-ordinates before and after transformation should obey Hamilton's Canonical equations, But we know that ...
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Quantum canonical transformation

This post is very similar in content to this one. I'm looking for a quantum implementation of the transformations $$ x_i \to x_i + f(p) p_i, $$ $$ p_i \to h(p) p_i. $$ In these, the subindex $i$ ...
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What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$?

I would like to calculate the Wigner representation of the operators $\left(\hat{x}^2+\hat{p}^2\right)^3$ and $\left(\hat{x}^2+\hat{p}^2\right)^4$. I know at least two ways to do it, but both rely on ...
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Density of states in microcanonical ensemble for discrete and continuum energy spectrum

I'm introducing myself to statistical mechanics using two books: Introduction to Statistical Physics by S. Salinas, and Statistical Physics of Particles, by Mehran Kardar. Both textbooks work on an ...
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What is a flow generated by a function in phase space?

I am reading the book "Quantum Field Theory" by Jean-Bernard Zuber and Claude Itzykson. I encounter great difficulties from page 457 section 9-3-1, which introduces Dirac's constrained ...
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Ergodic hypothesis definition confusion

In the Wikipedia article about the Ergodic hypothesis (https://en.wikipedia.org/wiki/Ergodic_hypothesis) this is what the hypothesis say: over long periods of time, the time spent by a system in some ...
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Understanding ergodicity and what an ergodic system is

I am trying to understand the concept of ergodicity/ergodic system in physics, but because my understanding of phase space, its elements is a bit unclear,I have trouble understanding the former. ...
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Time evolution of a microstate

In phase space each point represents a microstate of the system. And for a system in a certain macrostate, depending whether we have a MCE,CE, or GCE, is a mixed state of pure states, the microstates ...
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How to draw the phase plane of this equation?

Using various computational tools, it's possible to draw a phase plane from two first-order ODEs or a single second-order ODE. However, when there is a parameter in the equation and we don't know the ...
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Microcanonical ensemble probability density distribution

In microcanonical ensemble the probability density function is postulated as $\rho(q,p)=const.\times\delta(E-E_0)$ so the probability of an ensemble being in an element of phase space $\mathrm{d} q \...
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Time dependent canonical transformations (a problem in Arnold's classical mechanics textbook)

I am stuck on a problem on page 242 of Arnold's book "Mathematical Methods of Classical Mechanics". The problem statement is as follows: Let $g(t): \mathbb{R}^{2 n} \rightarrow \mathbb{R}^{...
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Liouville's theorem and uniform probability density

In Kardar's book on statistical physics it is claimed that Liouville's theorem gives support for the common assumption that the points in phase space compatible with the hamiltonian are all equally ...
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Example of a system which configuration space is a cross $K$ in a plane

In Jet Nestruev's book "Smooth Manifolds and Observables" the author emphasises that a configuration space of a systems can be modeled well by a smooth manifold. One point of the book is to ...
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A coupled nonlinear dynamical system in four dimensional phase space

I have come across a coupled nonlinear dynamical system given below $$ r\, \ddot{x} + \dot{x} = \sin y~,$$ $$ r\, \ddot{y} + \dot{y} = \sin x~,$$ where $r$ is some real number and $\dot{x}$ denotes $\...
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Calculating the average state occupation number of particles (in momentum space)

The average state occupation number of particles in the momentum space (I think "state" refers to a cell in the momentum space) is given by $$\mathcal{N} = n\frac{(2\pi)^3}{\frac{4\pi}{3}(m\...
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What's wrong with this "proof" that all Hamiltonians are time independent?

The time evolution of a system in classical mechanics is given by the solution of Hamilton's equations of motion, which tell us that $$\frac{\mathrm{d}p}{\mathrm{d}t}=-\frac{\partial H}{\partial q},\...
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Reconciling two descriptions of the Wigner function

A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$: $$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
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System of $N$ non-interacting particles with a linear restoring force acting

I am given a system of $N$ non-interacting particles with a linear restoring force acting on them. What will be there phase space. I understand that each particle can be thought of as a linear ...
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Generating Functions for Extended Canonical Transformations

From Goldstein we have that, for non-extended ($\lambda =1 $), the generating function of third type is $$F = F_3(p,Q,t) + q_ip_i.\tag{1}$$ Although I found it hard to see if that would hold true also ...
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Phase space density for Bose-Einstein condensation

A figure of merit for Bose-Einstein condensation is the phase space density which can be defined as $$\rho=n\lambda_T^3,$$ where $n$ is the number density of atoms and $\lambda_T$ the thermal de ...
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How to find the probability of finding quantum particles within a certain region [closed]

How to evaluate the probability of finding a quantum particle within a certain radius $R$ from the origin ? I have not been provided with any radial distribution functions, and I'm not sure how to ...
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Phase space and uncertainty

In phase space every point represents both position and momentum. Isn't it against of Uncertainty principle?
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Canonical / invertible transformation

What distinguishes a canonical transformation from an ordinary invertible Coordinate transformation? I understand what canonical is but searching the difference between canonical and invertible didn't ...
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How do I calculate the Altarelli-Parisi splitting function for $g \rightarrow q\bar{q}$?

I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for ...
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Can every canonical transformation be broken down into a large number of infinitesimal canonical transformations?

Consider a canonical transformation from $(q,p)$ to $(Q,P)$ depending upon a continuous parameter $\alpha$ such that: $$Q_i=Q_i(q,p,t,\alpha), \space P_i=P_i(q,p,t,\alpha)$$ where $q$ and $p$ ...
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What does integrating the probability density function over all phase space gives us?

For a system of N-3D particles, we have 6N D.O.F and therefore a 6N dimensional phase space. I know that one point in phase space represents a possible state of the system. I also understand that a ...
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Phase space, ensemble of systems and probability density function

I am trying to understand the concept of phase space in statistical mechanics. I can understand that a system, with $N$ total degrees of freedom, can be in different states, which correspond to ...
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