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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39 views

Solving Liouville's Equation for the Harmonic Oscillator and Fluctuating energy

I am trying to solve the Liouville eqaution of the classic harmonic oscillator with fluctuating energy and arbitrary initial condition $\rho_0$. I want to approach the problem using the method of ...
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2D Harmonic Oscillator trajectory and Ergodicity [closed]

For 2D Harmonic Oscillator $H(p,q) = \frac12(p_x^2 + p_y^2 + x^2 + y^2)$ For a fixed energy, the motion of the system is uniquely determined by the initial conditions $(p(0), q(0)) = (p_0, q_0)$. I ...
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Jacobian rules with canonical transformations

If we consider a canonical transformation from $(q,p)$ to $(Q,P)$, it is stated in several sources that by Jacobian rules, $$ \frac{\partial(Q,P)}{\partial(q,p)} = \frac{\partial(Q,P)/\partial(q,P)}{...
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Phasor Notation Convention - Which Projection to Consider?

In the following video (not necessary to watch)... https://www.youtube.com/watch?v=K_vWYEjgVRg&list=PLdnqjKaksr8pXF2SpDyyD7ouAVlz96_Ra&index=23 ...the creator takes the projection of the ...
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Bounds on the locality of single-particle coherence

It's well known in the multi-partite setting that entanglement can't be generated by local operations and classical communication; indeed, this is often taken as one of the defining properties of ...
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Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

I am very new to the subject, so please forgive my very naïf question. I learned that there are some non-hamiltonian systems which can become hamiltonian, just by a change of coordinates. I was given ...
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Geometric Intuition for the Moyal Product

I've recently been reading into deformation quantization as another formulation of quantum mechanics. I have focused on understanding the Moyal product in particular, as it contains the seeds for the ...
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Strange fixed point in state space [migrated]

I'm studying the following dynamical system, \begin{align} \dot{x} &= y \,\, , \\ \dot{y} &= \frac{\left(-4 x^3+33 x^2-78 x+54\right) y^2+(x-3) (2 x-6)^2}{(3-2 x) (2-x) x (2 x-6)} \,\, , \end{...
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53 views

Contradiction in canonical transformation

The problem I'm supposed to solve is finding $Q$, such that $(p,q)\rightarrow(P,Q)$ is a canonical transformation. In this case $\mathcal{H}=\frac{p^{2}+q^{2}}{2}$ and the new hamiltonian $\mathcal{K}$...
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Hydrodynamic force in complex form for fluid structure interaction

I am studying force being acted from a viscous fluid surrounding a cantilever cylindrical rod, the rod is vibrating and its velocity causes the fluid around it to flow in the plane perpendicular to ...
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Justification for taking the new Hamiltonian (Kamiltonian) as zero in deriving the Hamilton-Jacobi Equation

I was reading MG Calkin's text, Lagrangian and Hamiltonian Mechanics. On p147 of his text, he derives the Hamilton-Jacobi equation using the type 2 generating approach. He proceeds to the point that ...
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Question about a 2D Harmonic Oscillator with incommensurate frequencies and Integrability

In Classical Dynamics by José & Saletan [section 4.2.2] they give the example of a 2D Harmonic Oscillator whose equations of motion are \begin{equation} \ddot{x}_i+\omega_i^2x_i=0 \ \ \ \ \ \text{...
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Calculate new Hamiltonian which has to be independent on time

If after making a canonical transformation I get a new Hamiltonian $K(X,P,t)$ in terms of these three coordinates and initial velocity, if I want to make the Hamiltonian independent on time, do I have ...
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Does the Hamilton-Jacobi equation imply that there are always $N$ conserved quantities for any system with $N$ degrees of freedom? [duplicate]

I'm reviewing the Hamilton-Jacobi equation because I'm working on a research project about Kerr black holes and the geodesics of particles gravitating them (This is not really relevant to the question,...
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Classical analog of state vector

I can’t believe I don’t know the answer to this question. What is the classical analog of the state vector of quantum mechanics?
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Time-independent Canonical transformation conditions

A solution manual to Goldstein's $9.1$ states that for an explicit time-independent transformation (for a system with two degrees of freedom) to be canonical it must satisfy $$\frac{\partial Q}{\...
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Change of variables in an expression involving differentials

I'm trying to derive Maxwell-Boltzmann's distribution using the statistics of the classical canonical ensemble. Doing some operations, I have found out that the probability that a molecule of a gas ...
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Are electric and magnetic fields canonical conjugates?

When quantizing the electromagnetic fields in the context of quantum optics and quantum field theory, we often go for the vector potential $\mathbf{A}$ and its canonical momentum, which turns out to ...
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About the $6N$ dimension when describing the Boltzmann equation

I have a question related with the formulation of the Boltzmann equation. In all the documents I read, it appears that the system is made of $N$ molecules, and so the phase space has $6N$ dimensions (...
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Calculating entropy in truncated Wigner

I'm trying to get some reasonable measure of the entropy of a system modelled by the truncated Wigner method. The Wigner function contains all the information about a density matrix. So, I figure it ...
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Why can a partial derivative be added to a hamiltonian in canonical transformations?

In canonical transformations, how come we allow hamiltonian to change by a partial derivative of time? $$H'(P, Q, t) = H(p, q, t) + \frac{\partial F}{\partial t}.$$ Here $F$ is the generating function....
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Determinisitic system with a probablisitic initial condition [closed]

Consider a deterministic system like a spring mass damper. Lets say we do not know the exact initial condition but we are given a probability distribution function (PDF), $p(x,v,t = 0)$ of the mass's ...
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Geometrical description of canonical commutation relation [duplicate]

Does there exist any geometrical description of canonical commutation relation of quantum mechanics $$[\hat{x},\hat{p}]=i\hbar$$ maybe in phase space? What I meant by geometry is along the lines ...
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Geometric description of canonical commutation relation

Does there exist any geometrical description of canonical commutation relation (CCR) of quantum mechanics $$[\hat{x},\hat{p}] = i \hbar \, ,$$ e.g. in phase space? The commutator, along with the ...
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Confusion regarding properties of Poisson Brackets

I have just started learning about Poisson Brackets, and came across the following property $$\{q_i,q_j\}=0$$ And $$\{p_i,p_j\}=0.$$ Where $p$ and $q$ are respectively the momentum and position ...
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Why is the change in Hamiltonian for an active infinitesimal canonical transformation defined the way it is?

I'm trying to understand infinitesimal canonical transformations and conservation theorems (section 9.6 Goldstein ed3). My specific problem is with understanding eq 9.104, $\partial H = H(B) - ...
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Transform from Cartesian coordinates to Delaunay elements

Place a central mass $M = 1/G$ at the origin $(0,0,0)$. We know that for a test particle moving in the $xy$-plane, we can transform $(x,y,p_x,p_y)$ to the Delaunay elements $(\theta_\lambda, \theta_\...
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3-Body Phase in a $2\rightarrow 3$ scattering [closed]

I am trying to calculate Bethe-Heitler cross section for pair production. I am starting from the well known formula \begin{equation} d\sigma=\cfrac{1}{2E_1 2E_p}\vert M_0\vert^2(2\pi)^4\delta^4(p_1+p-...
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Canonically conjugate variable of rapidity [closed]

Does the rapidity $\theta \in \mathbb{R}$ have a canonically conjugate variable? More specifically, for some smooth function $f \in \mathcal{S}(\mathbb{R})$, by Plancherel's theorem we have (up to ...
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Extension of classical Liouville operator

Let us consider a classical Hamiltonian system described by the Hamiltonian \begin{equation} H(q,p) =\frac{p^2}{2m}+V(q) \end{equation} where we stick to the case of single particle for simplicity. I ...
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Can Liouville's theorem describe the passage from nonequilibrium to equilibrium?

Let an isolated system starts in a nonequilibrium state at a time $t=0$. Then it is left undisturbed so that at a later time $t>0$, it comes to equilibrium. When it reaches equilibrium, the number ...
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How are the authors obtaining the asymptotic form of the sympletic form for the Maxwell + massive field system?

I've been studying the paper "Asymptotic symmetries of QED and Weinberg’s soft photon theorem" by Campiglia & Laddha and there is one step in their analysis I'm being unable to understand. I shall ...
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Why are positions and momenta independent variables in the Hamiltonian formulation?

Lifshitz and Landau's Vol. $1$ explicitly states that $$ \cfrac{\partial{q_k}}{\partial{p_i}} = 0$$ And seems to imply also that $$ \cfrac{\partial{p_k}}{\partial{q_i}} = 0.$$ I guess that whenever ...
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$P$ representation of a general Gaussian state

Let $\rho$ be the density operator of a Gaussian quantum state on $M$ modes. This implies that its Wigner function can be written as $$ W_{\text{Gaussian}}\left(\boldsymbol{q},\boldsymbol{p}\right)=\...
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The Functional Methods in Peskin and Schroeder (page 280)

I'm working on the Functional Methods in Peskin (page 280) However, I canʻt obtain Eq.(b) and Eq.(c) from Eq.(a) Consider Eq.(a) \begin{align} \left\langle q_{k+1}|f(q)| q_{k}\right\rangle&= ...
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A question on Liouville's theorem and time-dependence of the Hamiltonian

The condition of equilibrium in statistical mechanics is $\frac{\partial \rho}{\partial t}=0$ where $\rho$ is the phase space density. By virtue of Liouville's theorem, this is equivalent to the ...
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Hamiltonian dependence of variables

How can one say that in Hamilton mechanics the $q$'s are independent of the $p$'s while if I have the Lagrangian $L = \frac{1}{2}\dot{x}^2 + \frac{1}{2}x^2\dot{y}^2$ then $p_y = \frac{\partial{L}}{\...
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Is it useful to define canonical pressure?

Pressure is defined as: $$ P = \frac{\partial U}{\partial V}$$ where $V$ is the volume and $U$ is the internal energy. Does the quantity $P'$: $$ P' = \frac{\partial U}{\partial V'}$$ where $V'$ ...
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Particle states per phase space volume of a quantum gas

Consider a quantum gas in a box with volume $V=L^3$ and temperature $T$, consisting of spinless bosons. Explain why the number $dN$ of "single-particle states" in a "momentum space volume" $d^3p$ is ...
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Can any sum of infinitesimal canonical transforms on phase space be obtained from evolution under a static Hamiltonian?

Suppose I have a canonical transformation on phase space, which is obtained by evolving a classical Hamiltonian system from time $t=0$ to $t=T$, with some arbitrary time-dependent Hamiltonian $H(t)$. ...
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Application of non-commutative geometry in quantum mechanics [closed]

What are some of the applications of Connes' non-commutative geometry in quantum mechanics? Is it useful in defining and studying phase structure of a quantum system since we have $[\hat{x},\hat{p}]=...
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Generating function for canonical transformation

Short version: I've been reading through some notes on integrable systems/Hamiltonian dynamics, and am stuck on a problem: Show that the coordinate transformation derived via the generating function ...
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Are probabilities equivalent to a metric in phase space?

Say $P(A)=|\Psi(A)|^2$ is the probability of event $A$. So to add up many probabilities we would have: $\int P(A)dA$. But $P(A)$ looks like some sort of weight on the state $A$. This looks a lot like ...
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Would a world-sheet through phase-space make sense?

Imagine every possible configuration of the Universe as points in a phase-space. A path from one point A to another point B represents a 'history'. This is a one dimensional idea. But imagine ...
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Reversibility of Hamiltonian dynamics

I'm trying to understand a very basic property of Hamiltonian dynamics. I don't have a physics background but I do know some mathematics. I want to understand why negating the momentum is equivalent ...
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Replacing a sum with an integral in the limit of a large system

I have the expression $$\sum_{\vec{x}} \sum_{\vec{p}} e^{-\beta \, h(x,p)}$$ for a single particle in a box with volume V and the energy h(x,p). Then I have to bring it in an exercise to the form $$\...
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Rigorous canonical coordinate definition

I would like to know what is the definition of a canonical coordinate. Let's assume we have a Lagrangian $\mathcal{L}(q,\dot{q})$. Are the canonical coordinate simply the set $(q,p)$ where $p=\frac{\...
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Phase space formulation: “Representation” vs “function” vs “quasi-probability distribution”

In the phase space formulation, the terms "representation", "function, and "quasi-probability distribution" (as in Glauber–Sudarshan P representation, $P$-function) seem to be used interchangeably. I ...
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Phase space cell volume

Iam studying decay rates using QFT by Lahiri and Pal. At this point I stuck. The book says that "Phase space can be devided into cells of volume $(2\pi \hbar)^3$ and putting 1 state into each ...
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Why I find two possible $S$ of a free particle by solving the Hamilton-Jacobi Equation?

For a free particle, the Hamiltonian is $$H(p)=\frac{p^2}{2m}.$$ The corresponding H-J equation thus can be written as $$\frac{1}{2m} \left(\frac{\partial S(q,t)}{\partial q}\right)^2=- \frac{\...

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