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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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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)+\...
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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 ...
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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 ...
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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^{...
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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 $...
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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 ...
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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\...
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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} \...
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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 ...
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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 ...
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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^{...
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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 ...
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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 ...
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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 ...
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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}} \...
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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{\...
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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 ...
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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]{...
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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 ...
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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 ...
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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}{\...
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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. ...
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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 $\...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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?
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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Does Liouville’s theorem hold for any ensemble?

I’m having some difficulties with my Statistical Mechanics course. In particular, I have some doubts about Liouville’s theorem and the various ensembles. Consider, for instance, the Canonical ...
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What if we set Hamilton-Jacobi mechanics as an axiom?

We postulate principle of least action then we get Lagrange mechanics, after we can get Hamilton mechanics either from postulate or lagrange mechanics. Then we get HJE. But what if we have HJE as ...
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Hamilton-Jacobi equation and method of solving it [duplicate]

So, this equation we get when we find canonical transformation that makes new hamiltonian=0. There are 4 main transformations: F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t). On practice and in every book ...
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Fluid mechanics, symplectic structure, the Hamiltonian, and vorticity

Consider an inviscid irrotational fluid in two dimensions. There are some explicit connections with symplectic geometry that I do not understand. I am not well versed in the later topic, so please ...
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Level set of Hamiltonian are the orbits?

Just a small question : If $x(t)=(p(t),q(t))$, then the position $x(t)$ of a particle is given by $$\dot p=-H_q(x(t))\quad \text{and}\quad \dot q=H_p(x(t)).$$ In particular, if $x$ solve the previous ...
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What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
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Poincaré recurrence theorem with irrational frequencies?

The Poincaré recurrence theorem states that, for a bound phase space, the system will return to a state very close to the initial conditions, in some finite time $\tau$. For example, let's say I have ...
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The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
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Hamiltonian description of a system

I know that phase space is the Hamiltonian description of a system, where we deal with position and momentum in equal footing. My question is in this phase space are those position and momentum are ...
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Canonical coordinate

Sorry for my broken English. I'm a physics undergrad and quite poor at math. While reading a mechanics textbook, I've found something I cannot understand. There are coordinates, $(q,p,t)$ $\...
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Phase space as differential manifold

Generally we "draw" phase space as typical coordinate system, where $q$s and $p$s are treated like perpendicular axes. Why do we then regard phase space as generall differential manifold while it ...