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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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### Question about ergodicity and the evolution of the probability distribution under Liouville's theorem

According to Liouville's theorem, the probability distribution function $\rho$ evolve in phase space with $$\frac{d \rho}{d t} = \frac{\partial \rho}{\partial t}+\left\{\rho,H\right\}_{P.B} =0$$ ...
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### Lorentz invariance of volume element from the four-volume element: why on-shell?

In Srednicki's QFT book, in chapter 3 (eqn. 3.16 onwards) he talks about the lorentz invariance of the volume element. For this he writes $d^3k/f(k)$ should be invariant under lorentz transformations. ...
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### Difference between conserved quantities and constants of motion?

In Hamiltonian mechanics, consider extended phase space, the trajectory followed by a particle in that space is formed by an intersection of different 2n dimensional surfaces, all of these surfaces ...
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### Hamilton-Jacobi theory vs Hamiltonian formalism

I'm writing some notes on Hamilton-Jacobi Theory and I'd like to find an example of a system that is quite difficult to integrate in the usual Hamiltonian formalism, but quite easy in the Hamilton-...
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### Boundary conditions for calculus of variations in phase space and under canonical transformations

This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ ...
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### How can momentum and position be combined into a phase space when they have different units?

Elaboration of the question: What is the geometrical interpretation of units? As in, a unit of length is a choice of scaling of the coordinate systems i.e. it is a choice of diffeomorphism, but then ...
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### Mean free path for $3\rightarrow 1$ scattering

I want to calculate the mean free path of an antineutrino in nuclear matter where it can undergo the reaction $p+e^-+\bar{\nu} \rightarrow n$, which I imagine will involve calculating the rate of that ...
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### Why exactly is the Husimi-Q distribution not a real probability distribution?

From this question I understood that the uncertainty principle is causing a problem because two points $x,p$ and $x',p'$ in phase space can be confused. Why exactly is this a problem? I don't grasp ...
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### Mixed canonical transformation

Wikipedia and most authors denote four types of canonical transformations: $F_1(\mathbf{q},\mathbf{Q})$ , $F_2(\mathbf{q},\mathbf{P})$, $F_3(\mathbf{p},\mathbf{Q})$ and $F_4(\mathbf{p},\mathbf{P})$. ...
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### How can a pendulum have amplitude greater than $\pi$?

How can a pendulum have amplitude angle greater than $\pi$? I've been reading about phase plots, which are graphs of the $\frac{d\theta}{dt}$ on the $y$ axis and $\theta$ on the $x$ axis, shown below. ...
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### Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
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### Is a phase space a function? [closed]

I saw a graph of a phase space of a pendulum and it looks like an $x-y$ plane with a spiral representing the speed and position (I assume from the origin). Are all phase spaces two dimensional, or is ...
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### What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid. However, in the phase space formulation ...
We can consider a dynamical theory to be a "transport theory" if it can be described entirely by a series of continuity equations of the form: $$\frac{\partial \rho}{\partial t} + \nabla \cdot \left({... 2answers 910 views ### How do contact transformations differ from canonical transformations? From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval dt can be described by an infinitesimal contact transformation generated ... 1answer 60 views ### Symplectic Standard map [closed] I have come across this map, which the notes call standard symplectic map. Why is it symplectic? How do I show it? Are those action-angle variables? I(t+1)=I(t)+K\sinθ(t) θ(t+1)=θ(t)+I(t+1) \quad ... 2answers 171 views ### Physical interpretation of differences between classical and quantum ensemble dynamics The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. ... 1answer 151 views ### Lagrangian and Hamiltonian dynamics, momentum and canonical transformations I am relatively new to Lagrangian and Hamiltonian dynamics. I am aware of how to form the equations of motion using the Legendre Transformation. I, however, have one fundamental question and I was ... 1answer 99 views ### Linear canonical transformation represented by a unitary operator I am reading a paper on Squeezed states which mentions the following fact "a linear canonical transformation can be represented by a unitary transformation" and then used a operator \hat{U} for ... 0answers 112 views ### Wigner 's unreasonable effectiveness of mathematics in natural sciences [closed] This question is related to Wigner's problem, related to the unreasonable effectiveness of mathematics in natural sciences. Understanding a phenomenon means constructing a mathematical model , and ... 1answer 215 views ### Density function in phase space What does density function in phase space physically mean? How does it indicate, the more familiar density that we are accustomed to ( an analogy may be), in phase space? 1answer 56 views ### Problem with the regularity condition for a constraint So, I'm considering Lagrangian:$$L=\frac{1} {2}e^{q_1}\dot{q}_2^2. $$I obtain the primary constraint \phi=p_1=0. The canonical Hamiltonian is H_c=\frac{1} {2}p_2^2e^{-q_1} , and the total ... 0answers 56 views ### The formula for the average number of fermions \langle N \rangle In the context of Fermi gases (or fluids in general), one would typically in the grand-canonical formalism use the formula \langle N \rangle = -\frac{\partial \psi}{\partial \mu}, where \psi is ... 0answers 50 views ### Four body decay rate I would like to calculate a 4-body decay rate and I'm stuck. I have a massive scalar particle (with mass M) decaying to four massless particles (2 fermions and 2 scalars). I am not sure how to ... 1answer 808 views ### Poisson Brackets And Angular Momentum Components Related: Poisson brackets of angular momentum When Poisson Brackets are taught as part of an Analytical Mechanics courses, examples are commonly shown which anticipate analogue results in QM. One ... 2answers 156 views ### One dimensional system a Hamiltonian system? I have the following equation of motion:$$ \dot x = \beta x y  with $y=1-x$. I would like to see if it is Hamiltonian or not. Due to it being one dimensional, I think it should be locally ...
A Hamiltonian system of $n$ interacting atoms, each of mass $M$, is confined within a cubical box of sides $V$. The average initial speed of each particle is $v$. How do I estimate the timescale for ...