# 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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### Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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### A question on Lagrangian dynamics an the velocity phase space

I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
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### Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0$$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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### Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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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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### When can action-angle variables be defined?

According to Goldstein, "We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (...
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### Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
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### Showing time-dependent transformations are canonical

In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$Q = Q(q,p,t) \tag{9.4a}$$ and $$P = P(q,p,t) \tag{9.4b}$$ such that if $H$ ...
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### 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 ...
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### Invariant Phase space volume under canonical transfromation

I have a volume element in phase space: $$d\omega = \prod _{i=1}^{N}(dq_{i},dp_{i})$$ Now I should show the invariance of this product under canonical transformations. I think first I would have to ...
I will use the example of one dimensional harmonic oscillator to explain my doubt. $$H=\frac{p^2}{2}+\frac{q^2}{2}.$$ We allow, $S=W-\alpha t.$ We get, $$\frac{1}{2}\left(\frac{\partial W}{\partial ... 0answers 114 views ### Any model integrable but not separable? In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable? 0answers 56 views ### Is the reversal of the movement of all particles in phase-space, the expansion of space and the development of quantumfields equal to time reversal? In an earlier question, which was called a duplicate, I already asked a similar question. But I state my question differently now and add something to it. Suppose "we" would reverse the direction of ... 1answer 63 views ### How to find the right constraints on a canonical transformation? I have a 1-dof Hamiltonian in the variables p,q. I am not sure on how to find the constraints on a canonical tansformation \psi(P,Q)=(p,q) such that the Hamiltonian in the new variables P and Q... 0answers 94 views ### Does an stereographical projection of a physical plane have any physical meaning? Mathematically, an arbitrary 2D plane can be mapped onto a sphere by stereographical projection. Each line on the plane is equivalent to each line on the sphere. If the sphere rotates under the SO(3)... 1answer 245 views ### Can the Wigner function be described using coherent states? The Wigner function for a wave function \Psi(\vec{r}) is$$ W(\vec{r},\vec{k}) = \frac{1}{2\pi} \int dy e^{-i \vec{k} \cdot \vec{y}} \Psi^{*}(\vec{r}-\vec{y}/2) \Psi(\vec{r}+\vec{y}/2) . \tag{1} $$... 0answers 68 views ### Commutator implies division of phase space in cells of area h? There is a detail from a very well-written answer here that interested me. Unfortunately Springer is charging 41 Euros to access the paper cited by the author. I wonder if someone could elaborate on ... 0answers 103 views ### Canonical transformations I was searching for some more information about canonical transformations. Thus consider canonical coordinates (p_i,q_i) and an other set of canonical coordinates (P_i, Q_i). Both sets will ... 0answers 426 views ### Phase plane of simple pendulum I'm trying to create a phase plane of simple pendulum motion by plotting \dot\theta against \theta in Matlab. I have the equation \ddot\theta + \sin\theta = 0, then by integrating I come to the ... 0answers 137 views ### What physical system does the Hamiltonian  H = \frac{q^4p^2}{2\mu} + \frac{\lambda}{q^2} describe? My question is basically in the title: I'm training myself on exams of Hamiltonian mechanics and I had this Hamiltonian$$ H = \frac{q^4p^2}{2\mu} + \frac{\lambda}{q^2},$$without any interpretation. ... 0answers 222 views ### Probability conservation versus Liouville's theorem This question arises to my mind while studying notes on Kinetic theory written by Prof. David Tong. There he derived the Liouville’s equation. The outline of his derivation goes as follows: Our ... 0answers 354 views ### What is the phase space volume in terms of angular momentum? Given a rigid rotor Hamiltonian, defined along the principle axes as$$ H = \sum_{i=1}^3 \frac{L_i^2}{2I_i}  say we would like to compute the classical partition function of this system. Is the ...
I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...