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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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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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33 views

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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49 views

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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59 views

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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52 views

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 ...
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231 views

Why do we care only about canonical transformations?

In Hamiltonian mechanics we search change of coordinates that leaves the Hamilton equation invariant: these are the canonical transformations. My question is: why we want to leave the equations ...
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60 views

Gaussian State Spread [closed]

A measurement device which can be represented by a 1D quantum system (with canonical observables $X$ and $P$) 'is prepared in a Gaussian state with spread $s$' $$\vert \psi \rangle = \frac{1}{(\pi^2s^...
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Hamiltonian as differential manifold

I know that phase space $(q^i, p^i)$ can be treated as manifold. But for me defining hamiltonian as a function also leads to new manifold $(q^i, p^i, H(q^i,p^i))$. Like map from open set $(q^i, p^i) \...
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States of classical general relativity

In Classical Mechanics a state of a system is either a pair $(q,p)$ or $(q,\dot{q})$ depending if we formulate the theory on the tangent or cotangent bundle of the configuration space. The evolution ...
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Why is the partition function an integral over momentum and position?

I am learning statistical mechanics through the series of online lectures from Prof Leonard Susskind, and the partition function derived is $$Z = \sum e^{-\beta E_i} .$$ I understand this to be ...
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Surjectivity of momentum mapping

I have to show that the following mapping of momenta is surjective. The mapping $\{p_i^{\mu},p_j^{\mu},p_k^{\mu}\}\rightarrow\{\tilde{p}_{ij}^{\mu},\tilde{p}_k^{\mu}\}$ is given by $$ \tilde{p}_k^{\...
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3-particle phase space in $d$ dimensions

recently I came across a problem concerning the 3-particle phase space. I am trying to show, that the 3-particle phase space for massless particles with momenta $p_1$, $p_2$ and $k$ is given by $$ d\...
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1answer
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Hamiltonian formalism and the phase space

In my book, it says that Hamilton's equations of motion are equations of the first order in the time and that they describe the motion of the system in the $2S$-dimensional phase space. Could someone ...
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66 views

Poincare return map as area-preserving map

I'm trying to get some intuition into how the Poincare return map is area-preserving (when there are two momenta and two positions). Suppose $H=H(q_1,q_2,p_1,p_2)$, and let's suppose the system is ...
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45 views

Coordinates change for separating the Hamiltonian of a quantum system

Are there general methods, tips or tricks for choosing the correct change of coordinates so that the Hamiltonian of a quantum system becomes separable? Referring to Shankar's Principles of Quantum ...
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40 views

Why are canonical transformations made in the first place?

Is it to get cyclic coordinates in the Hamilton's equation?
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29 views

How can one construct a phase space for the time evolution trajectories in Hamiltonian?

I was wondering if conjugate momenta and position can be the variables of the phase space or not. I have $\frac{\mathrm{d}x_1}{\mathrm{d}t}$, $\frac{\mathrm{d}x_2}{\mathrm{d}t}$, $\frac{\mathrm{d}...
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Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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2answers
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Use of generating function in canonical transformation

In the theory of Canonical transformations, initially we use the fact that the new and the old system of $(q_i, p_i)$ with the Hamiltonian $H$ satisfy the modified Hamilton's principle. Now here, the ...
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Poisson Brackets in inhomogenous magnetic field [closed]

This question came in my classical mechanics paper and I still can’t solve it. A particle of mass $m$ and electric charge $e$ is moving under the influence of an inhomogeneous magnetic field $\vec{...
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Derivation of Hamilton-Jacobi theory using canonical transformations

The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function. Is it possible to use a another type of generating function, ...
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Generating function in action-angle method and Hamilton-Jacobi theory

I think that in action angle method, generating function which generates such a canonical transformation does not explicitly depend on time, so new and old hamiltonians are equal. But in H-J method, ...
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1answer
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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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2answers
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Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
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Phase space in Statistical mechanics

I'm currently struggling with the concept of phase space in Thermodynamics/Statistical physics. In particular I have trouble understanding the use of the "one-particle phase space". If we look at a ...
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1answer
72 views

Quantum micro canonical ensemble

In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (...
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Discrepancy regarding Husimi Probability distribution calculation

I am trying to simulate a system of j qubits and for visualization of the dynamics considering the Husimi distribution of the state. To carry out the projection onto coherent states I have proceeded ...
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Why does the Boltzmann equation deal with single-particle phase space density?

Why does the Boltzmann equation deal with single-particle phase space density $\rho_{1}(\textbf{r}_1,\textbf{p}_1,t)$ rather than the N-particle phase space density $\rho(\{\textbf{r}_i,\textbf{p}_i,t\...
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Derivation of Proportionality of Phase Space Volume log(Γ)∝N

In the derivation of extensivity of entropy for the micro-canonical ensemble, we assume an ensemble of two systems with the energies $E_1$ and $E_2$. The total energy is given as $E<E_1+E_2<E+\...
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Problem on deriving canonical transformation condition

I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with $$\omega=\stackrel{...
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Lorentz invariance of volume element

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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Expansion to show $g$ is conserved if $H$ is invariant

On Shankar QM page 99 it says that If $H$ is invariant under the following infinitesimal transformation $$q_i\rightarrow\bar{q_i}=q_i +\epsilon\frac{\partial{g}}{\partial{p_i}}$$ $$p_i\...
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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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1answer
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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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1answer
61 views

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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2answers
96 views

Time-independent canonical transformations

Lie's criterion tells us that $(q,p) \to (Q,P)$ is a canonical transformation, for a system with Hamiltonian $H$ and "Kamiltonian" $K$, if and only if the identity $$\sum_k p_k dq_k -Hdt = \sum_k P_k ...
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Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis. There was an important equation called Cauchy-Riemann equation. When I saw it at first, I recalled a book's sentence stating ...
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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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Do compact symplectic manifolds play a role in physics?

In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact. Do compact symplectic manifolds have physical meaning? ...