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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38 views

Expected value in usual quantum mechanics vs quantum information

In standard Quantum Mechanics, one computes the expected value of an operator $A$ (arbitrary state $|\Psi\rangle$) as $$ \langle\Psi|A|\Psi\rangle. $$ This has the virtue that we can compute for ...
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Examples of Weyl's Law in WKB [closed]

Can someone please provide an example or two of the usage of Weyl's Law in WKB as stated below? For me, Weyl's Law/Rule was stated as follows: The number of eigenstates with energy $\leq E$ $\approx \...
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How did Glauber come up with his definition of non-classical states?

In their paper, Titulaer and Glauber state The results [...] are derived only for fields with positive-definite P functions. Those are, in fact, precisely the quantum fields which may be described in ...
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Constraints are not functional relations!

I am reading a Wikipedia article on Dirac brackets. At the bottom of the page "illustration on example provided" the article states that for a system with constraints: $$ \phi_1 = p_x + \...
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What is a time-dependent symmetry in Hamiltonian mechanics?

I've read something from John Baez which I don't understand: If we consider a single nonrelativistic free particle - in one-dimensional space, to keep life simple - and describe its state by its ...
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Curvature in phase space

As I understand it, one way to describe a wave function is as a probability density distribution in phase space. The equations of motion for the wave function would describe how that density ...
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38 views

Relation between the canonical symplectic form on phase space and the Hamiltonian in GR

I am working on the Hamiltonian formulation of the Einstein equations of motion in General Relativity, where the aim is to find the Hamiltonian generating the dynamics from the Einstein equations (...
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Splitting (in horizontal and vertical subspace) of the thermodynamic phase space in contact geometry

Im currently studying thermodynamics in a contact geometric context and I've stumbled upon a comprehension problem on my side. We have the equilibrium connection $\Gamma_p$, which generates the ...
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Wigner functional for fermionic fields (QFT in phase space)

I'm curently studying the Wigner functional formulation of Quantum Field Theory, which is derived from the Schrödinger picture: the operators which act on the states of the Fock space are functions of ...
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The mathematical model of the Galilean transformation in Hamiltonian mechanics

In my previous question, I asked about the Galilean invariance of the Hamiltonian. I've got already two answers, probably good but I have difficulties interpreting them. Both answers write the ...
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Overlap of Liouville densities

I am having a little trouble understanding the meaning of the fact that the convolution of two Liouville densities $\rho_{1}(p,q,t)$ and $\rho_{2}(p,q,t)$ (i.e. the classical probability overlap) ...
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A strange consequence of Isotropy in Phase Space

In Goodstein, States of Matter, page 66 the following is stated: For a single particle the number of states in a region $d^3pd^3r$ of its phase space is: $$\frac{d^3pd^3r}{(2 \pi \hbar)^2} \tag{1.3....
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Are Hamilton's equations really form invariant under canonical transformations?

Let under a canonical transformation $(q,p,t)\to(Q,P,t)$, the Hamiltonian is changed from $H(q,p,t)\to \tilde{H}(Q,P,t)$. But in general, the functional forms of $H$ and $\tilde{H}$ are different. ...
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In canonical transformation, how the new coordinate is the old momenta?

I am studying Canonical transformation using Goldstein (3ed), Ch.9. I do understand everything he does in the first section and why we do need a generating function, $F$. The problem I am facing is ...
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Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)

I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase ...
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Upper bounds on phase space momenta

Suppose I wish to calculate the phase space volume for the process $\overline{X}X \to A_1 A_2 A_3 A_4 A_5$ in the CM frame of $\overline{X}, X$ so that $\sqrt{s} = 2m_X$. The volume is given by $$ V \...
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Why the volume of a cell in phase space should be equal to $(2\pi \hbar)^s$?

We want to properly define the concept of entropy using the Boltzmann's Definition of it. But there is a big problem: the coarse graining problem (Id est: How do we count the number of microstates in ...
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Why there is no commutator term in the pre-sympletic density?

In this post I'm considering the Covariant Phase Space (CPS) formalism as presented by Lee & Wald in "Local symmetries and constraints ". In the CPS formalism we take the Lagrangian form ...
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$n$-Body Phase Space Recurrence Relation

On slide 23 of these slides, it is stated that an $n$ body phase space element $d\Phi_n(P; p_1, \ldots, p_n)$ may be decomposed according to the recurrence relation \begin{align*} \mathrm{d} \Phi_{n}\...
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Are Poisson brackets preserved during a canonical transformation?

Fix a Hamiltonian $H(q, p, t)$. Definition: A transformation $(q, p, t)\mapsto (Q(q, p, t), P(q, p, t), t)$ is said to be canonical iff for the Kamiltonian $K$ defined as $H(q, p, t)=K(Q(q, p, t), P(q,...
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Clever phase space parametrization of $n$-body processes

Consider the process $X\to A_1 A_2 A_3$ where $X$ in the frame where $X$ is at rest with mass $m$. It is well known from Dalitz that we may choose a frame to evaluate the Lorentz invariant phase space ...
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117 views

Hamilton-Jacobi method with time dependent Hamiltonian

I have the following phase space $$ M = \{ (p, q) \in \mathbb{R}^2 \mid q \geq 0 \} $$ and the Hamiltonian $H = q^2p^2t$. How does one solve for $q(t)$, with $q(0) = q_0 > 0, p(0) = p_0$ using the ...
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51 views

Multiplicative inverse of Weyl symbol and invertibility of operator

If the Weyl symbol $A_W$ of an operator $\hat{A}$ has a multiplicative inverse at every point of the phase-space, can I conclude that $\hat{A}$ is invertible?
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Can a Hamiltonian include explicitly the derivative of the conjugate momentum, especially after a canonical transformation?

Can a Hamiltonian expression, say $H$ with $(q,p)$ as conjugate variable pair, include the total derivative of $p$ explicitly? That is, can we have $H=H(q,p,\dot{p})$? And, if so, what does it imply ...
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How is the relationship between the old and the new canonical variables justified?

In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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The relationship between symplectomorphism, canonical transformations, and the symplectic group

This is a follow up to this question. In the answer by Qmechanic, they state that the symplectic group, $Sp(2n,\mathbb{R})$, is the group of linear, time-independent canonical transformations. If we ...
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Two consecutive symmetry transformation generated via Poisson brackets

Question If an infinitesimal symmetry transformation parametrized by Killing field $f^\mu(x)$ $$ \delta_f\phi=\phi'(x)-\phi(x)=f^\mu\partial_\mu\phi\tag1 $$ can be generated via Poisson bracket $$ \...
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How to evaluate the equal time Poisson bracket $\{ \phi(x), \vec{\nabla}_y\phi(y) \cdot \vec{\nabla}_y\phi(y)\}$?

I learned that for a classical scalar field theory in 4 dimensions, we can use the equal time Poisson brackets $$\{ \phi(x), \phi(y) \}_{x_0=y_0}=0$$ $$\{ \pi(x),\pi(y)\}_{x_0=y_0} =0$$ $$\{\phi(x),\...
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A question about the Boltzmann calculation in the reply to the Zermelo's recurrence objection

I'm reading the Boltzmann "Reply to Zermelo's Remarks on the Theory of Heat" (Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, Annalen der Physik 57, pp. 773-84 (1896)...
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Noether's Theorem and Liouville's Theorem

Liouville's theorem states that for Hamiltonian systems the phase space volume $V(t)$ is a conserved quantity, i.e., $\frac{d}{dt}V(t)=0$. This is related to the fact that trajectories in phase space ...
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Why a $e^{i \pi/4}$ (or $e^{-i \pi/4}$) causes a phase shift of $\pi/2$ instead of $\pi/4$ in the case of a quarter-wave plate?

Given a spinor $\begin{pmatrix} E_x \\ E_y \end{pmatrix}$, I learned that if we place a quarter-wave plate with its fast and slow axes in the x- and y-direction, the relative phase shift in the x- and ...
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What happened to phase space?

I need a simple explanation, as I am still a BSc student. I learned about Hamiltonian formalism and also I know much about quantum mechanics. As far as I have heard/read, we can quantize the classical ...
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Connectedness in phase-space

In my statistical mechanics lecture, it was claimed that a volume of phase-space cannot be split into two separate volumes as time evolves. I suspect that this is a topological fact that I am not ...
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Calculating the Decay Rate for a Three-Body Decay

As the question states, how does one calculate the decay rate for a three-body decay $a \rightarrow 1 +2 +3 $? For a two-body process, the answer is in the center-of-mass frame (Thomson, page 67, Eq. (...
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Explicit independence of Hamiltonian phase-space variables from the time parameter

In general, we have for a Hamiltonian flow $H$ of some "time" parameter $t$, the following relation for any function $f=f(q,p;t)$ of the phase-space generalized position ($q$) and conjugate ...
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Period behavior near separatrix in Hamiltonian system

Given the periodic potential Hamiltonian $H=\frac{p^2}{2} - \omega_0^2 \cos(q)$ I would like to show that near the separatrix the period has this behavior: $T(E)\sim |\log(\delta E)|$ with $\delta E=|...
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Canonical transformation such that Hamiltonian of a freely falling body becomes $H'(P,Q)=P$

Can someone please help me with this problem I am unable to find a suitable generating function? The question says: To find a canonical transformation such that Hamiltonian of a freely falling body ...
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Wigner transform & convolution

I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
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Why this integral represents an area in the phase space?

In classical mechanics, and mainly when studying the Hamiltonian formalism, the quantity $$ \oint p \ dq $$ is usually referred as an area in the phase space, and can be computed graphically as such. ...
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Phase space for a damped harmonic oscillator [closed]

Given the equation $x''+2 \beta x'+ \omega^2 x=0$ for a damped oscillator, I can get to the equation of motion x(t) and deriving it with respect to time: $x'(t)$. I am asked to plot the phase plane $\...
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How do I intuitively draw phase portrait from pseudopotential vs $x$ plot?

I am struggling with the process of deducing a phase portrait from pseudopotential vs $x$ plot. Is there a resource that would be helpful in understanding it better? Given the pseudopotential(V) vs $x$...
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On the Hamiltonian vector fields of classical Hamiltonian mechanics

Notation: I denote phase space as the symplectic manifold $(M,\omega)$, in which $\omega=\sum_i\mathrm dp_i\wedge\mathrm dq_i$ in canonical coordinates. In definitions of Hamiltonian vector fields I ...
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Growing phase space volume

If we consider a flow $f$, the evolution of the phase space volume is connected to $\mathrm{div} f$ (or its time average): If the divergence is zero, the phase space volume is conserved (e.g. in ...
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Liouville CFT Poisson Brackets

I have been given an action of the form: $$S = \frac{1}{4\pi}\int d^2\sigma \ \sqrt{-g}\left(\frac{1}{2}\partial_\mu\phi \partial^\mu\phi + \frac{1}{\zeta}\phi R + \frac{\mu}{2\zeta^2}e^{\zeta\phi} \...
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How can a classical phase space be unquantizable?

On page 2 of the paper "2 + 1 dimensional gravity as an exactly soluble system" Witten claims that: Depending on its topology, a finite-dimensional phase space might be unquantizable, How ...
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How does the phase space volume change in the presence of magnetic field and Berry curvature?

I was looking at this paper which describes how the phase space volume changes with time in case a non-zero magnetic field and Berry curvature is present. On the first page, the authors state that the ...
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Invariance of Particle Distribution in Heavy Ion Colissions

I'm reading about about Heavy Ion collisions from the book by Csernai and i can't make sense of the proof that the particle distribution $f(x,p)$ is an invariant scalar. Before i quote the book: The ...
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Notation for the position and momentum differentials in a system of $N$ particles and $d$ dimensions

I am a little confused with the notation used in Statistical Mechanics for the differentials of position and momentum in the phase space. For instance, I have found different notations in different ...
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$N$-body phase space for Fermi golden rule

I was following along Mark Thomson's Modern Particle Physics, and stumble upe the derivation of d$n$ of Fermi golden rule on page 62: "... For the decay of a particle to a final state consisting ...
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General quantum operator

Is it true that any operator can be expressed as (e.g. in one dimension) $$\hat{A}=\sum_{n=0, \, m=0}^{\infty}c_{n,m}\hat{x}^n\hat{p}^m \, ?$$ It seems true because any classical observable is a ...

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