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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Problems with Kerson Huang's derivation of NVE ensemble entropy

So I'm currently studying statistical mechanics from different textbooks, but my professor suggested Kerson-Huang for a general derivation of entropy in microcanonical ensembles. In chapter 6.2 is ...
Matteo Grandinetti's user avatar
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Why $q,p,Q,P$ are Independent Variables when Using Generating Functions?

In Hamiltonian formalism, specifically generating functions, why do the variables $q, p, Q, P$ are treated as independent when finding the equations that arise from the generating function? I ...
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Why do we need a Poisson bracket structure?

Let me start by asking why we need a Poisson bracket like structure on the Hamiltonian phase space? Say we have a constraint, why do we go through the trouble of defining a Dirac bracket structure on ...
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Regarding Poisson and Dirac brackets [duplicate]

The question starts with why Poisson brackets (in constrained systems) gives different relation if we substitute the constraints before or after expanding the bracket, and why this difference in ...
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Regarding Poission structure of Hamiltonian phase space

Why exactly do we need $$ \{q^i,p_j\}=\delta^i_j,$$ where $\delta^i_j$ is Kronecker delta and $\{\cdot,\cdot\}$ is the Poisson bracket? What happens to the phase space structure if these fundamental ...
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Decoherence model of two qubits interacting with correlated multimode fields - open quantum system

I read paper on open quantum system, that talk about non-Markovian process and memory effects. they described the system as a generic decoherence model of two qubits interacting with correlated ...
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Constrained Hamiltonian problems [closed]

What happens to the poisson bracket structure of Hamiltonian phase space if We have some constraints in $p$ and $q$. What physical aspects this structure represents?
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Question about canonical transformation and generating functions

In Goldsteins' Mechanics, page 371 (relevant part appears below), it follows from what he states in the first yellow part that the equations of transformation: $$Q = Q(q, p,t), \quad P = P(q, p,t)\tag{...
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Can we get quasiprobability distributions other than $P,Q,W$ from generalised characteristic functions?

It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as $$ W(\alpha) = \int\frac{d^2\...
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Canonical Transformations and Poisson Brackets - Sufficient and Necessary Condition for Canonical Transformation [duplicate]

I am currently taking analytical mechanics, My professor directed us in the lecture to Hand and Finch, problem 6.9, to prove ourselves (as he didn't have time) that the equation $$[Q,P]_{Q,P} = [Q(q,p)...
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Is there a probability distribution associated with fermionic Gaussian states

I am writing this as a mathematician trying to understand fermionic Gaussian states. Up to global phase, a quantum state can be faithfully represented in terms of a quasi-probability distribution on ...
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Role of macrostates in formulation of phase space density

This is from page 58 of Kardar's "Statistical Physics of Particles": Therefore, there must be a very large number of microstates corresponding to the same macrostate $M$. This many-to-one ...
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Explicit construction of action-angle variables for the two-fixed-centers problem

After studying action-angle variables and Eulers two-fixed-center problem in a course on mechanics and symplectic geometry, I understand that a two-fixed-center system is Liouville integrable and ...
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Relation of entropy given in terms of phase space volume vs. multiplicity

I find in the liuterature (e.g. Landau & Lifshitz [1]) that the entropy in a microcanonical ensemble is given as: $S = k_B \log(\Omega),$ where $\Omega$ is the mutiplicity of microstates (Landau ...
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How to Find the Phase Space Integral Bounds with a Regulating Photon Mass?

In Schwartz' "Quantum Field Theory and the Standard Model", chapter 20, the author calculates the cross section for a process to produce a muon-antimuon pair as well as an extra photon ...
Leuca Patmore's user avatar
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Geometry of anticommutation relations

I am asking this question as a mathematician trying to understand quantum theory, so please forgive my naivety. Systems satisfying the canonical commutation relations are naturally modeled with ...
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Unitary evolution of composite system in phase space

Given a quantum state $\rho$ in a Hilbert space $\mathcal H_S$, we can always write it in terms of the displacement operator $D_\alpha$ using the characteristic function $\chi_\rho(\alpha)=\text{Tr}[\...
B. Baker's user avatar
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Irreversibility and organisation of trajectories in the phase space

I am currently thinking about the irreversibility paradox. I am not working in this area and my question is certainly not original but I couldn't see it stated in that form yet. I can't grasp how are ...
Chevallier's user avatar
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Describe the characteristics of a Hamiltonian System to a non-scientist

A Hamiltonian system is a dynamical system driven by a Hamiltonian $H$, i.e. $$ \dot{q}=\nabla_p H,~~~~ \dot{p}=-\nabla_q H. $$ These systems have nice properties like being symplectic as well as the ...
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Can (extended) canonical transformation involve change of time?

A map from $(q,p)$ to $(Q,P)$ is called an extended canonical transformation if it satisfies $$ \lambda(pdq-H(q,p,t)dt)-(PdQ-K(Q,P,t)dt)=dF $$ Here, to include the change of $t$, let us use $$ \lambda(...
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An example of symplectomorphism that is not a canonical transformation

I want to check my understanding on the difference between symplectomorphism and canonical transformation. This is a follow-up of my previous post. (A) A map $(q,p)$ to $(Q,P)$ is called a ...
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Partition function with Hamiltonian depending on parameter

Given a Hamiltonain $H(p,q)$, I know that the classical partition function for a single particle is given by an integral over the phase space $$ Z_1 = \frac{1}{h^3} \int e^{-\beta H(p,q)} d^3pd^3q $$ ...
Franz Bauer's user avatar
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How can we design the global structure of the phase space?

I want to know how to design a classical mechanical system that has a phase space $M$ with a nontrivial global topology. If I naively consider a system in which the generalized coordinate $q_1,\cdots,...
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Any concrete example of symplectomorphism that is not a canonical transformation? [duplicate]

I want to understand the relation between several different definitions of canonical transformation. I am studying the answer by Qmechanic in this post Let us define a canonical transformation as a ...
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Why do distributions tend to the one that maximizes entropy?

It appears to be a well-known fact that a probability distribution on phase space will tend towards the distribution that maximizes entropy. The Wikipedia article on maximum entropy states: "The ...
Jackson Walters's user avatar
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Phase space of the $n$-vector model

The classical Heisenberg model is described in terms of the three-component unit vector $S_a(x)$, which is a function of position, $$H=\int d^dx\frac{1}{2}\sum_{a,i}\left(\partial_i S_a(x)\right)^2.$$ ...
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Phase space for systems with varying number of particles

I'm currently studying statistical mechanics and having a hard time going conceptually from the canonical ensemble to the gran canonical ensemble. Up until now I've been studying only ensembles with a ...
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When is the charge generating isometries given by a symplectic product?

I am reading the following paper https://arxiv.org/abs/2309.15897 and have the following confusion: The authors look at the covariant phase space of linearised general relativity after one expands ...
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What goes wrong when we quantise a classical system without using $[X,P]=i\hbar$?

Let's say we have a classical system with a Poisson bracket. We quantise this system to get a quantum theory where we choose some variable to operator replacement : $x\rightarrow X, p\rightarrow P$, ...
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Canonical transformations in the covariant phase space formalism

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good ...
P. C. Spaniel's user avatar
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Equipartition theorem contradiction for coordinate transformation

The equipartition theorem states the following: Let $H$ be the Hamiltonian describing a system and $x_i, x_j$ be canonical variables. Then, for a canonical ensemble with temperature $T$, it follows ...
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The Lorentz-non-covariance of the Wigner Function

What does the fact that the Wigner function is not Lorentz-covariant imply? My analysis so far led me to the (probably naive) understanding that there really is nothing special about it, just that it ...
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Operating with the Weyl transform on a wave function

I'm very new to studying quantum mechanics in phase space, so I'm trying to demonstrate some results that I see in books to get used to the formalism. I recently got stuck when i applied the Weyl map $...
Wagner Coelho's user avatar
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Quantization of $x$ and $p$ through the Weyl transformation

I have a question about the development of the integral for calculating the quantization of the classical variables $x$ and $p$ using the Weyl transformation method. The notation that the textbook I'm ...
Wagner Coelho's user avatar
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Regarding the phase space density of stars and the Maxwell-Boltzmann distribution

I was reading a paper where the authors effectively made the following equality when talking about stellar populations: $$\frac{\mathrm{d} N_* }{ \mathrm{d} m_* \mathrm{~d}^3 \mathbf{x} \mathrm{d}^3 \...
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Concept of Volume of a System in Statistical Mechanics

I was reading my professor's notes on the microcanonical ensemble and I've been having some trouble understanding some of the concepts introduced. First of all, I'll start with some necessary ...
Claudio Menchinelli's user avatar
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Why does the phase space theory of relativistic particles work out but not the corresponding quantum theory with a position operator?

As we know, we can formulate the phase space theory of relativistic free particles using the Hamiltonian $H=\sqrt{p^2+m^2}$ and the Poisson bracket $[x,p]=1$. So there are no problems with Lorentz ...
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Changing the time parameter and finding the corresponding hamiltonian

I'm dealing with a problem where I have a (classical) Hamiltonian $H(q,p)$ such that, for any scalar function $f(p,q)$, $$ \dot{f} = \frac{\mathrm{d} f}{\mathrm{d}t} =\{ f,H \} $$ If I change the time ...
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$C^{\ast}$-algebra approach to classical mechanics

Can someone please help me understand how classical mechanics (for example in terms of Hamiltonian formalism) can be described in terms of $C^{\ast}$-algebras? I read usually that in this case the ...
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Weyl Quantization Integral

I have some doubts when calculating the integral for Weyl Quantization symbol. If I understand correctly, quantization using the Weyl symbol takes a function in phase space and takes it to an operator ...
Wagner Coelho's user avatar
1 vote
3 answers
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Evolution of volumes in Phase-Space

Liouville's theorem states that the volume occupied by an ensemble does not change as the ensemble evolves. My question regards the volume of the smallest sphere that contains the ensemble. Is there a ...
Antonio Bernardo's user avatar
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Amplitude of the process $q \bar q \rightarrow \tau^+ \tau^-H$

I want calculate the cross section of the process $q \bar q \rightarrow \tau^+ \tau^-H$, where the Higgs takes the $vev$. My question is: if the Higgs takes the $vev$ the amplitude of the process is ...
Andrea's user avatar
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Integral on phase space

I'd like to integrate the square amplitude in the phase space( in $d^3k/(2\pi)^3 2E_k$ and $d^3k'/(2\pi)^3 2E_k$) where $p$ and $p'$ are the 4-momentum of the input particles, $k$ and $k'$ 4-momentum ...
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Questions on Symplectic approach to canonical transformations

Reading section 9.4 of Classical Mechanics by Goldstein, I got a question in my mind. That is, it says that for restricted canonical transformation, we have the new Hamiltonian equal to the old one. I ...
Ting-Kai Hsu's user avatar
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2 answers
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What does it mean for an operator to depend on position or momentum?

While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
Lourenco Entrudo's user avatar
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Variable Dependence of Quantum Operators and Commutator Relationships [duplicate]

EDIT: After doing some digging, I am convinced that the approach taken in this paper was simply an incorrect approach to deriving a quantum version of Hamilton's equations (also related to Ehrenfest's ...
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How does the Stratonovich-Weyl operator kernel, used to find the Wigner function, work?

Recently during my studies, I came across an alternative construction of the Wigner function. This construction starts from the notion of the Stratonovich-Weyl operator kernel. I saw this construction ...
Wagner Coelho's user avatar
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1 answer
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Non-formal expression for the classical propagator

I'm studying classical molecular dynamics and have come across an object called the classical propagator in the following context. Let $\mathcal{A}(t) = \mathcal{A}(\vec{x}(t))$ be a function on phase-...
Christoph90's user avatar
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Gauge symmetry and the volume in phase space

Recently, I am reading a paper about the soft theorem and large gauge symmetry which is non-zero in the boundary. In section 6, the author introduces the covariant phase space method to illuminate why ...
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Non-symplectic Hamiltonian systems

I'm wondering when the phase space of a Hamiltonian system looses its symplectic structure. I think it happens when the Hamiltonian $H$ depends on a set of other variables $S_1,...,S_k$ as well as on ...
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