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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Can the hamiltonian be derived from phase space evolution?

Given the phase space evolution of a system, $x(t)$ and $p(t)$, is there any way of getting the hamiltonian to make a later study of the system under the hamiltonian formalism? My first thought was ...
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Question about transformation's generator

We need to find the $g$ transformation's generator under a rotation on the phase space, where the Hamiltonian is equal to: $H = \frac{p^{2} + x^{2}}{2}$. At first i made the expression of an ...
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The connection between classical phase space and quantum multiplicity

I am aware of the relationship $N = V/h^n$ where $N$ is the quantum multiplicity, $n$ is the number of position (or momenta) degrees of freedom, $V$ is the volume of classical phase space and $h$ is ...
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Is the Wigner function a signed measure?

I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be $\sigma$-additive, but I don't know in which sense this is true. If I have a Wigner ...
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How do i read a pendulum phase diagram?

I'm trying to understand how the intermediate axis theorem works. And in one of the works that I found, they used a pendulum phase diagram, but idk how to read it. Can anybody help please? The work ...
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What is the difference between a macrostate and multiplicity?

The entropy of a system of an ideal gas depends on the external parameters $U, V, N$. I always thought entropy is defined by a certain macrostate, which is a set of given external conditions like ...
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Why does the number of accessible microstates always increase?

If we consider two systems with number of accessible microstates A and B and internal energies of E(A) and E(B) that are exchanging an small amount of heat Q from A to B we get the following: A total ...
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What is the analogue for symplectic structure in case of spin variables?

According to some (e.g. Haroche and Raimond in Exploring the quantum: atoms, cavities and photons), the quantum world consists (mainly) of spins and harmonic oscillators. For harmonic oscillators (i.e....
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Hamilton-Jacobi Equation & Canonical Transformation

I am attempting to solve the Hamilton-Jacobi Equation in the case of a simple harmonic oscillator, to recover the associated generating function and the generated canonical transformation. Consider ...
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Prove that $f_\psi(x,p)$ is the Wigner Function of a pure state iff $H\star f_\psi= E f_\psi$

Given a pure state $|\psi\rangle$ with position wavefunction $x\mapsto\psi(x)$, define its Wigner function as $$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2) \equiv \frac{1}{2\...
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Wigner Function and Spin in the Classical Limit?

This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system: $$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n) $$ Now, let's say this system obeys has spin. As far ...
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Problem with microstates and corresponding volume in the phase space

I've been struggling without any result on a formal problem about the relation between microscopic states of a system and the volume occupied by a macrostate in the phase space. I'm now very confused ...
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How to obtain commutation relations from symplectic potential?

I am studying the notes on susy qm in 1 dimension of David Skinner (http://www.damtp.cam.ac.uk/user/dbs26/SUSY.html) (which itself follows the mirror symmetry book by Vafa and Hori (relevant pp. 206 - ...
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Question concerning phase curves and Lissajous figures

I want to draw the "orbits" of a spherical pendulum under small oscillations. In this case its equations are given by $\ddot{x}_{1}=-x_{1}$ and $\ddot{x}_{2}=-x_{2}$. Of course the potential ...
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Paths in phase space can never intersect, but why can't they merge?

Page 272 of No-Nonsense Classical Mechanics sketches why paths in phase space can never intersect: Problem: It seems to me this reasoning only implies that paths can never "strictly" ...
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Doubt on the need for Topological Manifolds [closed]

I happen to be trying to motivate physically and intuitively the need to use topological spaces. So without using emerging concepts such as differentiable manifolds or extremely formal concepts as ...
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71 views

How does the Weyl transform take into account which quasiprobability distribution was used?

I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
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Why does the star product satisfy the “Bopp Shift relations”: $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?

In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts": $$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
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What are the Fock-state probabilities of general Gaussian states?

A general (pure) Gaussian state has the form $\newcommand{\on}[1]{\operatorname{#1}}\newcommand{\ket}[1]{\lvert #1\rangle}\ket{\alpha,\xi}\equiv D(\alpha)S(\xi)\ket{\on{vac}}$, with $\ket{\on{vac}}$ ...
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The “real butterfly effect”

This question stems from the confusion that I feel after reading this popular blog post by Sabine Hossenfelder. It is based on this paper which is paywalled, unfortunately. The claim is the following: ...
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Why does the separatrix in phase portraits have infinite period and pass through at least one unstable equillibrium point?

In the case of 1D Hamiltonians not explicitely dependent on time, our professor claims that the "period" of the separatrix is necessarily infinite and must pass through an unsable ...
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Bounding derivatives of the Wigner function using phase-space tails

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} ...
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Poisson Bracket $\{\delta_{ij}, g\}$ and partial derivative of Kronecker delta

I am currently working through Shankar's Princeiple of Quantum Mechanics Exercise 2.8.2 is to verify that the infinitesimal transformation generated by any dynamical variable g is a canonical ...
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Changing to action-angle variables to reduce the degrees of freedom

I have a Hamiltonian with two degrees of freedom, and when I change to action angle variables, one of the action variables does not appear in the final Hamiltonian. The reason seems to be because the ...
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Poisson Bracket of $\{Q, P\}$ in the original coordinate $(q, p)$

For simplicity, I use $(q,p)$ and $(Q,P)$ instead of $(q_i,p_i)$ and $(Q_i,P_i)$. I know that we should get $\{Q, P\} = 1$ for a canonical transformation $(q,p)\rightarrow(Q,P)$. But we also know from ...
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Why does a point transformation in configuration space imply that $P = \frac{\partial q}{\partial Q} p$ in phase space?

What the book demonstrates In No-Nonsense Classical Mechanics, the author spends some time discussing how point transformations in configuration space correspond with canonical transformations in ...
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Interpretations of abstract dynamics - single system vs field

As I understand the approach to abstract dynamics it involves a vector valued function $f(x,t)$ of a vector valued state variable $x$ and the real valued time $t$. The dynamics of "a system"...
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What is the “secret ” behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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Are all time-independent Hamiltonian systems related locally via time-independent canonical transformation?

So recently I've been doing some self-study on canonical transformations and relating together different Hamiltonian systems. I've found this paper (PDF) with a remarkable result showing that any two ...
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What is the process of finding a good canonical transformation to describe a system? How do I choose the correct generating function?

Supposedly, canonical transformations are used to provide a general procedure to transform a Hamiltonian such that all coordinates in the new frame are cyclic. I have done the proofs and derivations, ...
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Why is $Q=p$, $P=-q$ a canonical transformation from the perspective of 2 variational principles satisfying boundary conditions? [duplicate]

This is to ask a more general question: Landau-Lifshitz say that for the variational principles $$\delta\int_{t_1}^{t_2}p\mathrm{d}q-H\mathrm{d}t =0$$$$ \delta\int_{t_1}^{t_2}P\mathrm{d}Q-H'\mathrm{d}...
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Composite systems [closed]

Studying QM attracted my attention to the concept of a composite system. I erroneously thought, following classical reasoning, that a composite system can be i.e. a system formed by two qubits and ...
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How is Hamilton's first equation useful in solving mechanics problems? [duplicate]

Here is the first Hamilton equation: $\frac{\partial H}{\partial {p}_q} = \dot{q}$ Let's use it. Imagine a ball rolling down a frictionless hill (ignore the friction vector in the image). As time goes ...
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Addition of a constant to the operator due to quantization

Groenewold in his book On the Principles of Elementary Quantum Mechanics (1946, Springer Netherlands) page 45, maps the canonical momentum $p^2$ in the classical phase space to a general canonical ...
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Name for region in phase space with no outgoing or incoming flows?

I've been looking for a term online but couldnt find it: suppose we have a subset $X$ in phase space, such that for all $q\in X$, the path starting at $q$ never exits $X$ either forward or backward in ...
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Is The Seiberg-Witten Map Unique?

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions $[x, y]=i \...
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Sanity check on meaning of Liouville's theorem

I've been studying Liouville's theorem lately. The statistical mechanics textbook by Kardar proves the theorem by showing that $dq\cdot dp$ is unchanged for each coordinate for an infinitessimal ...
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Sanity check: can we define “arbitrary” hamiltonians?

I just want to do a sanity check on my understanding of Hamiltonian mechanics: My understanding is: For any number $n$, take the phase space $\mathbb R^{2n}$, and take any arbitrary differentiable ...
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How does symplectic geometry relate to classical hamiltonian mechanics?

I just found out about symplectic geometry in the context about this question on volume preservation in phase space. It seems somewhat complicated and I am not sure what to do with the notation $\...
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Liouville's theorem for the submanifold of given conserved quantities?

Liouville's theorem states that phase space volume is conserved over time with respect to the dynamical system generated by the Hamiltonian and Hamilton's equations. However, any given point in phase ...
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Rovelli's relativistic phase space

I'm asking for help to understand the definition of relativistic phase space given by Rovelli in his book Quantum Gravity. At chapter 3, he states those following definitions The relativistic phase ...
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What is wrong with Weyl-Wigner representation?

The Weyl-Wigner representation is a useful tool to study QM from a semiclassical, phase-space point of view. My question is simple: if this method is so close to classical mechanics, why don't we use ...
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Liouville's Theorem derivation

As an example of the book "Introduction to Quantum Mechanics Schrodinger Equation and Path integral" by Harald J. W. Muller. We have to prove Liouville's theorem. Here I show the proof as ...
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About the group of canonical transformations and the matrices representing them

Recently I have come to know that for a system with $2n$ dimensional phase space, the set of all canonical transformations form a group ${\rm Sp(2n, R)}$. But in contrast to other Lie groups e.g. ${\...
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One parameter-groups and coordinate transformations in phase-space

I have given a function $$G=p_1q_1 - p_2q_2$$ on a 4-dimensional phase-space. This function $G$ commutes with the Hamiltonian $$H= \frac{p_1p_2}{m} + m\omega^2q_1q_2.$$ It generates a flow $$(\vec{q},\...
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${\rm 2D}$ isotropic oscillator: Is ${\rm SO(4)}$ a subgroup of ${\rm Sp}(4,{\rm R})$?

Consider the ${\rm 2D}$ isotropic oscillator. The hamiltonian is $$H=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)$$ and the phase space is $4$ dimensional. In this case, the set of all linear canonical ...
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How can I express velocity as a function of position in a damped oscillation? [closed]

In a damped oscillation that obeys $x(t)=Ae^{-bt/2m}\cos(ωt)$ which shows the position of the oscillating object as a function of time, how can I express the velocity of the oscillating object as a ...
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How to find the canonical transformation?

Knowing that the Hamiltonian of a system is $H=\frac{1}{2}(q^{4}p^{2}+\frac{1}{q^{2}})$ The Hamiltonian after the canonical transformation is $K=\frac{1}{2}(P^{2}+Q^{2})$ How do I know what ...
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How are the two definitions of Canonical Transformations related/equivalent? [duplicate]

I am aware of two definitions of canonical transformations which I state below. Definition $1$ We go from old set of $\{q_i,p_i,t\}$ of $2n$ phase space variables to a new set $\{Q(q_i,p_i,t),P(q_i,...
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Itzykson Zuber Quantum Field Theory: meaning of integrable system

Here is a part of the book Quantum Field Theory by Itzykson and Zuber: I have two questions: what does the author mean that equation (1-30) form and integrable system, and why? what is the ...

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