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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Phase space and uncertainty

In phase space every point represents both position and momentum. Isn't it against of Uncertainty principle?
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Canonical / invertible transformation

What distinguishes a canonical transformation from an ordinary invertible Coordinate transformation? I understand what canonical is but searching the difference between canonical and invertible didn't ...
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How do I calculate the Altarelli-Parisi splitting function for $g \rightarrow q\bar{q}$?

I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for ...
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Can every canonical transformation be broken down into a large number of infinitesimal canonical transformations?

Consider a canonical transformation from $(q,p)$ to $(Q,P)$ depending upon a continuous parameter $\alpha$ such that: $$Q_i=Q_i(q,p,t,\alpha), \space P_i=P_i(q,p,t,\alpha)$$ where $q$ and $p$ ...
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What does integrating the probability density function over all phase space gives us?

For a system of N-3D particles, we have 6N D.O.F and therefore a 6N dimensional phase space. I know that one point in phase space represents a possible state of the system. I also understand that a ...
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Phase space, ensemble of systems and probability density function

I am trying to understand the concept of phase space in statistical mechanics. I can understand that a system, with $N$ total degrees of freedom, can be in different states, which correspond to ...
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Operator ordering (for the Groenewold-van Hove theorem)

Suppose we have the (un)usual Schrödinger representation $\pi'(\cdot)$ of the Heisenberg algebra, along with the extension in $\mathbb{sl}(2,\mathbb{R})$ for the quadratic polynomials. It is assumed ...
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Is the definition of a canonical transformation symmetric in the specification of old and new coordinates?

Consider the following transformation: $$q=P^\alpha \cos(\beta Q)$$ $$p=P^\alpha \sin(\beta Q)$$ for $\alpha=1/2$ and $\beta=2$. Now, by convention, one takes $(q,p)$ to be the old coordinates and $(Q,...
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Do physical systems have intrinsic degrees of freedom that are independent of its representation?

Considering just the Newtonian case, suppose we have a system described by $n$ canonical position-momentum pairs, $(p_1,q_1),\dots,(p_n,q_n)$, and a Hamiltonian $H$. If we "scrubbed" all the ...
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Can someone help me understand this example of a phase space plot of an inverse SHO?

I can't quite grasp this particular phase plot. I am used to the momentum vs displacement plot that usually results in an ellipse, but I think this is trajectory vs displacement which I don't really ...
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Ergodic theorem with more conserved quantities

Ergodic theory is constructed by fixing the dynamics on a surface of the phase space with constant energy. In case a non-integrable system conserved more additional quantities apart from the energy, ...
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Does the complex version of Hamilton's equations have any connection with the Schrodinger's equation?

I thought that Hamilton's equations could be written like this: $i\frac{dq}{dt}=\hat{H(q)}$ By $q$, I mean the complex function $x+ip$ By $\hat{H}$, I mean the complex function $$\frac{\partial H}{\...
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How can you confirm that two variables are canonically conjugate using Poisson brackets?

Suppose you have two conjugate variables $q$ and $p$ that are canonically transformed into two other variables $Q$ and $P$. What needs to hold true for these variables in terms of Poisson brackets? I ...
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Inconsistency of numbers of $d p$ and $d q$ in path integrals over phase space

I am new to QFT. In books like Fradkin's QFT an integrated approach, and Stefan's Gauge field theories 2nd Ed., they derive the path integral from first writing down the integral over the phase space, ...
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Change in Hamiltonian under restricted canonical and symmetry transformations [closed]

The following problem appeared in our test: Given the Hamiltonian $$H=\frac{p^2}{2m}+\frac{m\omega^2q^2}{2},$$ and the generating function $$F=-\frac{Q}{q},$$ what should be the Hamiltonian in the ...
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Are there methods to determine the equilibrium points of the general three bodies problem?

Starting from the three second-order differential equations, I have written the problem in the form of a system (from a modelling point of view): $$\mathbf{\dot{x}}=\mathbf{f(\mathbf{x})}$$ where $\...
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A Hamiltonian with a potential depending on the momentum

Imagine we have a Hamiltonian, whose potential depends on velocities (and hence on the momentum), like, for example, $$ H= \frac{p^{2}}{2m}+ V(x,p)$$ then how can I quantize that?
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How is Phase Space defined in Statistical Mechanics?

What do we mean by Phase Space in Statistical Mechanics. How can we define it?
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Is there a classical limit that connects classical volume/phase space with the scalar product of momentum and position eigenstates?

The book Understanding Molecular Simulation: From Algorithms to Applications, 2002, Daan Frenkel and Berend Smit, states the following $$ \langle r|k\rangle \langle k|r\rangle = 1/V^N $$ where $|r\...
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How prove that the 2-form of the hamiltonian formalism is a sympletic form?

The form What I say is $$\omega= dp_{\mu}\land dq^{\mu} $$ (I wrote anyway because the question is quick) how prove that are sympletic, i prove that $$d\omega=0$$but how prove that this is non-...
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Parity transformation: mistake or puzzle in Sakurai's "Modern Quantum Mechanics"?

In Sakurai's Modern Quantum Mechanics, p.270, he wrote an equation the parity transformation $\pi$ (where $\pi = \pi^\dagger = \pi^{-1}$) as $$\pi \left(1- \frac{i p \cdot d x'}{\hbar}\right) \pi^\...
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Quantum corrections in the phase space formulation

I'm trying to reconcile the following two statements: Quantum Mechanics gives physical predictions which are different than the predictions that are obtained in the $\hbar \rightarrow 0$ limit, that ...
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How does coarse-graining lead to irreversibility?

This is how I used to understand how coarse-graining leads to irreversibility. Suppose that we start with a coarse-grained phase space and two initial conditions belonging to two different phase cells....
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In classical physics, by knowing the present, can we always uniquely construct the past? [duplicate]

In classical mechanics, by knowing the present, is it always possible to uniquely reconstruct the past? By knowing the phase space point at present i.e., the set of coordinates $\{q_i(0),p_i(0)\}$, ...
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Can we make a phase space plot for any quantum mechanical system?

Can we make a phase space plot for any quantum mechanical system? If we can then lets say at a particular point represented by $(q,p)$ does not it violate the Heisenberg Uncertainity as we ...
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Limit Cycles of a Simple Pendulum

In this pdf file the dynamical behavior of a simple pendulum is discussed. The equation of motion for a pendulum with no dissipation is: $$\dot{\theta}=\omega, \qquad \dot{\omega}=-\frac{g}{l}\sin\...
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Is the motion of a particle in the surface of a torus always periodic?

I am trying to see if there are ballistic trajectories in the surface of the torus that are not periodic and to what extent. Maybe it is not only quasiperiodic but chaotic. I guess there are ...
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Arguments of specific Hamiltonian, always conserved?

I'm studying an introductory course in theoretical physics, I stumbled upon something I really can't understand. So, in my book there is written the following statment: Consider a Hamiltonian system $...
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Are all canonical transformations unitary transformations?

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates $(q, p, t) \rightarrow (Q, P, t)$ that preserves the form of Hamilton's equations. Now in quantum mechanics ...
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Would a non-separable position momentum wavefunction really violate the uncertainty principle? [closed]

I've seen it claimed on here that a position momentum wavefunction would violate the uncertainty principle. I would interpret as saying that position momentum wavefunctions that are not separable ...
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Is ensemble a set of all microstates of a system?

Definition of ensemble says it's a collection of identical mental copies of a system in which microscopic parameters can differ. Now, the phase space of the $N$ particles in the system in 3 ...
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Poisson bracket of the angular momentum and a scalar function

In the context of the Hamiltonian mechanics, I am trying to demonstrate the following statement: For any scalar function $f$, just as the dot product $\boldsymbol{q}·\boldsymbol{p}$, the Poisson ...
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Condition for a time-dependent transformation to be a canonical transformation [duplicate]

I'm looking for a sufficient condition to determine if a given transformation is a canonical transformation. I have found two conditions, but they are only valid for the case that the transformation ...
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Simple (classical, non-damped) 1D harmonic oscillator - how to represent phase space elliptical trajectory in polar form for an ellipse?

Considering a classical, non-damped 1D harmonic oscillator (e.g. mass $m$ oscillating along $x$-axis attached to spring with constant $k$) -- described by Hamiltonian (for constant energy $E$) $$E=p^2/...
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Is 2D rectangular billiard an integrable system? What's the form of explicit solution?

Suppose the free particle moving inside 2D box. $$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+V$$ where the potential is zero inside the box and infinite outside the box. It's clear that $p_x,p_y$ are not ...
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Phase space, Algebra of observables and Hamiltonian flow

I read in this paper: https://arxiv.org/abs/gr-qc/9406019 the following In classical Hamiltonian mechanics, the states are represented as points $s$ of a phase space $Γ$, and observables as elements $...
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Does the integral of a Wigner function over a finite region mean anything?

I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
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Fourier transform of a vector

For a function $f(t)$ the Fourier transform $\mathcal{F}[f(t)] = \tilde{f}(\omega) = \int_{-\infty}^{\infty} e^{-i \omega t} f(t)$ of its derivation i.e., $$\mathcal{F}\left[\frac{d}{dt} f(t) \right] =...
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Can we consider position space and momentum space of any state $\psi$ as two subspaces (subsystems) and find joint entropy between them?

For any quantum or classical state e.g coherent state, if we write it in position space and momentum space. Can we consider position space and momentum space of that state as two subspaces (subsystems)...
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How is angular momentum defined on symplectic space?

In classical mechanics on a flat 3D vector space (which we will refer to as configuration space), we can define position and momentum vectors $x$ and $p$. We can then define angular momentum as $L = r ...
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Why does a transformation on $p$ imply the inverse transformation on $q$ in phase space?

I'm reading A conceptual introduction to Hamiltonian Monte Carlo. On pages 30-31 it is stated that applying a transformation to the momentum p in phase space, implies the opposite transformation to ...
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Intuition about conservation of volume in phase space (Liouville's theorem)

I'm reading a text that says "Conservative dynamics in physical systems requires that volumes are exactly preserved". I'm assuming this means volumes in phase space, since this seems to me ...
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Independence of momentum and position for Hamiltonians

This question might be a replication but I could not find an answers till now. Why are $(x,p)$ independent in the Hamiltonian formulation? I'm interested in the independence of $(x,p)$ in the ...
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How can the Wigner function of squeezed states be non-negative?

It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
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Alternative derivation of NVT partition function

I wanted to use an approach based on the Liouville equation to formulate the partition function of the NVT ensemble from its conservation laws. You can see some more about this approach in https://...
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Integrating out the angular dependence of a correlation function

So I have to do an integral on 2 phase spaces $d^{3}k_1$ and $d^{3}k_2$ (which at the end has to be computed numerically): $$\iiint\!\!\!\!\iiint\!\!d^{3}k_1 d^{3}k_2~~ \frac{1}{(\vec{k}_1 - \vec{k}...
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Classic analog of quantum mechanics when dealing with Hamiltonian operator

I am reading The Principles of Quantum Mechanics by Dirac, in chapter 28 Heisenberg's form for the equations of motion, there is ...
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Why do Action-Angle Variables form an invariant Torus?

I've been casually reading up on Hamiltonian Mechanics and integrable systems and one term that is used a lot of "invariant torus" where bounded orbits live. KAM theory is also mentioned as ...
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Vanishing mean flux

Given a phase-space distribution function in special relativity, we can define the following Lorentz invariant vector - $$S^a(x^i) = c\int \frac{d^3p}{E_{p}}P^a f(x^i, p).$$ The spatial component of ...
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Annihilation Scattering Cross-Section $N_{n}+N_{n} \rightarrow \nu_{i} + \nu_{j}$ [closed]

I'm calculating the annihilation scattering cross section for fermionic dark matter candidate. I got the following result for the squared amplitude, $|\mathcal{M}|^2= \frac{h_{ji}f^{*}_{ij}}{(m^{2}_{{...

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