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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Why is second harmonic Intensity periodic in coherent length?

Solving for the intensity of second harmonic generation we get that intensity is $sinc^2(\pi/2*L/L_{coherent})$. How is calculated that the intensity is periodic in L_{coherent} (coherent length, L ...
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28 views

Interpretation of contourplot pendulum

I've made this plot of a function that evaluates the size of the angle on the x-axis, and the velocity of the angle for the pendulum on the y-axis. I'm having a hard time interpreting the meaning of ...
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65 views

Hamiltonian from a differential equation

In my differential equations course an example is given from the Lotka-Volterra system of equations: $$ x'=x-xy$$ $$y'=-\gamma y+xy.\tag{1}$$ This is then transformed by the substitution: $q=\ln x, ...
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After positron is produced after the target,what will the magnetic strength of the solenoid change phase space of the particle?

When positrons are generated from a particle beam hitting a dense target, solenoids are used to focus them, so what is the relationship between the magnetic fields of the axial direction(parallel to ...
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104 views

Symbolic dynamics of a multidimensional system

Let $x_t = F(x_{t-1})$ be a discrete-time dynamical system in the chaotic regime. Starting from an initial condition $x_0$, we can generate a time series $(x_t)$ where $t =1,2,...,T$ indicates the ...
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51 views

Possible duality between Harmonic oscillator and free particle?

There is some connection between classical non-interacting harmonic oscillator (OH) and the free particle in higher dimensions? I was studying statistical mechanics and I came across the idea that ...
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53 views

Quantum versus classical computation of the density of states

If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy ...
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85 views

Difference between phase space and Hilbert space? [closed]

Why is the phase space of classical mechanics not a vector space, but Hilbert space of QM is?
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57 views

Ambiguity in True Quantum Phase-Space Distribution

In this paper, the following is stated: It is well known that the uncertainty principle makes the concept of phase space in quantum mechanics problematic. Because a particle cannot simultaneously ...
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102 views

Understanding the Mathematics of Wigner function [duplicate]

I fully understand that Wigner function provides the complete information of a state of a quantum system, i.e. quantum phase space, while not violating Uncertainty principle. But can anyone tell me ...
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32 views

Uhlhorn's “On the connection between transformations in classical mechanics and …”?

I've looked around online for quite a while for the following paper, but have only been able to find people citing it. And, as you can see, the journal itself hasn't made their papers available, so... ...
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When is an attractor meaningful?

I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of ...
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53 views

Motivation for covariant phase space

The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
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85 views

What does Liouville's Theorem actually mean?

Basically, the mathematical statement of Liouville's theorem is: $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial ...
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36 views

Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?

Consider a canonical transformation $(p,q) \rightarrow (P,Q)$ under a generating function $F$. The condition for form invariance of Hamiltonian equations of motion looks like : $$\sum_{s}P_s\dot{Q_s} ...
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30 views

Phase Portraits Given Hamiltonian

Given a Hamiltonian say $$ H = 5p^2 $$ What is the correct procedure for producing a phase portrait. My initial thoughts were to solve the system of equations $\frac{dq}{dt} = 0$ and $\frac{dp}{dt} ...
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Angular Momentum Addition in Phase Space QM

In my very limited understanding of geometric quantization, we quantize spin by choosing as our phase space $S^2$ with a suitably normalized area form as the symplectic form. Depending on the ...
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24 views

Canonical Transformation [duplicate]

How can I prove that the following transformation is canonical: $\begin{cases}\overline{q}_i=\dfrac{q_i}{Q} \\ \overline{p}_i=Qp_i-2Pq_i \end{cases},\ i\in\overline{1,n}$ where $Q=\sum_{i=1}^n ...
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76 views

Wigner functions, symmetry

I'm trying to get more insight into quasiprobability distributions, as for example the Wigner function. There are some Wigner functions, which are symmetric. Symmetric: Fock state Thermal states ...
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241 views

Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
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114 views

Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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56 views

What is the phase space volume in terms of angular momentum?

Given a rigid rotor Hamiltonian, defined along the principle axes as $$ H = \sum_{i=1}^3 \frac{L_i^2}{2I_i} $$ say we would like to compute the classical partition function of this system. Is the ...
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190 views

How to show period is defined by $T=dS/dE$ (V.I. Arnold Mathemtical Physics)

I'm looking at a book by VI Arnold on mathematical physics and I've hit a roadblock pretty early on. I'll quote the question: "Let $S(E)$ be the area enclosed by the closed phase curve ...
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67 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = ...
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Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...
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58 views

Canonical transformation question

Let $(\vec{r},\vec{p})$ denote set of canonical variables. Assume a system is described by the following Hamiltonian $$H(r,p) = \frac{1}{2m}(p_1^2 + (p_2 - \beta*x_1)^2 + p_3^2),$$ where $\beta$ ...
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60 views

Harmonic oscillator and cyclic coordinates

I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows $$H = \frac{1}{2m}(p^2 ...
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332 views

Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
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89 views

Volume as a choice of measure in phase space

For equilibrium systems, we expect the Liouville theorem to hold. This theorem states that the density function of the states of the system is a constant of motion, which in turn can be translated ...
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245 views

What is the correct relativistic distribution function?

General Statement and Questions I am trying to figure out the proper way to model a velocity/momentum distribution function that is correct in the relativistic limit. I would like to determine/know ...
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34 views

Phase space orbit for a projectile [closed]

After playing around with drawing the phase space orbit for a harmonic oscillator I started wondering about the case for a free falling object. So the equations of motion are: $$ P = P_0 + mgt $$ ...
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198 views

Area of phase space of Harmonic oscillator

We all know that the phase trajectory of an undamped linear harmonic oscillator is an ellipse. But when we calculate the area of the ellipse we find it does not depend of mass of the particle. Why is ...
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Preservation of phase space volume: the extension from “small” times to generic times

Having a classical system whose evolution is described by \begin{equation} \dot{\phi_t}(x) = f(\phi_t (x))\\ \phi_0 (x) = x \end{equation} denoting with $\phi_t (x)$ the evolution for a time t of the ...
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120 views

Density of states of classical harmonic oscillator in phase space

Since all classical harmonic oscillators are ellipses in phase (position-momentum) space, and since the entire phase trajectory of a given system (with a fixed rigidity and mass factor) can be ...
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101 views

Calculating the number of particles in phase space

I'm looking at the first part of question 7 here (I'm a mathematician trying to self teach some physics, this isn't a homework assignment so I'm just in need of hints)! I'm struggling to make sense of ...
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214 views

Planck's constant and phase space in quantum mechanics

During my undergrad physics classes, I've come across several seemingly related phenomena dealing with $h$ and phase space in quantum mechanics. Let $T_x$ be a translation operator by $x$ in ...
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69 views

Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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91 views

When is the phase space diagram an ellipse?

For a two dimensional dynamical system, when does the phase space diagram give an ellipse? I know about the examples for damped and undamped harmonic oscillators, but our instructor said that the ...
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67 views

Differential cross section $d\sigma/dp^{\gamma}_{T}$?

Why we care about $d\sigma/dp^{\gamma}_{T}$? What the physical meaning of it? Why not plot $\sigma$ follow $p^{\gamma}_{T}$?. As in this picture.
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108 views

Discontinuity of paths in phase space path integrals

Berezin's famous paper "Feynman path integrals in a phase space" discusses the space of paths on which the phase space path integral is concentrated. In particular, these paths are known to be ...
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85 views

What is the effect of squeezing on the Husimi phase space representation or Q-function?

The effect of the squeezing operator \begin{equation} S = e^{- r (a^2 + a^{\dagger 2}) / 2} \end{equation} on a Wigner phase space representation or W-function of a system with density matrix $\rho$ ...
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126 views

Angular momentum and the Units

I'm just curious about why many physical identities build relationship with the same units as angular momentum like the action, Lagrangian$\cdot$time, Hamiltonian$\cdot$time, phase space area etc?
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2k views

What is the difference between configuration space and phase space?

What is the difference between configuration space and phase space? In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's ...
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643 views

How do we find the phase space density from the Hamiltonian?

How do we find the phase space density from the Hamiltonian? For example: Consider a classical gas made of N identical non-interacting particles in 1d. Each molecule is characterised by centre mass ...
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48 views

Time responses (position and speed) of system

This is a basic question regarding state space representation and differential equations. I want to find the time response of states $x_{1} = x$ and $x_{2} = \dot{x}$ of the following system: $$ ...
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205 views

What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with Wigner quasi-probability distributions on the phase space and the path integral ...
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179 views

Necessary and sufficient conditions for a function to be the Wigner function of state

For any quantum state defined with a continuous position, the Wigner function is a quasiprobability distribution on phase space. It has many properties, such as that its marginal are probability ...
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69 views

Counting classical microstates

In my notes it states that the convention for summing over the classical states is $$\sum_{\Gamma} \longrightarrow \frac{1}{N!}\int \prod_{i=1}^N \frac{d^3q_id^3p_i}{h_0^3} \tag1$$ Now I know that ...
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55 views

Do non-Gaussian states always show negativity in phase space? [closed]

According to Hudson’s theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. In cases, in which the existing well-known Hudson theorem immediately tells that ...
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458 views

Liouville's Theorem and Boltzmann equation for plasma

The Boltzmann equation for a plasma can be thought of as coming from a continuity equation in the 6 dimensional phase space of the plasma with coordinates $\left\{x,y,z,v_x,v_y,v_z \right\}$. So ...