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.

learn more… | top users | synonyms

4
votes
1answer
63 views

Significance of symplectic form in classical field theory

I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions. Given two solutions $\phi_1$, $\phi_2$ of the ...
0
votes
0answers
16 views

How can we defined the distance for a pair quantum states in phase space

In condense matter physics, we know that two degenerate ferromagnetic ground states $|\uparrow\uparrow ...\uparrow \rangle$ and $|\downarrow\downarrow...\downarrow\rangle$ are far from each other in ...
2
votes
0answers
42 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
0
votes
0answers
10 views

Phase space - Diameter of attractor

For a dynamical system, with a phase space reconstructed from single scalar measurement, how do I calculate the diameter of the attractor (such as the one shown in figure below)?
1
vote
3answers
73 views

Meaning of the phase space in statistical physics

I have a silly question about the phase space. I am confused with the meaning of points in phase space. Does the each point in phase space represent concrete particle of the system, or does it ...
1
vote
0answers
37 views

Definition for a trajectory in phase-space

When we say "a trajectory in phase space", when the parameter is time, do we mean the set of points in phase-space corresponding to a continuous segment in time? Does it have to be continuous? Does it ...
1
vote
1answer
51 views

Why does time-independent Hamiltonian not depend on angle variable?

In Landau and Lifshitz Mechanics, $\S50$ Canonical variables a time-independent Hamiltonian is considered, and a canonical transformation is done such that adiabatic invariant $I$ becomes the new ...
6
votes
2answers
668 views

Is there any uncertainty between mass and proper length or time?

I was trying to naively draw a parallel between special relativity and the Heisenberg uncertainty principle. I try to understand uncertainty principle as a consequence of 4-position and 4-momentum ...
2
votes
1answer
67 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, ...
0
votes
1answer
33 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... ...
0
votes
1answer
21 views

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 ...
0
votes
0answers
29 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 ...
0
votes
0answers
13 views

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 ...
1
vote
1answer
105 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 ...
7
votes
1answer
99 views

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 ...
1
vote
1answer
52 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 ...
3
votes
1answer
55 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 ...
1
vote
1answer
93 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?
3
votes
1answer
60 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 ...
2
votes
1answer
131 views

Current density in phase space

$\newcommand{\dd}{{\rm d}}$ I have a question which arises from looking at the impact free Boltzmann equation. Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The ...
0
votes
1answer
92 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 p_i}\,\...
1
vote
1answer
71 views

Confusion about the use of the term “Phase Space” in Strogatz text

I've just started learning about Hamiltonian mechanics, and from the definition given in Taylor's classical mechanics, phase space must always have an even dimension. However, I recall from reading ...
2
votes
1answer
79 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 ...
2
votes
1answer
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: $$ m\...
0
votes
2answers
134 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 ...
7
votes
1answer
241 views

Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
0
votes
1answer
124 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 ...
2
votes
2answers
35 views

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 ...
4
votes
3answers
724 views

Why Liouville's theorem is obvious?

In Florian Scheck's Mechanics, he stated the local form of Liouville's theorem as follows: Let $\Phi_{t,s}(x)$ be the flow of the differential equation $$-J\frac{d}{dt}x=H_{x}.$$ Then for all $x,t,s$ ...
4
votes
0answers
60 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 ...
1
vote
1answer
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} ...
0
votes
1answer
32 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} ...
2
votes
5answers
2k views

Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
1
vote
0answers
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 $Q=\sum_{i=1}^n q^2_i$...
0
votes
2answers
133 views

According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
2
votes
4answers
603 views

Books on Hilbert space and phase space?

Can you recommend books or papers that highlight or discuss extensively, or at least more than average, the similarites/differences between phase space and Hilbert space? I am primarily interested in ...
6
votes
2answers
336 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 ...
5
votes
2answers
264 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 ...
1
vote
3answers
257 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), physically?...
0
votes
1answer
115 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 $(q_i(t), \,p_i(t))$ ...
1
vote
0answers
59 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 ...
3
votes
1answer
193 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 ...
3
votes
0answers
72 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}] = \...
1
vote
0answers
54 views

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 ...
0
votes
0answers
61 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 $\beta$ is ...
4
votes
3answers
485 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 ...
0
votes
1answer
62 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 +...
2
votes
1answer
96 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 ...
1
vote
1answer
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: $$ P = P_0 + mgt $$ ...
6
votes
3answers
195 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 ...