Phase space Definition and 133 Threads

If you produce three massive particles with m1=/=m2=/=m3 near threshold (beta -> 0), the cross section of the production is supressed by a factor beta^4, where beta = sqrt(1-(M_tot)^2/s) and s is COM energy. I have been trying to prove this statement, but I can't seem to manage. Could anybody...

Homework Statement The Hamiltonian for a single diatomic molecule of identical atoms is given as $$H=\dfrac{\vec{p_1}\cdot\vec{p_1}}{2m}+\dfrac{\vec{p_2}\cdot\vec{p_2}}{2m}+\dfrac{K}{2}(\vec{r_1}-\vec{r_2})\cdot(\vec{r_1}-\vec{r_2})$$. Find the grand canonical partition function for a gas of...

Hi, I am trying to fully understand the meaning and usage of phase space in the various contexts it's used. For example particle physics, classical mechanics, statistical mechanics, thermodynamics, relativity. Also, there is configuration space, parameter space, and state space. How are all of...

First post ! I hope that my question will not make some long time physicists laugh. It is about geometrical quantization and the phase space in which we use : z=1/sqrt(2)(q+ip) My question is simple what is this 1/sqrt(2) ? And what is it is interpretation ? Thank you !

Hello, Surfing across the internet, I learned that the volume of a sphere in n dimensions can be expressed by V(n) = (Π^(n/2)) / Γ((n/2)+1), where n is the number of dimensions we are considering But if we consider n=0, then we get 1. So, how do we interpret this? I mean what does volume in zero...

Dimension of Phase Space of a rigid diatomic molecule is 10. Shouldn't it be 12?

Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively Continue reading the Original PF Insights Post.

Is the phase space of the universe, including it's partitioning, defined and calculated from the the SM?

There is phase space, phase point, etc. So what is the exact meaning of phase? I only understand the definition of phase in wave. Beside what is canonical coordinate? What does this canonical mean?

Homework Statement Dear all, I am desperately trying to solve the following exercise, but unfortunately can't find any resources how to properly calculate the phase space volume. Given is a system of ##N>>1## classical particles that are allowed to move in a cylinder with a Radius of ##R##...

First, two definitions: let ## \varrho (M)## be the probability density of macro states ##M ## (which correspond to a subgroup of the phase space) and ## \mathrm{d} \Gamma ## be the volume element of a phase space. In my lecture notes, the derivation for continuity equation of probability...

In Lagrangian/Hamiltonian mechanics, what is it that makes phase space special compared to configuration space? As a simple example, if I use ## q ## as my generalized position and ## v = \dot{q} ## as my generalized momentum, then the Hamiltonian H = \frac{1}{2} v^2 + \frac{1}{m} V(q) gives...

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space. I see how they both lead...

Currently learning about Statistical Mechanics and just wanted to check my understanding. Am I right in saying that a point in phase space is just a specific microstate of the system?

Homework Statement For a double pendulum, how do we plot the phase space for ##\theta_2## (the lower of the pendulum), i.e. the plot ##\theta_2, \ \dot{\theta}_2?## ##x## = horizontal position of pendulum mass ##y## = vertical position of pendulum mass ##\theta## = angle of pendulum (0 =...

Homework Statement A mass m = 750 g is connected to a spring with spring constant k = 1.5 N/m. At t = 0 the mass is set into simple harmonic motion (no damping) with the initial conditions represented by the point P in the phase space diagram at the right. **(This phase space diagram has...

Hi PF I read the definition of the displacement operator: ##D(\lambda) = e^{\lambda a^\dagger - \lambda ^* a}## but i did not find how this operator can be implemented say in a cavity with a photonic state inside. Could you give me links? thanks.

I have some 3-D model output for a river system that is tidally forced at the entrance. Right now, I'm trying to perform some linear regression on the harmonic constants of various tidal constituents at for several locations along the river compared to the observed tidal data. A linear...

I want to know lagrange mechanics work in phase space or in coordinate system.Leonard Susskind talked about the least action and he said If we know two point we can define trajectory but I don't know the diagram that he drow its a phase space or coordinate system (x,y,z,t) 19 min or...

In several books I have seen the statement that due to Heisenbergs principle no particle can be localized into a region of phase space smaller than ##(2 \pi \hbar)^3##. However, Heisenbergs uncertainty principle states that ##dx dp \geq \hbar/2## -- so a direct translation of this should imply...

Wieland gave his ILQGS talk yesterday, 16 September. Audio and slides are on line. Title: Covariant loop quantum gravity: Its classical action, phase space and gauge symmetries http://relativity.phys.lsu.edu/ilqgs/wieland091614.pdf http://relativity.phys.lsu.edu/ilqgs/wieland091614.wav The...

In classical mechanics we can get a nice overview of the dynamics of a system by looking at its position-momentum phase space. Is there a useful analogue of this concept in special relativity? Can the dynamics of a relativistic system be represented by its phase space in the same way as is done...

Hi, I found out this paper http://www.pha.jhu.edu/~javalab/pendula/pendula.files/users/olegt/pendulum.pdf with this animation http://www.pha.jhu.edu/~javalab/pendula/pendula.files/users/olegt/pendula.html At first there is written there, that the area of possible states in some range...

What exactly is phase space and how is it different from real space i.e. 3 co-ordinate system? what does it mean when someone says "dynamics occurs in phase space"? I'm very new to all this so pls take that into account to

Hi guys, I was hoping someone may be able to clear something up for me. I have been reading a paper on Quantum decoherence and was curious about what particular point (it could easily just be me misunderstanding) It is commonly noted that the superposition of waves represents all of the...

So I noticed we can define entropy in two very different ways: 1) quantum mechanically S = -k Tr(\hat{\rho}\ln{(\hat{\rho})}) 2) classically S = -k \int \rho \ln{(\rho)} d\Gamma where Tr is the trace and d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i} is the phase...

To what extent do phase space trajectories describe a system? I often see classical systems being identified with (trajectories in) phase space, from which I get the impression these trajectories are supposed to completely specify a system. However, if you take for example the trajectory...

If I understand well like in kinematics where we could have eq of motion ##x=x(t)##, ##y=y(t)## and we get eq of trajectory with elimination of time. In dynamics we have ##x=x(t)##, ##p=p(t)## and with elimination of time we get eq of phase trajectory. Am I right?

Is there any relationship between phase space and entropy? regarding to decay of particles

Homework Statement A classical gas consists of N molecules; each molecule is composed of two atoms connected by a spring. Identify the dimensionality of the phase space that can be used to describe a microstate of the system. The Attempt at a Solution I believe the answer is 12, but...

This is about a specific property of the Wigner distribution in phase space. My professor mentioned the other day that the Wigner distribution treats all functions of momentum and space on the same footing as momentum itself or at least that's what I recall.He mentioned a specific problem where...

Classically a single particle will have 3 position coordinates and 3 momentum coordinates, and so it "exists" in a 6-dimensional phase space and moves around this space in relation to time (known as the phase trajectory). However I've read that when we have N classical particles, their...

In statistical mechanics, nearly all the textbooks say that the volume of the smallest cell in the phase space of a N-particle system is h^{rN} where h is the Planck Constant, r is the degree of freedom. Also these books say that this comes from the uncertainty principle. However, the...

Title says it all, confused as to how I'm supposed to define the phase space of a system, in my lecture notes I have the phase space as {(q, p) ϵ ℝ2} for a 1 dimensional free particle but then for a harmonic oscillator its defined as {(q, p)}, why is the free particles phase space all squared...

1. Sketch the phase space of a weight free falling along the z coordinate (no motion in other directions). Sketch the trajectory of the free fall including impact on the ground. 2. Calculate the density of states, entropy, and temperature (all as a function of energy) for the following model...

Hi, this is my first post. I did a search and in this sub-forum I found the most related threads for what I'm looking for. I need some guidance or where or how to learn all this mathematics for velocity-phase space integrals that appear in Maxwellian distributions. I'm an Engineer in...

In general, how do you prove that a given trajectory in phase space is closed? For example, suppose the energy E of a one-dimensional system is given by E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4, where ε is a positive constant. Now, I can easily show that all phase...

Can I graph the phase space of a 2D harmonic oscillator in R^2 in the following way? Let one vector in R^2 represent for position of the point mass and let another vector represent momentum. Together these two vectors in R^2 can represent a single vector in R^4? Do we loose any "information" in...

Homework Statement Ok I have attached the pdf file and I have a problem with velocity phase spaces (Question 3a). Honestly, the lecture notes were not very helpful and looking online and in textbooks, they talked about solving Lagrange's equations but nothing to deal with the problem of Q3...

Homework Statement if i wanted to obtain an "exact" solution for flow s(t|k) k=(q,p) with a hamiltonian H(k) = x(ak) use the fact aJa = 0 where J is the poisson matrix Homework Equations The Attempt at a Solution I hate obscure proofs... i like actual question so I'm...

Coordinate System of Coupled Oscillators and "4D" Phase Space representation So, I've modeled the interaction between two cantilever beams with the kinetic and potential energies shown in the above figure. The cantilevers are very stiff and have a small oscillation amplitude, so they can be...

Hello, I thought the statistical definition of entropy for an isolated system of energy E (i.e. microcanonical ensemble) was S=k \ln \Omega where \Omega is the volume in phase space of all the microstates with energy E. However, if you take a look here...

The phase space trajectories of an autonomous system of equations don't intersect. Can this be proved mathematically. Also what is the physical significance of this statement. What happens if they intersect?

X = [0, 1] \bigcup (2,3) is phase space. Show that (2, 3) open and closed set of X . the question is like that but I think it is false because it is not close, right?

In my lecture they give the phase space picture for a simple pendulum http://mathematicalgarden.files.wordpress.com/2009/03/pendulum-portrait3.png?w=500&h=195 and then say that adjacent trajectories never diverge and therefore evolution is predictable. I wanted to ask, is the statement that...

How can we go from the configuration space of the system to the phase space when velocity can be expressed in terms of momenta?

can you construct (or if yes, is it regularly done) a Hamiltonian in curved spacetime? If you took a system and moved it into a strong gravitational field or accelerated it to relativistic speeds can you still do Hamiltonian mechanics...

Erik Verlinde gave this great talk at Perimeter on Wednesday last week, which is online as video. It was at a recent Holo Cosmo workshop they had there http://pirsa.org/C11010 It is a very exciting talk. He is actively grasping for what many people dream about: a concrete way to think of...

Why can't trajectories in phase space intersect?

I would like to understand phase space better, spec. in relation to the quantum Liouville theorem. Can anyone point me to a decent online resource? I am most interested in conceptual understanding to begin with. Liouville's theorem says that if you follow a point in phase space, the number of...