Phase space Definition and 118 Threads

Hi, Do you know if there is an explicit formula for the integrated 3 body relativistic phase space of 3 particle with the same mass? I.e. M->3m Or an approximate one? Thank you!

I'm working on a visualizer of sorts for a system: x_{n+1} = sin(a y_n) - cos(b x_n) y_{n+1} = sin(c x_n) - cos(d y_n) with a,b,c,d \in [-2.5, 2.5] So for whatever initial (x_0,y_0) I give the system, I know the next iteration will have both x and y between -2 and 2, and that will be...

Let X=(x1, x2, x3) be an element of the vector space C^3. The dot product of X with itself, X·X, is (x1x1+x2x2+x3x3). Note that if x1=a+ib then x1x1=x1^2 = a^2 - b^2 + i(2ab), rather that a^2+b^2, which is x1 times the conjugate of x1. Let the real part of C represent the position of a...

Can we make a connection? Consider the phase space of a point particle in R^3. Six numbers are required, three for position and three for velocity. Now consider an isotropic vector, X, in C^3 with X*X = 0. X = (x1,x2,x3), X*X = (x1*x1 + x2*x2 + x3*x3), x1 = c1 + i*c2, x1*x1 = (c1*c1 +...

I just want to clarify the geometrical interpretation of these objects as encountered in the basic theory of ODEs. For discussion let's use the simple set of differential equations found in classical mechanics for a free falling particle: \dot{x} = v;\ \ \dot{v} = -g; Now in phase space the...

Phase Space -- does each point have a unique time associated with it? Hi all, If I have an autonomous system: dx/dt=f(x)The k-dimensional state vector x lives in a k-dimensional phase space. Does each point in the k-dimensional phase space have a UNIQUE time associated with it?I don't think...

the question is if we have a classical phase space (p,q) the idea is using Heisenberg's uncertainty could we generalize the usual 'geometry' to a non-commutative phase space ? for example we could impose the conditions [ x_i , x_j ]= iL_p \hbar where L_p means Planck's Energy scale and...

Hello ladies and gentlemen Why can't flows in phase space cross? Would it imply that the system may be at the same state at some future time and then follow a different trajectory? That is to say that the identical initial condition gives a different final condition. To my mind, flows in...

Can somebody help me out? I'm reading about formulas for cross sections for spin1 particles but I don't understand the delta functions, in calculating the 2particle pahse space psi For example the interaction; A+B -> C+D has the formula; psi= (2pi)^2 delta(Pa+Pb-Pc-Pd) d3Pc d3Pd / 4EcEd...

Hi guys, I have a volume integral in 3D phase space that looks like: \int \frac{4\pi p^2 dp}{h^3} Now, I want to generalize to N dimensions. How does this look: \int \frac{\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}p^N dp}{N!h^{3N}} Essentially, I've changed the 4 pi (which I...

To make a long story short, the problem has an elliptical ring from width E to E+dE in phase space (p on y axis, x on x axis). This is a harmonic oscillator, so the standard equations apply (E=p^2/2m + kx^2/2)... now for the question I need to find the total area in the ring of the ellipse in a...

Consider a simple harmonic oscillation in 1 dimension: x(t)=Acos(wt+k). If the energy of this oscillator is btw E and E+\delta E, show that the probability the the position of the oscillator is btw x and x+dx is given by P(x)dx=\frac{1}{\pi}\frac{dx}{\sqrt{A^2-x^2}} Hint: calculate the volume...

Does the history of wave packets translate exactly onto infinite phase space, or is phase space incompletely (or redundantly) covered by quantum mechanics?

can anyone handle this one? when deriving a distribution function using a purely statistical approach, Boltzmann uses some kind of a phase space, that is one with 6N dimentions, 3 for position, 3 for momentum. i see some really short descripions of it but not enough to understand it. all...

it's just not sinking in.. i know a cell in phase space has 6 dimensions, 3 for momentum and the other 3 for position. but i'd like to understand it(phase space). can someone give me an example maybe or tell me why this constuct is needed?? or a link to a very good description?

I just read Penrose's explanation of entropy in his book "The Emperor's New Mind". His explanation is completely saturated in an extended discussion of "phase space" . Is this concept of "phase space" absolutely necessary in order to explain or understand entropy ? Celal

This should be an easy general question to someone out there. My "quarks and Leptons" book by Halzen and Martin introduces the term "phase space" 50 pages before the index reference, and never seems to define it. What is phase space in this context? Thanks

Construct a phase space where every point is center to a circle of radius h, Planck's constant. Particular to such a given point, outside its radius lies conventional phase space and inside, conventional phase space inverted through h - together potentially doubling the effective...