Phase space Definition and 118 Threads

The force two static electric charges exert on each other fulfills (velocity of light set unity) \begin{equation} F r^2 = N_1 N_2 \alpha \hbar \;, \end{equation} where ##F## is the force, ##r## is the mutual distance, ##\alpha## is the dimensionless fine structure constant, ##\hbar## is Planck's...

Why do we treat velocity and coordinates as independent variables until the very end, where we then assume the dependence of velocity on coordinates via a time derivative? That is, let the Lagrangian of a given system be simply $$\mathcal L=\frac12mv^2$$ Now, plugging this into the...

I've a doubt regarding the definition of canonical coordinates in phase space. As far as I can tell, phase space ##T^*M## is the cotangent bundle of the system configuration space ##M##. ##M## is assumed to be a differential manifold with atlas ##A=\{ U_i, \phi_i \}##. Call ##q_i## the...

Hey all, I am reading David Tong's notes on Chern-Simons: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf and he makes the following statement that doesn't make much sense to me: "Because the Chern-Simons theory is first order in time derivatives, these Wilson loops are really parameterising...

At the moment, I do not have any supporting material.

The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$ $$d\eta=dq_1.....dq_ndp_1......dp_n$$ $$d\rho=dQ_1.......dQ_ndP_1....dP_n$$ and we know that both of them are related to each other by the absolute value of the determinant of...

I encounter a function that I don‘t know in the calculation of Relativistically invariant 2-body phase space integral: in this equation, ##s##is the square of total energy of the system in the center-of-mass frame(I think) I don't know what the function ##\lambda^{\frac{1}{2}}## is. There are...

Can someone please explain to me how can we obtain this integral in eq. 5.27 from eq. 5.26? I quite do not understand how is it possible to make this adjustment and why the (p_(f))^2 appeared there in the numerator and also why a solid angle appeared there suddenly.

Hi, I am currently preparing for my exam and have just watched a video about motion in phase space. From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory. Here are the pictures quadratic potential phase trajectory Regarding phase...

what is the reason for that the energy hypersurfaces in a phase space, which belong to systems with constant energy are closed? ( see picture )

*Pathria, Statistical mechanics*"The microstate of a given classical system, at any time, may be defined by specifying the instantaneous positions and momenta of all the particles constituting the system. Thus. If ##N## is the number of particles in the system, the definition of a microstate...

Hello all. I am studying stat mech from Pathria's book. It says a system is completely described by all positions and momenta of all the N particles. This maybe represented by a single point in 6N-D gamma space. So, each point is a (micro)state. Now if we restrict the system (N,V,E to E+ΔE)...

Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...

I attatched an example plot where I created the histogram for the differential distribution with respect to the energy of the d-quark produced in the scattering process. My conception is that the phase space generator can "decide" how much of the available energy it assigns to the respective...

I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...

The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...

In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist...

I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem: Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential...

I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...

In Henley and Garcia's Subatomic Physics, they introduce phase space in chapter 10 by considering all the possible locations a particle can occupy in a plot of ##p_x \ vs. x##, ##p_x## being the momentum of the particle in the x direction. They next consider an area pL on this plot, and state...

Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...

For a system of interacting particles, is it possible to define single-particle phase spaces? If not, why?

I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...

Phase space trajectories can't intersect each other is it due to the fact that at the intersection point there will be more than one possible path for the system to evolve with time??

Homework Statement I have phase space coordinates (x0,y0,z0,vx,vy,vz)=(1,0,0,0,1,0). I need to analytically show that these phase space coordinates correspond to a circular orbit. Homework Equations r=sqrt(x^2+y^2+z^2) maybe? The Attempt at a Solution My core problem here is maybe that I...

I was looking at the following proof of Louviles theorem : we define a velocity field as V=(dpi/dt, dqi/dt). using Hamilton equations we find that div(V)=0. using continuity equation we find that the volume doesn't change. I couldn't figure out the following : 1- the whole point was to show...

Homework Statement I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space d^3q \ d^3p and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J...

Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space. Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space. Could anyone please explain the difference between configuration space and phase space. Thank you in advance for...

In the field of medical physics, specifically in monte carlo simulation of radiation beams produced by electron accelerators, people call ‘phase space’ to a file that contains the data of a large number of particles when they traverse a reference surface in the machine (usually a plane), i.e...

Homework Statement From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion: 14.13: Decide if the energy surfaces in phase space are bounded for the following cases: i.) The two-body gravitation problem with E<0 ii.) The two-body gravitation problem viewed from the...

Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Reduced Phase Space Continue reading the Original PF Insights Post.

Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Phase Space Continue reading the Original PF Insights Post.

In state space, the coordinates are the state variables of a system.So,each point in state space represents a specific value of state variables.Thus,state space representation represents the changes in a dynamical system. The state variables are the minimum number of variables which uniquely...

Hi, i would like to know how can i plot a three dimentionnal phase space (mathematica), for this kind of differential equations: x'= (z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/(z^2+x^2-2) y'=y(y-3)+z(4y-z)+3(1-x^2)-2(y-3z)(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)...

What is a symplectic manifold or symplectic geometry? (In intuitive terms please) I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more...

Hi, I'm recently reading "Particle Physics in the LHC Era" and there is a part about the phase space factor that confuses me. When giving the Lorentz invariant phase space, they wrote: d3p / 2E = θ(E) δ(p2 - m2) d4p This is very confusing as it equates a three dimensional differential to a...

Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...

In quantum mechanics, the velocity field which governs phase space, takes the form \begin{equation} \boldsymbol{\mathcal{w}}=\begin{pmatrix}\partial_tx\\\partial_tp\end{pmatrix} =\frac{1}{W}\begin{pmatrix}J_x\\J_p\end{pmatrix}...

I always had this doubt,but i guess i never asked someone. What's the main difference between the Classical phase space, and the two dimensional Hilbert Space ?

I have trouble understanding why we classify an inviscid adiabatic incompressible flow along a streamline as isentropic I understand this from a Thermodynamic definition/explanation $$dS = dQ/T$$ Adiabatic Invsicid $$dQ =0= dS$$ So no heat added or lost no change in entropy I'm fine with that...

Hello! I was reading a paper on formulation of QM in phase space (https://arxiv.org/abs/physics/0405029) and I have some doubts related to chapter 5. It seems to me that there is a transformation to modified polar coordinates (instead of radius there is u which is square of radius multiplied by...

I'm asking if space space is subject to change, if not, why not, and if so, then would it be subject to a subsequent phase space that describes that? Let us make a phase space for a deterministic time evolved dynamic system of three spatial dimensions. Since the actions of the system and the...

Hi, So I am aiming to derive the continuity equation using the fact that phase space points are not created/destroyed. So I am going to use the Leibiniz rule for integration extended to 3-d: ## d/dt \int\limits_{v(t)} F dv = \int\limits_{v(t)} \frac{\partial F}{\partial t} dV +...

Homework Statement [/B] (a) Verify explicitly the invariance of the volume element ##d\omega## of the phase space of a single particle under transformation from the Cartesian coordinates ##(x, y, z, p_x , p_y , p_z)## to the spherical polar coordinates ##(r, θ, φ, p_r , p_θ , p_φ )##. (b) The...

In statistical physics, the partition function should be calculated in the whole phase space.This is finally an integral over the phase space, like ∫d3Nqd3Np... The problem is that the integral covers some cases that different particles have the same generalized coordinates and momenta. So, is...

Why do we take a particle's position ##x## and its velocity ##\dot{x}## as independent variables in a phase space when they are dependent in the sense that given the function ##x(t)##, we can get the function ##\dot{x}(t)##? I'm thinking they are independent variables but not independent...

Can Phase Space be break down into different regions, different regions that are not mixed up with one another, if we do so, the different regions can raise to any conservation of physical quantity?

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...