Peeter Joot's (OLD) Blog.

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Posts Tagged ‘velocity space volume element’

A final pre-exam update of my notes compilation for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on April 22, 2013

Here’s my third update of my notes compilation for this course, including all of the following:

April 21, 2013 Fermi function expansion for thermodynamic quantities

April 20, 2013 Relativistic Fermi Gas

April 10, 2013 Non integral binomial coefficient

April 10, 2013 energy distribution around mean energy

April 09, 2013 Velocity volume element to momentum volume element

April 04, 2013 Phonon modes

April 03, 2013 BEC and phonons

April 03, 2013 Max entropy, fugacity, and Fermi gas

April 02, 2013 Bosons

April 02, 2013 Relativisitic density of states

March 28, 2013 Bosons

plus everything detailed in the description of my previous update and before.

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Velocity volume element to momentum volume element

Posted by peeterjoot on April 9, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Motivation

One of the problems I attempted had integrals over velocity space with volume element d^3\mathbf{u}. Initially I thought that I’d need a change of variables to momentum space, and calculated the corresponding momentum space volume element. Here’s that calculation.

Guts

We are working with a Hamiltonian

\begin{aligned}\epsilon = \sqrt{ (p c)^2 + \epsilon_0^2 },\end{aligned} \hspace{\stretch{1}}(1.1)

where the rest energy is

\begin{aligned}\epsilon_0 = m c^2.\end{aligned} \hspace{\stretch{1}}(1.2)

Hamilton’s equations give us

\begin{aligned}u_\alpha = \frac{ p_\alpha/c^2 }{\epsilon},\end{aligned} \hspace{\stretch{1}}(1.3)

or

\begin{aligned}p_\alpha = \frac{ m u_\alpha }{\sqrt{1 - \mathbf{u}^2/c^2}}.\end{aligned} \hspace{\stretch{1}}(1.4)

This is enough to calculate the Jacobian for our volume element change of variables

\begin{aligned}du_x \wedge du_y \wedge du_z &= \frac{\partial(u_x, u_y, u_z)}{\partial(p_x, p_y, p_z)}dp_x \wedge dp_y \wedge dp_z \\ &= \frac{1}{{c^6 \left( { m^2 + (\mathbf{p}/c)^2 } \right)^{9/2}}}\begin{vmatrix}m^2 c^2 + p_y^2 + p_z^2 & - p_y p_x & - p_z p_x \\ -p_x p_y & m^2 c^2 + p_x^2 + p_z^2 & - p_z p_y \\ -p_x p_z & -p_y p_z & m^2 c^2 + p_x^2 + p_y^2\end{vmatrix}dp_x \wedge dp_y \wedge dp_z \\ &= m^2 \left( { m^2 + \mathbf{p}^2/c^2 } \right)^{-5/2}dp_x \wedge dp_y \wedge dp_z.\end{aligned} \hspace{\stretch{1}}(1.5)

That final simplification of the determinant was a little hairy, but yielded nicely to Mathematica.

Our final result for the velocity volume element in momentum space, in terms of the particle energy is

\begin{aligned}d^3 \mathbf{u} = \frac{c^6 \epsilon_0^2 } {\epsilon^5} d^3 \mathbf{p}.\end{aligned} \hspace{\stretch{1}}(1.6)

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