The single particle density of states (Statistical physics)

[SingBluSilver]

Homework Statement

I'm having a little bit of trouble getting started with this problem. Can I get a little help?

Using: (number of states in the six-dimensional region d$$^{3}$$x d$$^{3}$$3p) = (d$$^{3}$$x d$$^{3}$$p)/h$$^{3}$$
Which provides a convenient route to the single-particle density of masses.

a) Integrate over space (of volume V) and over the direction of the momentum p to determine D(p)dp, where D(p) denotes the number of states per unit interval of momentum magnitude.

b) Adopt the non-relativistic relationship between kinetic energy and momentum, ε = p$$^{2}$$ / 2m, and determine the number of states per unit energy interval, D(ε). Do you find agreement with our previous result?

c) Consider the relativistic relationship between the total energy and momentum, ε$$_{rel}$$ = (p$$^{2}$$c$$^{2}$$+ m$$^{2}$$c$$^{4}$$)$$^{1/2}$$. determine the number of states per unit interval of total energy, D(ε$$_{rel}$$)

(The rel after epsilon is supposed to be a subscript, not sure why it went superscript)

Homework Equations

(number of states in the six-dimensional region d$$^{3}$$x d$$^{3}$$3p) = (d$$^{3}$$x d$$^{3}$$p)/h$$^{3}$$

The Attempt at a Solution

For part a I'm not exactly sure what I'm supposed to be integrating. Do I just convert d$$^{3}$$x to a volume and use polar coordinates to convert d$$^{3}$$p to 4pi*p^2dp and integrate that?
I'm also not understanding what parts b and c are asking. Can anyone push me in the right direction?

Any help would be appreciated.

Thank you

 

[kloptok]

This is probably too late now, but I think I can help you.

You should do just as you yourself suggested - integrate to get rid of all spatial dependence (assuming no x-, y-, or z-dependence the spatial part is just the volume, V) and then integrate to get rid of the angular part of the momentum (this is assuming spherical symmetry of course, so the integration over the solid angle just gives $$4\pi$$). What is left is something like
$$4 \pi * V * p^2 dp / h^3$$
where h is Planck's constant.

For b) and c) of the problem. This is just a change of variables, like in ordinary calculus. Thus, you replace all p:s with p(E) and switch dp for dE, with the approprate "weight factor" ($$dE= (p/m) dp$$ for a free particle with $$E=p^2 /2m$$.