Re: Accessible States

[Daniel Schroeder <DSCHROEDER@CC.WEBER.EDU>]

John has already answered a lot of this, but let me add that you don't
have to use a basis of minimum-uncertainty wave packets.  You could
just as well use a basis of particle-in-a-box energy eigenfunctions,
which overlap each other completely in space.  The point is just to
count the dimensionality of the vector space of accessible states.
Any basis will do.  However, some bases are more convenient than
others.  In particular, I don't know how to use the definite-energy
states in a microcanonical treatment like Jim describes.  If anyone
out there does, please let me know!  (Canonical is no problem...)

Dan Schroeder

Jim Green writes:
> Consider a box of volume V which contains N atoms of an ideal gas which
> have a total energy E.
>
> Now I start to worry:  Assume that each atom is contained in one of a
> number of
> 6-dimensional cells of
>
> V(6) = DxDyDzDpxDpyDpz = h^3
>
> The total number of states in the box for each atom is
>
> Omega(1) = (total 6-d volume)/cell volume
>                  = V(real)V(momentum)/DxDyDzDpxDpyDpz
>                  =  VrVp/h^3
>
> Now Vp = (4/3)pi x p(max)^3    &   p(max) = sqrt(2mE)
>
> So Omega(1) = (4/3)pi(2m)^(3/2) x V E^(3/2)
>
> Now Omega(N) =Prod[Omega(i)]
>
> So Omega(N) = Const x V^N x E^(3N/2)    Cf big Rief Eqn 2.5.19
>
> The result seems OK (assuming I typed it correctly), but I have problems:
>
> 1) Is it ok to say that the atom fills a Heisenberg volume?  ie that the
> possible 6-d positions of the atoms are this size? (This is what the
> "Particle Position" thread has warmed me to.)  Is it ok to find Omega by
> just dividing the total 6-d volume by the cell volume?  This seems to be in
> accord with the similar approach of Stowe, but alas I think that he makes
> mistakes elsewhere.  I would feel a lot better if there were some
> concurrence by the list here.  At least this approach arrives at the usual
> answer.
>
> 2) Is there a way to get this result via some method not using modern
> concepts -- none of the usual texts do.  What was Boltzman's thinking here
> for S=k ln Omega?