Calculating Phase Space Volume for Canonical Ensemble

[mkbh_10]

In calculating entropy in micro canonical ensemble we use KlnW where W is the no. of accessible micro states to the system , now when we move on to canonical ensemble ,we calculate the partition function and from there derive the thermodynamics of the system , and divide it by factor of h to make it dimensionless , now if one is asked to calculate phase space volume , it means we calculate the partition function by integration of exp[B(H(q,p))] dqdp and not dividing this by h . Is this correct ??

 

[RedX]

In calculating entropy in micro canonical ensemble we use KlnW where W is the no. of accessible micro states to the system , now when we move on to canonical ensemble ,we calculate the partition function and from there derive the thermodynamics of the system , and divide it by factor of h to make it dimensionless , now if one is asked to calculate phase space volume , it means we calculate the partition function by integration of exp[B(H(q,p))] dqdp and not dividing this by h . Is this correct ??

I think that's correct, except the exponential should have a negative sign.

I think you should remember that the whole reason for dividing by "h" is because you are calculating something classically and want it to work quantum mechanically (it's not just a dimension thing). There is no such thing as an integral over dqdp in quantum mechanics (except in the path integral formalism) as you can't know both the position and momentum at the same time. So you coarse grain the phase space by a box of size "h" and hope this uncertainty in phase space gives you the correct classical answer.

addendum:

there are some good problems in a textbook by Pathria on statistical mechanics that illustrates coarse-graining. a simple one is to solve the harmonic oscillator both classically (where you need to insert "h" by hand) and quantum mechanically (where "h" comes naturally) and compare the two results in the limit of high energy. another one involves solving a gas of particles subject to the condition that if you take a snapshot of the gas, they are all within a n-dimensional regular polyhedra - calculate this classically and compare it quantum mechanically and you'll see that they give more or less the same result.

if you have a good teacher she will go through coarse-graining in class as it's important to know when classical mechanics works and when you need quantum mechanics. unfortunately, I didn't have a good teacher, so I learned from those two problems.