Chemical Potential for Bosonic Particles

[YAHA]

Homework Statement

I am working Problem 5.29 *** (b) in Griffiths QM. We are asked to show that m(T) monotonically increases as T decreases, assuming N and V are constants. m(T) - is chemical potential.

Homework Equations

Too many to list, probably easier to look in the book if you have it.

The Attempt at a Solution

Honestly, I played with this for 2 hours. I also have the solutions manual. Even after looking there, the logic is completely incomprehensible. Specifically, he concludes that as T->0, E->0 and thus, m(T) must be negative. This step is not evident at all (as are many others in Griffiths books and solutions manuals).

 

[member 553083]

I would really appreciate some help. The solution goes something like this: m(T) = (2/V)[NkTln(2)+(2π2/3)N2/2V2/3T2/3] Differentiating with respect to T: dm/dT = -(2/VT)[N(2π2/3)N2V-2/3T-1/2]Using the fact that dE/dT > 0 we get: dm/dT > 0 This means that m(T) is an increasing function of T As T--> 0, E --> 0 and thus m(T) must be negative.