Mean field theory for Ising model and Weiss approximation.

See attached file.

Bose-Einstein condensate of atoms in a potential well.

This exercise is for a Bose-Einstein condensate of indistinguishable atoms which do not interact with each other and are in a 3-dimensional harmonic well. The system is described by the following Hamiltonian (see attached file).

Taylor Expansion

Three Point Forward Difference Formula. See attached file for full problem description.

Consider two spin systems A and A' placed in an external field H.

I need to know how to start the problem, just how to do part a), and then I'll probably be able to do the rest on my own. Problem : Consider two spin systems A and A' placed in an external field H. System A consist of N weakly interacting localized particles of spin 1/2 and magntic moment u. Similarly, system A' consists of N' weakly interacting localized particles of spin 1/2 and magnetic moment u'. The two systems are initially isolated with respective total energies bNuH and b'N'u'H. They are then placed in thermal contact with each other. Suppose that |b|<<1 and |b'|<<1 so that the simple expressions of Problem 2.4c can be used for the densities of stats of the two systems. ... click for more

Expansions of a Perfect Gas

Please see attached file for full problem description. I started doing this problem but I could way off here. I think that for part C it is the Joule-Kelvin expansion. So the final temperature would be T1 plus a hairy integral. For part B) it seem like it is just the Joule expansion. However, I am not sure if I manipulated the partial derivatives in the right way. Part A I can't figure out. I think I would start with the 1st law in the differential form. Drop the dS term and try to manipulate the derivatives again. However, I do not know what type of expansion this would be. If someone could do this problem from A to C I would really appreciate it.