How Do I Solve Part (b) of Problem 5.17 in Schroeder's Thermal Physics?

touqra
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This is really not a homework question. I'm learning thermal physics by Schroeder (2000) myself.
I'm stuck at Problem 5.17 page 160.

I can get Part (a) but Part (b), subtracting off the work that would be required even if the specimen were not present. I subtracted the work \frac{B}{\mu_0} dB but it wasn't correct...
 
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I'm sure someone has the book but I'd probably have an easier time if I had, you know, the problem
 
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This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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