Mathematical definition of critical point？(thermodynamics)

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Homework Statement


If a gas is described by Van der Waals eqn (p+a/v^2)(v-b)=kT, find the critical pint p_c, T_c,and v_c

Homework Equations




The Attempt at a Solution


Actually this is just one intermediate step I got stuck for my stat mech homework, I don't even know the mathematical definition for critical point(I didn't learn quite well in my thermodynamics course), the only relevant equation I found is in wiki http://en.wikipedia.org/wiki/Critical_point_(thermodynamics), but I don't think it is adequate for my problem.
 
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Em.. sorry, I made a silly mistake in my mind, equations given in wiki should be adequate for my 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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