I'm just trying to solve the Schrödinger equation for hydrogen. We went through the solution by hand in class but Mathematica doesn't do anything when I try to solve it (it just re-displays what I typed in):

DSolve[-h^2/(2*mu)*(1/r^2*D[r^2*D[y[r,theta,phi],r],r] + 
    1/(r^2*Sin[theta])*D[Sin[theta]*D[y[r,theta,phi],theta],theta] + 
    1/(r^2*Sin[theta]^2)*D[D[y[r,theta,phi],phi],phi]) - 
    k*q^2/r*y[r,theta,phi] == energy*y[r,theta,phi],y,{r,theta,phi}]
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    $\begingroup$ According to the help (tutorial/DSolveSecondOrderPDEs), this eigenvalue problem does not fall into the class of second order PDEs that DSolve can handle. Notice that energy is an unknown value to be found, not one of the three arguments of y. $\endgroup$ – whuber Dec 6 '12 at 9:10
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    $\begingroup$ The solution you found in class had some conditions at the centre and at infinity, fixing the eigenvalues. You haven't fixed these and mathematica does not solve this class of problems automatically anyway $\endgroup$ – acl Dec 6 '12 at 10:52

It seems there is a function in mathematica9 which can handle this problem. See http://www.wolfram.com/mathematica/new-in-9/parametric-differential-equations/eigenproblems.html

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    $\begingroup$ Welcome to MMA.stackexchange, and thanks for contributing an answer! Please consider registering your account so that you can keep track of comments, questions, and answers. Finally, could you please give an example of how you would use the new function to solve the problem at hand? $\endgroup$ – tkott Dec 6 '12 at 16:55
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    $\begingroup$ Yes, you can handle the radial equation numerically with NDSolve and the new parametric functionality. If the question is asking for analytical solutions, then it's much more difficult, see @acl's comment above. One would have to do at least some of the separation steps by hand. I guess that means we still need textbooks... $\endgroup$ – Jens Dec 6 '12 at 18:43

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