Why can we solve some QM problems?

Here is some old stuff back from Paulo Bedaque's 622.

Write the two operators



They are H.C.s of each other so lets call them A and  respectively. Write two Hamiltonians




If you have an eigenstate   of the second, then you automatically get an eigenstate of the first by taking  . Therefore the two Hamiltonians have the same spectrum except possibly at the bottom rung.

Expanding out the Hamiltonians gives the two potentials as .

Playing with the ground state of the Hamiltonian gives a formula for the superpotential, or the other way around if you were given the ground state instead.

Why is this supersymmetric? Take the direct sum of the two Hilbert spaces. A and  placed in blocks of a 2 by 2 in order to switch between the two pieces. If we call those two spaces, the fermionic and bosonic pieces then we just got the supersymmetry exchanging the two. The superalgebra here is SL(1,1)

This can be done in higher dimensions after passing to the effective 1 dimensional problem after including the angular momentum barrier pushing away from the origin.

Now we can get to the title. Why are certain potentials solvable?

This happens when you have the following condition on the partner potentials.



where the a's are parameters. If we have the above condition, you can construct a ground state for the first Hamiltonian and shape invariance guarantees that the spectra are just shifts of each other. You can build up the entire bound state spectrum from this by applying raising operators and bouncing back and forth between Hamiltonians.

Your favorite potentials that you have solved like harmonic oscillators and the hydrogen atom and their partners form shape invariant pairs. So they are exactly solvable as you knew because you solved them back in your misspent youth.
Shape invariance is a sufficient but not necessary condition for solvability.

For more:
http://arxiv.org/pdf/hep-th/9405029.pdf