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A particle is bound by

$$V(x) = \begin{cases}\infty,& x <0 \\ \frac{-32\hbar^2}{ma}, & x\le a \\ 0, & x \le a\end{cases}$$

a) how many states are there?

i'm attempting to solve this via raising/lowering operators, by taking $E_n = \langle \Psi|\hat{H}|\Psi\rangle$ then substituting the identity in for the Hamiltonian, but am confused on what to put for the wave function or states. Am I approaching this wrong, would it be easier to solve this as a scattering problem, and solving for coefficients A, B, & C and plugging it into the general solution form?

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you wrote x<=a twice, perhaps you meant x>=a the second time? – Prathyush Apr 5 '13 at 19:00
Hi Sean - this is a site for conceptual physics questions, not to get homework answers. Accordingly I removed the second part of your question, which is unrelated to the conceptual issue you posted about the first part. If, after trying part (b) you have a conceptual question about something you encounter there, feel free to post it separately. – David Z Apr 5 '13 at 19:40
There's a Wikipedia page that takes you through all the steps: – Vibert Apr 6 '13 at 8:41

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