- #1

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## Homework Statement

The eigenvalue problem H[itex]\psi[/itex]=E[itex]\psi[/itex] for [itex]\phi[/itex] becomes

-[itex]\phi[/itex]''+2x[itex]\phi[/itex]'+((a(a-1))/x

^{2})[itex]\phi[/itex]+(1-2E)=0

assume that [itex]\phi[/itex](x)=[itex]\sum[/itex]a

_{n}x

^{n+B}, determine B.

**2. The attempt at a solution**

As a first step I took the first and second derivatives of [itex]\phi[/itex]:

[itex]\phi[/itex]'=[itex]\sum[/itex](n+B)a

_{n}x

^{n+B-1}

[itex]\phi[/itex]''=[itex]\sum[/itex](n+B-1)(n+B)a

_{n}x

^{n+B-2}

and then substituted these back into -[itex]\phi[/itex]''+2x[itex]\phi[/itex]'+((a(a-1))/x

^{2})[itex]\phi[/itex]+(1-2E)=0; which is

-[itex]\sum[/itex](n+B-1)(n+B)a

_{n}x

^{n+B-2}+2x([itex]\sum[/itex](n+B)a

_{n}x

^{n+B-1})+((a(a-1))/x

^{2})([itex]\sum[/itex]a

_{n}x

^{n+B})+(1-2E)=0

And it's at this point (assuming I'm working correctly up to here) that I stop-short mentally; how do I go about solving this monster for B?