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I'm new to his forum. I'm having trouble with the following question on central potentials. It's an exercise from (Bransden and Joachain, Introduction to quantum mechanics).

1. The problem statement, all variables and given/known data

Suppose V(r) is a central potential, expand around r=0 as [tex]V(r)=r^p(b_0+b_1r+\ldots).[/tex] When p=-2 and [tex]b_0<0[\tex], show that physically acceptable solutions exist only when [tex]b_0>-\frac{\hbar}{8\mu}[/tex]

2. Relevant equations

R(r) is the radial component of the wave function

[tex]u(r)=r^{-1}R(r)=r^s\sum{c_kr^k}[/tex] is a solution of

[tex]-\frac{\hbar^2}{2\mu}\frac{d^2u(r)}{dr^2}+V_{eff}u(r)=Eu(r)[/tex]

3. The attempt at a solution

The effective potential is [tex]r^{-2}(b_0+\frac{l(l+2)\hbar^2}{2\mu}+b_1r+\ldots)[/tex]

When p>-2, the case that is discussed in the text book, they compare the lowest order terms in the radial Schroedinger equation. For p=-2, I get:

The lowest order term of [tex]\frac{d^2u}{dr^2}=s(s-1)r^{s-2}\sum{c_kr^k}[/tex]

The lowest order term of [tex]V_{eff}(r)u(r)=(b_0+\frac{l(l+2)\hbar^2}{2\mu})r^{s-2}\sum{c_kr^k}[/tex]

The difference is apparently that [tex]b_0[/tex] appears in the lowest order terms.

Now I need some constraint that excludes non-physical solutions. For p>-2, that is u(0)=0. But for [tex]b_0<0[/tex] that can't be used. Because the potential is attracting, I think the probability of finding a particle at the origin should be positive. The problem is I can't think of any constraint that should be imposed, that could limit the allowed values of [tex]b_0[/tex].

Any hints would be greatly appreciated.

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# Central potential in quantum mechanics

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