Central potential in quantum mechanics

Bavon
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Hi all,
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).

Homework Statement


Suppose V(r) is a central potential, expand around r=0 as V(r)=r^p(b_0+b_1r+\ldots). When p=-2 and b_0&lt;0[\tex], show that physically acceptable solutions exist only when b_0&amp;gt;-\frac{\hbar}{8\mu}<br /> <br /> <br /> <h2>Homework Equations</h2><br /> R(r) is the radial component of the wave function<br /> u(r)=r^{-1}R(r)=r^s\sum{c_kr^k} is a solution of<br /> -\frac{\hbar^2}{2\mu}\frac{d^2u(r)}{dr^2}+V_{eff}u(r)=Eu(r)<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> The effective potential is r^{-2}(b_0+\frac{l(l+2)\hbar^2}{2\mu}+b_1r+\ldots)<br /> <br /> When p&gt;-2, the case that is discussed in the textbook, they compare the lowest order terms in the radial Schroedinger equation. For p=-2, I get:<br /> <br /> The lowest order term of \frac{d^2u}{dr^2}=s(s-1)r^{s-2}\sum{c_kr^k}<br /> The lowest order term of V_{eff}(r)u(r)=(b_0+\frac{l(l+2)\hbar^2}{2\mu})r^{s-2}\sum{c_kr^k}<br /> <br /> The difference is apparently that b_0 appears in the lowest order terms.<br /> <br /> Now I need some constraint that excludes non-physical solutions. For p&gt;-2, that is u(0)=0. But for b_0&amp;lt;0 that can&#039;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&#039;t think of any constraint that should be imposed, that could limit the allowed values of b_0.<br /> <br /> Any hints would be greatly appreciated.
 
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I solved this myself. It was so easy after all!
 
Dear Bavon
Would u please help me to solve this problem too?
thanks
I am looking forward for ur reply. It is an emergency situation!
 
Last edited:
If I decyphered my notes from almost 3 years ago correctly, I constructed an equation in s, and then expressed that it should have real solutions.
 
How can I undrestand that an equation have real solutions?
thanks
 
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