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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).
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<0[\tex], show that physically acceptable solutions exist only when b_0&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>-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>-2, that is u(0)=0. But for b_0&lt;0 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 b_0.<br /> <br /> Any hints would be greatly appreciated.
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<0[\tex], show that physically acceptable solutions exist only when b_0&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>-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>-2, that is u(0)=0. But for b_0&lt;0 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 b_0.<br /> <br /> Any hints would be greatly appreciated.