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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 [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]<br /> <br /> <br /> <h2>Homework Equations</h2><br /> R(r) is the radial component of the wave function<br /> [tex]u(r)=r^{-1}R(r)=r^s\sum{c_kr^k}[/tex] is a solution of<br /> [tex]-\frac{\hbar^2}{2\mu}\frac{d^2u(r)}{dr^2}+V_{eff}u(r)=Eu(r)[/tex]<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> The effective potential is [tex]r^{-2}(b_0+\frac{l(l+2)\hbar^2}{2\mu}+b_1r+\ldots)[/tex]<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 [tex]\frac{d^2u}{dr^2}=s(s-1)r^{s-2}\sum{c_kr^k}[/tex]<br /> 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]<br /> <br /> The difference is apparently that [tex]b_0[/tex] 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 [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].<br /> <br /> Any hints would be greatly appreciated.[/tex]
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 [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]<br /> <br /> <br /> <h2>Homework Equations</h2><br /> R(r) is the radial component of the wave function<br /> [tex]u(r)=r^{-1}R(r)=r^s\sum{c_kr^k}[/tex] is a solution of<br /> [tex]-\frac{\hbar^2}{2\mu}\frac{d^2u(r)}{dr^2}+V_{eff}u(r)=Eu(r)[/tex]<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> The effective potential is [tex]r^{-2}(b_0+\frac{l(l+2)\hbar^2}{2\mu}+b_1r+\ldots)[/tex]<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 [tex]\frac{d^2u}{dr^2}=s(s-1)r^{s-2}\sum{c_kr^k}[/tex]<br /> 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]<br /> <br /> The difference is apparently that [tex]b_0[/tex] 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 [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].<br /> <br /> Any hints would be greatly appreciated.[/tex]