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cscott
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Homework Statement
Consider the time-independent Schrodinger equation in spherical polar coordinates for a free particle, in the case where we have an azimuthal quantum number [itex]l=0[/itex].
(a) Solve the radial equation to find the (unnormalized) radial wavefunction [itex]R(r)[/itex].
(b) Normalize [itex]R(r)[/itex], using the definition of the dirac delta function [itex]\delta(k'-k)[/itex].
Homework Equations
Radial Equation:
[tex]u(r) = rR(r)[/tex]
[tex]-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[V + \frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right]u = Eu[/tex]
The Attempt at a Solution
For a free particle, [itex]V=0[/itex], and with [itex]l = 0[/itex] the radial equation reduces to,
[tex]\frac{d^2u}{dr^2}=-k^2u,~~k=\frac{\sqrt{2mE}}{\hbar}[/tex],
with solution,
[tex]u = A\sin(kr) + B\cos(kr)[/tex],
but [itex]u(r)=rR(r)[/itex], so [itex]B=0[/itex] for a normalizable wavefunction (considering r->0). Therefore,
[tex]R(r) = \frac{A}{r}\sin(kr)[/tex]
and to normalize,
[tex]\int_{0}^{\inf} r^2|R(r)|^2~dr = 1[/tex]
[tex]|A|^2 \int_{0}^{\inf} \sin^2(kr)~dr=1[/tex]
How do I use the dirac delta function?
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