(a) Since V(r) = 0 in the region r ≤ a, the radial Schrödinger equation (6.57)
-\frac{\hbar ^{2} }{2M} \frac{d^{2}U_{nl}(r) }{dr^{2} } +V_{eff} (r)U_{nl}(r)=E_{n} U_{nl}(r), (6,57)
becomes
-\frac{\hbar ^{2} }{2m}\left[\frac{d^{2}U_{nl}(r)}{dr^{2}}-\frac{l(l+1)}{r^{2}} U_{nl}(r) \right] =EU_{nl}(r), (6.241)
where U_{nl}(r)=rR_{nl}(r). For the case where l = 0, this equation reduces to
\frac{d^{2}U_{n0}(r)}{dr^{2}} =-k^{2}_{n} U_{n0}(r), (6.242)
where k^{2}_{n}=2mE_{n} /\hbar ^{2}. The general solution to this differential equation is given by
U_{n0}(r)=A\cos (k_{n}r )+B\sin (k_{n}r ) (6.243)
or
R_{n0}(r)=\frac{1}{r} (A\cos (k_{n}r )+\sin (k_{n}r )). (6.244)
Since R_{n0}(r) is finite at the origin or U_{n0}(0)= 0, the coefficient A must be zero. In addition, since the potential is infinite at r = a (rigid wall), the radial function R_{n0}(a) must vanish:
R_{n0}(a)=B\frac{\sin k_{n}a}{a} =0; (6.245)
hence ka=n\pi ,n=1, 2, 3, …. This relation leads to
E_{n} =\frac{\hbar ^{2}\pi ^{2} }{2ma^{2}} n^{2}. (6.246)
The normalization of the radial wave function R(r), ∫^{a}_{0} \left|R_{n0}(r)\right| ^{2} r^{2}dr=1, leads to
1=\left|B\right| ^{2} ∫^{a}_{0} \frac{1}{r^{2}} \sin ^{2} (k_{n}r )r^{2}dr=\frac{\left|B\right| ^{2}}{k_{n}} ∫^{k_{n}a}_{0} \sin ^{2}\rho d\rho =\frac{\left|B\right| ^{2}}{k_{n}}\left(\frac{\rho }{2} -\frac{\sin 2\rho }{4} \right) \mid ^{\rho =k_{n}a}_{\rho =0}
=\frac{1}{2} \left|B\right| ^{2}a; (6.247)
hence B=\sqrt{2/a} . The normalized radial wave function is thus given by
R_{n0}(r)=\sqrt{\frac{2}{a} } \frac{1}{r} \sin \left(\sqrt{\frac {2mE_{n} }{\hbar ^{2}} }r \right). (6.248)
(b) For l = 7 we have
E_{1} (l=7)>V_{eff}(l=7)=\frac{56\hbar ^{2}}{ma^{2}} =\frac{28\hbar ^{2}}{ma^{2}}. (6.249)
The second lowest state for l = 0 is given by the 3s state; its energy is
E_{2} (l=0)=\frac{2\pi ^{2}\hbar ^{2} }{ma^{2}} , (6.250)
since n = 2. We see that
E_{1} (l=7)>E_{2} (l=0). (6.251)
(c) Since the probability of finding the electron in the sphere of radius a is equal to 1, the probability of finding it in a sphere of radius a/2 is equal to 1/2.
As for the probability of finding the electron in the spherical shell between r = a and r = 3a/2, it is equal to zero, since the electron cannot tunnel through the infinite potential from r < a to r > a.