Physics 471 Problem Set 12 Fall 2006
49. Solve the radial equation for a spherical well of depth −V0and radius a
V(r) = (−V00≤r≤a
0r > a
with E < 0 and `= 0.
(a) Show that the bound state energies are determined by the equation
κnacot(κna) = −s2mV0a2
¯h2−κ2
na2with κ2
n=2m
¯h2(En+V0).
(b) Determine the ground state energy for the case 2mV0a2/¯h2= 4. Express your result
as a factor times V0.
(c) Is there a bound state for every value negative value of V0?
50. The radial probability density P(r) for a hydrogenic state with quantum numbers n, ` is
given by r2|Rn`(r)|2.
(a) From the general result for the form of the radial wave function,
Rn`(r) = An`ρ`e−ρv(ρ), ρ =r
na ,
determine An` for the states with `=n−1.
(b) Obtain P(r) for these states.
(c) Show that the most probable value of rfor the states with `=n−1 is given by the
Bohr result
r=n2a .
51. An electron in a hydrogen atom is in a state given by the wave function
ψ(~r) = 1
√3πa5re−r/a .
(a) What is the value of `for this state?
(b) Find the probability that a measurement of the energy of this electron will give the
value −13.6 eV. Here, ais the Bohr radius.