140 3 Quantum Mechanics – II
3.17 In Problem 3.16,
(a) Consider the case wherem=0. Make the change of variableμcosθand
consider a series solution to the equation for (μ). Derive a recurrence rela-
tion for the coefficients of the series solution.
(b) Explain why the series solution should be cut off at some finite term, give
a mechanism for doing this and hence derive another quantum numberl.
3.2.3 PotentialWellsandBarriers .........................
3.18 (a) The one-dimensional time-independent Schrodinger equation is
(
−
^2
2 m)
d^2 ψ(x)
dx^2+U(x)ψ(x)=Eψ(x)Give the meanings of the symbols in this equation.
(b) A particle of mass m is contained in a one-dimensional box of widtha.
The potential energyU(x) is infinite at the walls of the box (x =0 and
x=a) and zero in between (0<x<a).
Solve the Schrodinger equation for this particle and hence show that the
normalized solutions have the formψn(x)=( 2
a)^12
sin(nπx
a)
, with energy
En=h^2 n^2 / 8 ma^2 , wherenis an integer (n>0).
(c) For the casen=3, find the probability that the particle will be located in
the regiona 3 <x<^23 a.
(d) Sketch the wave-functions and the corresponding probability density dis-
tributions for the casesn= 1 ,2 and 3.3.19 Deuteron is a loose system of neutron and proton each of massM. Assuming
that the system can be described by a square well of depthV 0 and widthR,
show that to a good approximation
V 0 R^2 =(π
2) 2 ( 2
M
)
3.20 Show that the expectation value of the potential energy of deuteron described
by a square well of depthV 0 and widthRis given by
<V>=−V 0 A^2[
R
2
−
sin 2kR
4 k]
whereAis a constant.3.21 Assuming that the radial wave function
U(r)=rψ(r)=Cexp(−kr)
is valid for the deuteron fromr = 0tor =∞find the normalization
constantC.
Hence ifk= 0 .232 fm−^1 find the probability that the neutron – proton separa-
tion in the deuteron exceeds 2 fm. Find also the average distance of interaction
for this wave function.
[Royal Holloway University of London 1999]