140
3 Quantum Mechanics – II
3.17 In Problem 3.16,
(a) Consider the case where m = 0. Make the change of variable μcos θ and
consider a series solution to the equation for (μ). Derive a recurrence relation 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 number l.
3.2.3 Potential Wells and Barriers
3.18 (a) The one-dimensional time-independent Schrodinger equation is
−
2
2m
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 width a.
The potential energy U (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 form ψ n (x) =
2
a
1
2 sin
nπ x
a
, with energy
E n = h
2 n
2
/8ma
2 , where n is an integer (n > 0).
(c) For the case n = 3, find the probability that the particle will be located in
the region
a
3
< x <
2a
3
.
(d) Sketch the wave-functions and the corresponding probability density distributions for the cases n = 1, 2 and 3.
3.19 Deuteron is a loose system of neutron and proton each of mass M. Assuming
that the system can be described by a square well of depth V 0 and width R,
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 depth V 0 and width R is given by
< V >= −V 0 A
2
R
2
−
sin 2k R
4k
where A is a constant.
3.21 Assuming that the radial wave function
U (r ) = r ψ(r ) = C exp(−kr)
is valid for the deuteron from r = 0 to r = ∞ find the normalization
constant C.
Hence if k = 0.232 fm
−1 find the probability that the neutron – proton separation in the deuteron exceeds 2 fm. Find also the average distance of interaction
for this wave function.
[Royal Holloway University of London 1999]
3 Quantum Mechanics – II
3.17 In Problem 3.16,
(a) Consider the case where m = 0. Make the change of variable μcos θ and
consider a series solution to the equation for (μ). Derive a recurrence relation 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 number l.
3.2.3 Potential Wells and Barriers
3.18 (a) The one-dimensional time-independent Schrodinger equation is
−
2
2m
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 width a.
The potential energy U (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 form ψ n (x) =
2
a
1
2 sin
nπ x
a
, with energy
E n = h
2 n
2
/8ma
2 , where n is an integer (n > 0).
(c) For the case n = 3, find the probability that the particle will be located in
the region
a
3
< x <
2a
3
.
(d) Sketch the wave-functions and the corresponding probability density distributions for the cases n = 1, 2 and 3.
3.19 Deuteron is a loose system of neutron and proton each of mass M. Assuming
that the system can be described by a square well of depth V 0 and width R,
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 depth V 0 and width R is given by
< V >= −V 0 A
2
R
2
−
sin 2k R
4k
where A is a constant.
3.21 Assuming that the radial wave function
U (r ) = r ψ(r ) = C exp(−kr)
is valid for the deuteron from r = 0 to r = ∞ find the normalization
constant C.
Hence if k = 0.232 fm
−1 find the probability that the neutron – proton separation in the deuteron exceeds 2 fm. Find also the average distance of interaction
for this wave function.
[Royal Holloway University of London 1999]