182
3 Quantum Mechanics – II
< E >=< ψ|H |ψ >
=< (C 1 ψ 1 + C 2 ψ 2 )|H |(C 1 ψ 1 + C 2 ψ 2 ) >
=< C 1 ψ 1 + C 2 ψ 2 |C 1 H ψ 1 + C 2 H ψ 2 | >
=< C 1 ψ 1 + C 2 ψ 2 |C 1 E 1 ψ 1 + C 2 E 2 ψ 2 | >
= C
2
1 E 1 + C
2
2 E 2
=
C
2
1 ω
2
+
C
2
2 3ω
2
=
1
2
ω(C
2
1 + 3C
2
2 )
where ω =
k
m
1/2
3.33 The ground state is
ψ =
2
a
1/2
sin(π x/a)
The wave function corresponding to momentum p is
ψ i = (2π )
−1/2
k
C k e
ikx
The probability that the particle has momentum between p and p +dp is given
by the value of
|C k |
2 , where C k is the overlap integral
C k = (2π )
−
1
2
2
a
1/2 a
0
e
ikx sin
π x
a
dx
Itegrating by parts twice,
C k = (πa)
1
2
e
ika
+ 1
π
2
− k
2 a
2
−1
The required probability is
|C k |
2
= πa
e
ika
+ 1
e
−ika
+ 1
π
2
− k
2 a
2
−2
= 4π a cos
2
ka
2
π
2
− k
2 a
2
−2
3.34 The transmission coefficient is given by
T = e
−G
(1)
G =
2
b
a
[2m(U (r ) − E)
1/2 dr
(2)
Put
U (r ) =
z Ze
2
r
(3)
for the Coulomb potential energy between the alpha particle and the residual
nucleus at distance of separation r .