170
3 Quantum Mechanics – II
a
0
ψ
∗
n (x)ψ n (x)dx = 1
A
2
α
0
sin
2
nπ x
a
dx = 1
A
2
2
x − cos
2nπ x
a
a
0
= A
2 a = 1
Therefore, A =
2
a
1/2
(9)
The normalized wave function is
ψ n (x) =
2
a
1
2
sin
nπ x
a
(10)
Using the value of α from (7) in (3), the energy is
E n =
n
2 h
2
8ma 2
(11)
(c) probability p =
a
0 |ψ 3 (x)|
2 dx
=
2a
3
a
3
2
a
sin
2
3π x
a
dx =
1
3
(d) ψ(n) and probability density P(x) distributions for n = 1, 2 and 3 are
sketched in Fig 3.6
Fig. 3.6
3.19 The Schrodinger equation for the n – p system in the CMS is
∇
2
ψ(r, θ, ϕ) +
2μ
2
[E − V (r )]ψ(r, θ, ϕ) = 0
( 1 )
3 Quantum Mechanics – II
a
0
ψ
∗
n (x)ψ n (x)dx = 1
A
2
α
0
sin
2
nπ x
a
dx = 1
A
2
2
x − cos
2nπ x
a
a
0
= A
2 a = 1
Therefore, A =
2
a
1/2
(9)
The normalized wave function is
ψ n (x) =
2
a
1
2
sin
nπ x
a
(10)
Using the value of α from (7) in (3), the energy is
E n =
n
2 h
2
8ma 2
(11)
(c) probability p =
a
0 |ψ 3 (x)|
2 dx
=
2a
3
a
3
2
a
sin
2
3π x
a
dx =
1
3
(d) ψ(n) and probability density P(x) distributions for n = 1, 2 and 3 are
sketched in Fig 3.6
Fig. 3.6
3.19 The Schrodinger equation for the n – p system in the CMS is
∇
2
ψ(r, θ, ϕ) +
2μ
2
[E − V (r )]ψ(r, θ, ϕ) = 0
( 1 )