PHY3708 Assignment 1 solutions 2026

PHY3708 Assignment 1 solutions 2026
UNISA
ATOMIC AND NUCLEAR PHYSICS

,Question 1

The wave function is
0, 𝑥 < −𝑎
1 3𝜋𝑥
𝜙(𝑥) = {𝐶 [ + cos⁡ ( )] , −𝑎 ≤ 𝑥 ≤ 𝑎
𝑎 𝑎
0, 𝑥>𝑎


We normalize the wave function using
∞
∫ 𝜙(𝑥) 𝜙 ∗ (𝑥) 𝑑𝑥 = 1
−∞



Since the function is real,

𝜙 ∗ (𝑥) = 𝜙(𝑥)


Thus,
𝑎
1 3𝜋𝑥 2
∫ 𝐶 2 [ + cos⁡ ( )] 𝑑𝑥 = 1
−𝑎 𝑎 𝑎


Expand the square:

1 3𝜋𝑥 2 1 2 3𝜋𝑥 3𝜋𝑥
[ + cos⁡ ( )] = 2 + cos⁡ ( ) + cos⁡2 ( )
𝑎 𝑎 𝑎 𝑎 𝑎 𝑎


Therefore,
𝑎
2
1 2 3𝜋𝑥 3𝜋𝑥
𝐶 ∫ [ 2
+ cos⁡ ( ) + cos⁡2 ( )] 𝑑𝑥 = 1
−𝑎 𝑎 𝑎 𝑎 𝑎


Now evaluate each integral.



First integral
𝑎
1 1 2
∫ 2
𝑑𝑥 = 2 (2𝑎) =
−𝑎 𝑎 𝑎 𝑎


Second integral

, 2 𝑎 3𝜋𝑥
∫ cos⁡ ( ) 𝑑𝑥
𝑎 −𝑎 𝑎
2 𝑎 3𝜋𝑥 𝑎
= [3𝜋 sin⁡ ( 𝑎 )]
𝑎 −𝑎
2 𝑎
= ⋅ [sin⁡(3𝜋) − sin⁡(−3𝜋)] = 0
𝑎 3𝜋


because

sin⁡(3𝜋) = 0




Third integral

Use
1
cos⁡2 𝜃 = (1 + cos⁡ 2𝜃)
2


Hence,
𝑎
3𝜋𝑥
2
1 𝑎 6𝜋𝑥
∫ cos⁡ ( ) 𝑑𝑥 = ∫ [1 + cos⁡ ( )] 𝑑𝑥
−𝑎 𝑎 2 −𝑎 𝑎
1
= [2𝑎+0] = 𝑎
2




Substitute all results:
2
𝐶 2 (𝑎 +𝑎) = 1


Thus,
1
𝐶2 =
2
𝑎+𝑎
𝑎
𝐶2 = 2
𝑎 +2
Therefore,

𝑎
𝐶=√
𝑎2 +2