PH424
Waves
Winter 2016
Worksheet #4
(Tuesday 02/16/16)
1. The particle is moving in the following potential:
∞, 𝑥 ≤ 0
0, 0 < 𝑥 < 𝑎
𝑉(𝑥) = {
𝑉1 , 𝑎 < 𝑥 < 𝑏
𝑉2 , 𝑏 < 𝑥 < ∞
Here V1 and V2 are positive quantities and V1>V2.
(a) (2 pts) Sketch the potential V(x)
(b) (8 pts) Write down the equation you need to solve to determine 𝜓(𝑥)and a functional
form for the wave function 𝜓(𝑥) in each region for an electron with energies E = E1,
E2, and E3, where 0<E1<V2, V2<E2<V1, and E3>V1. Give as much information as you
can in each case.
Page 1 of 3
(c) (6 pts) Sketch wave functions in each region in the case of E = E2 and E= E3. Explain.
1
1
2. The particle in an infinite square well is initially in the state |𝜓(0)⟩ = |𝜑1 ⟩ + |𝜑2 ⟩,
√2
√2
where |𝜑𝑛 ⟩ are eigenstates of the Hamiltonian.
(a) (2 pts) Write down the time-evolved state|𝜓(𝑡)⟩
(b) (2 pts) Write down the wave function representation of the time-evolved state 𝜓(𝑥, 𝑡)
(c) (2 pts) If energy is measured, what values are obtained and with what probabilities? Do
the measurement outcomes depend on time?
(d) (2 pts) What is the expectation value of the energy? Does it depend on time?
Page 2 of 3
(e) (6 pts) Find the expectation value of momentum <p> (t). You may find the following
integrals helpful:
𝐿
∫ 𝜑𝑛 ∗
0
𝐿
∫ 𝜑1 ∗
0
𝑑𝜑𝑛
𝑑𝑥 = 0
𝑑𝑥
𝑑𝜑2
𝑑𝑥 = −8/3𝐿
𝑑𝑥
Bonus (3 pts): Without solving integrals given above, rationalize why the first one is
zero and the second one is not.
Page 3 of 3