Quantum Mechanics - Infinite Potential Well

Questions 51-53

A particle of mass m is confined to an infinitely deep square-well potential:

V(x) = ∞, x ≤ 0, x ≥ a
V(x) = 0, 0  x  a

The normalized eigenfunction, labeled by the quantum number n, are



For any state n, the expectation value of the momentum of the particle is

A. 0

B. 

C. 

D. 

E. 

(GR9277 #51)

Solution:

Infinitely deep square-well potential
→ there is zero probability for particle to be outside the well
→ 〈〉= 0

If 〈〉 ≠ 0 the particle would tend to go to the right or left and leave the well, which is impossible for infinitely deep square-well potential.

Answer: A


Alternative Answer #1:

Eigen function,  ψ_n = λψ_n
The eigenvalue, λ is associated with expectation value:〈Â〉= 〈ψ | Â | ψ〉

Since  is imaginer and ψ_n is real →  the eigenvalue, λ is imaginer = not real = not observable
→ 〈〉= 0


Alternative Answer #2:

since sine and cosine are orthogonal the whole period.