Question

A particle in a strange potential well has the following two lowest-energy stationary states:

ψ1(x) = C1 sin^2 (πx) for − 1 ≤ x ≤ 1

ψ2(x) = C2 sin^2 (πx) for − 1 ≤ x ≤ 0

ψ2(x) = −C2 sin^2 (πx) for 0 ≤ x ≤ 1

The energy of ψ1 state is ¯hω. The energy of ψ2 state is 2¯hω. The particle at t = 0 is in a superposition state Ψ(x, t = 0) = 1/√ 2 [ψ1(x) + ψ2(x)].

1) Find the real positive coefficients C1 and C2 to normalize the wavefunctions ψ1(x) and ψ2(x). Show ψ1(x) and ψ2(x) are orthonormal.

2) What is the time-dependent wavefunction Ψ(x, t) of the particle? Draw the wavefunction at t = π/ω and t = 2π/ω .

3) Calculate the probability distribution of the particle as a function of time. Express it as a real function, and comment on its time dependence.

4) Calculate (x)(t).

5) If a measurement is made at t = 2π/13ω to probe the energy of the particle, what values might you get, and what is the probability of getting each of them?

Answer 1