EECS 428 Midterm PDF

EECS 428 Midterm Total: 30 points Name: Question 1 (9 points) For the square well potential below, one wall is an infinite potential barrier, and the other wall is finite with depth V0. a) Breaking the potential up into three regions (I, II, and II...

EECS 428 Midterm Total: 30 points Name: Question 1 (9 points) For the square well potential below, one wall is an infinite potential barrier, and the other wall is finite with depth V0. a) Breaking the potential up into three regions (I, II, and III) write down the boundary conditions for solutions (ψI (x), ψII (x) and ψIII (x)) in these regions. You are NOT asked to solve for the wavefunctions (3 points). b) Sketch the ground and first excited state wavefunctions on the potential above or next to it. Assume the energies are less than V0. (3 points). c) For a particle to be free, what must the condition on the energy, Ef , be (3 points)? 1 Question 2 (4 points) For the quantum harmonic oscillator the energies are given by En = h̄ω(n + 1/2) (n=0, 1, 2...). a) Can the measured energy be E = h̄ω? Explain (2 points). b) Can the expectation value of Ĥ, or ⟨Ĥ⟩, be h̄ω? Explain (2 points). Question 1 (17 points) In the quantum harmonic oscillator, consider a wavefunction that is a superposition of eigenstates |ψ⟩ = C [(1 + i)|0⟩ + |2⟩], or ψ(x) = C[(1 + i)ψ0 (x) + ψ2 (x)] where the oscillation frequency is ω. a) Find the normalization constant C (3 points). b) Find the expectation value of x̂, ⟨x̂⟩ (2 points). 2 c) Find the expectation value of the energy, ⟨Ĥ⟩, where Ĥ is the harmonic oscillator Hamil- tonian (4 points)? d) Find the time evolution for the state |ψ(t)⟩, or ψ(x, t)? (3 points) e) What is the probability that the wavefunction is in the ground state |0⟩ or ψ0 (x) (3 points)? f) What is the probability of measuring the energy as h̄ω/2? Hint: this is not the expectation value (2 points). 3