Assignment 10

Problem 1:

The one-dimensional square well shown in the figure rises to infinity at x = 0 and has range a and depth U1.

image

(a)  Derive the condition for a spinless particle of mass m to have two and only two bound states in the well.
(b)  Sketch the wave function of these two states inside and outside the well and give their analytic expressions.  These expressions may involve undetermined constants.

Solution:

Problem 2:

You are given a one dimensional potential barrier of height U which extends from x = 0 to x = a.  A particle of mass m and energy E < U is incident from the left.
(a)  Derive expressions for the transmittance T and reflectance R for the barrier.
(b)  Show that T + R =  1.

Solution:

Problem 3:

The wave function ψ(r) of a spinless particle is ψ(r) = Nz2exp(-r2/b2), where b is a real constant and N is a normalization constant.
(a)  If L2 is measured, what results can be obtained and with what probabilities?
(b)  If Lz is measured, what results can be obtained and with what probabilities?
(c)  Is ψ(r) an eigenfunction of L2 or Lz?

Solution:

Problem 4:

A particle moving in one dimension is confined to an infinite square well of width L.  At t = 0 its wave function is piecewise linear and symmetric about L/2 with a maximum height of A, as shown.
(a)  Normalize the wave function and find A.
(b)  Write down the normalized eigenfunctions in this potential.
(c)  Expand ψ(0) in terms of the eigenfunctions of part (b).
(d)  Write an expression for ψ(x,t).  How much time T must elapse in order for ψ(x,T) = ψ(x,0)?
(e)  Write down an expression for <E>, the expectation value of the particles energy in this state.

image
x sin(x)dx = sin(x) - x cos(x)
n=0, odd1/n2 = π2/8

Solution:

Problem 5:

Assume that the Hamiltonian for a spin ½ particle in a magnetic field B is given by H = g σB, where σ = (σx, σy, σz) is a vector for Pauli matrices and g is a positive real constant with the appropriate units.
The magnetic field can be written as B = B0(sinθ cosφ, sinθ sinφ, cosθ), with 0 ≤ θ < π,  0 ≤ φ < 2π.
(a)  Find eigenstates and the corresponding eigenvalues of H.

(b)  Let us denote the eigenstate with a negative eigenvalue by |->n.
Calculate A = (Aθ, Aφ), where Aα = in<-|∂/∂α|->n, for α = θ, φ.
Calculate Ωθ,φ = ∂Aφ/∂θ - ∂Aθ/∂φ.

(c)  Are A and Ωθ,φ invariant under a local gauge transformation |->n --> e|->n?

Solution: