Assignment 7

Problem 1:

Calculate the uncertainty product <Δx><Δp> for the ground state of the infinite square-well potential.   How can we tell if the ground-state wave function is or is not a minimum uncertainty state?

Problem 2:

Consider a spinless particle of mass m moving in one dimension in the presence of a delta-function potential well, U(x) = -λ δ(x),  λ > 0.
(a)  Evaluate the transmission coefficient T(E) as a function of the incident energy E > 0.
(b)  Find the energy of the bound state for this potential well.
(c)  Comment on the pole in the expression for T(E) from part (a) in light of your result
from part (b).

Problem 3:

Use what you know about the eigenfunctions of the harmonic oscillator in coordinate space to find the momentum-space eigenfunctions of a harmonic oscillator

Problem 4:

Consider a simple, small, but macroscopic LC circuit made from conventional superconducting material and kept at a temperature below the critical temperature (~1 K).  The circuit has no resistance.

(a)  Write down a second order differential equation describing the time evolution of the magnetic flux Φ = LI in the inductor. 
(Consider this “the equation of motion” of the circuit.)  Relabel C = m, k = 1/L, and compare this equation with the equation of motion of a simple harmonic oscillator.
(b) Write down a Lagrangian for the LC circuit.  (Lagrange’s equation then is the “equation of motion” of the circuit.)  Find the generalized momentum corresponding to the generalized coordinate in the Lagrangian, and write down the Hamiltonian for the LC circuit.
(c)  Assume that the circuit is cooled down to a temperature of near zero K (~1 mK).  At such a low temperature, only the lowest allowed energy states are accessible to the system and observable quantum-mechanical effects can appear.  Quantize the system and find the ground state energy of the system,
(d)  Qualitatively reason why excited states of such a system would be unstable even at very low temperature.

Problem 5:

Consider a one-dimensional crystal with primitive lattice translation a. 
Let {|n>} be a set orthonormal electron states, n = -∞ to +∞. 
Assume that in the subspace spanned by {|n>} the matrix elements of the electron Hamiltonian are given by

<n|H|n> = E0,  <n|H| n±1> = -A, <n|H| n±2> = B,  <n|H|m> = 0 for all other m, with
E0 = (7/8)ħ2/(ma2),  A = ½ħ2/(ma2),  and B = (1/16)ħ2/(ma2). 

(a) Assume the eigenstates of H are of the form |Φ> = ∑n b(xn)|n>. 
Write down the coupled linear equations for the b(xn).  Note that b(xn±1) = b(xn ± a).
(b)  Try solutions of the form b(xn) = exp(ikxn) and show that E as a function of k is given by
E(k) = [ħ2/(ma2)][7/8 – cos(ka) + (1/8) cos(2ka)].
(c)  Determine the effective mass at the bottom of the energy band and at the top of the band from a quadratic expansion of E in the departure of k from these points.