Assignment 7, solutions

Problem 1:

A one-dimensional potential barrier or square well problem is defined by the Hamiltonian H = (P2/2m) + U(x),> with U(x) = ζU0Θ(ℓ/2 - |x|). Here Θ(z) = 0 for z < 0 and Θ(z) = 1 for z > 0,  ζ = +1 for potential barriers and ζ = -1 for square wells.
(a)  Calculate the transmission coefficient for E = 1 eV incident electrons facing a potential barrier of U0 = 2 eV and ℓ = 1 Å.  What is the probability that a 1 eV protons will tunnel through the barrier?
(b)  Infer from (a) the general expression for the transmission coefficient T, and draw T as a function of ℓ for E = 2.25 eV electrons.
(c)  Calculate eigenfunctions and eigenvalues of the square well in the limit U0 --> ∞.

Solution:

Problem 2:

A one-dimensional potential well is given in the form of a delta function at x = 0, U(x) = Cδ(x), C < 0.  A stream of non-relativistic particles of mass m and energy E approaches the origin from one side. 
(a)  Derive an expression for the reflectance R(E).  
(b)  Can you express R(E) in terms of sin2(δ), where δ is the phase shift of the transmitted wave?

Solution:

Problem 3:

A particle of mass m moves in a one-dimensional potential given by
V(x) = -W for |x| < a,  V(x) = 0 for |x| ≥ a.
Demonstrate that this potential has at least one even bound state.

Solution:

 Alternative solution:

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.

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(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.

Solution:

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.

Solution: