Physics 4301: Homework 8. Problem 1 Consider a three-dimensional (3D) isotropic harmonic oscillator described by the Hamiltonian (no hats on operators) H=
p2 mω 2 r2 + , 2m 2
(p2 = p2x + p2y + p2z , r2 = x2 + y 2 + z 2 ).
(1)
As implied by our discussion in class, its energy spectrum is a simple equidistant ladder: the eigenvalues of the Hamiltonian (1) can be labeled as ) ( 3 En = n + ~ω, n = 0, 1, 2 . . . . (2) 2 (a) Explore the degeneracy of an arbitrary nth level in ladder (2) and find the analytical dependence d(n), that is, how many linearly independent states d correspond to the same level n. (b) When we were discussing 1D systems with symmetric potentials V (x) = V (−x), we realized that the stationary states ψ(x) can be conveniently described as either even (ψ(−x) = ψ(x)) or odd (ψ(−x) = −ψ(x)). In 3D systems with spherically-symmetric potentials V (r) = V (r), we generalize this idea to the symmetry with respect to the inversion transformation: r → −r. We define parity P = ±1 (even/odd) of state ψ(r) whenever ψ(−r) = P ψ(r). Find the parity of stationary states belonging to ladder (2). Is each energy level n “composed” of states with the same parity P ?
Problem 2 Probably still from our high-school days, we all remember the rules of reflection and refraction of light at a plane interface, say between vacuum and a medium with refractive index n: α = α′ , sin α/ sin β = n. (3) Here we denoted the incident angle as α, the reflection angle α′ and the refraction angle β. It is interesting to see if we have a similar situation for material quantum-mechanical waves. For this purpose, let us consider a particle of mass m moving in three-dimensional space with the potential energy step { 0, x < 0, U (x, y, z) = (4) U0 , x > 0. 1
The yz-plane is thus the interface separating two half-spaces. U0 can be either positive or negative. Consider a plane wave corresponding to the particle with energy E > U0 incident from x = −∞ towards the separating interface at angle α with the normal to the interface, say in the xy-plane. (a) Establish if reflection and refraction of the incident wave take place according to the “optical rules” (3). (b) If so, what quantity would be playing the role of the refractive index n? Does it depend on energy E of the particle? (c) The total internal reflection in optics occurs when there is no refracted wave, and the incident wave is fully reflected. Can we have the total internal reflection in our case, and what would be the condition for it then?
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