PHY 6646 - Quantum Mechanics II - Spring 2011
Homework set # 9, due March 16
1. Screened Potential: Consider a particle of mass μ in the central force potential
e2
r
e2
exp (−λ(r − R))
= −
r
V (r) = −
for 0 < r < R
for R < r < ∞ .
(0.1)
This potential differs from the Coulomb potential only in the region r > R , where the
Coulomb force is screened. The difference becomes negligible when λ → 0. Consider this
difference as a perturbation and calculate the first-order correction to the energy of the
ground state of the hydrogen atom.
2. Relativistic Harmonic Oscillator: A particle of mass μ moves in a one-dimensional
harmonic oscillator potential V (x) = 12 μω 2x2 . Allowing for relativistic effects, the kinetic
√
4
p2
− 8µp3 c2 . Treating the p4 term as a
energy is T = E − μc2 = μ2 c4 + p2 c2 − μc2 2µ
perturbation, calculate the first order shift in the ground state energy.
4. Problem 17.3.2 in Shankar’s book.
5. Problem 17.3.4 in Shankar’s book.
1