In a crystal, the electric field of neighboring ions perturbs the energy levels of an atom. As a crude model, imagine that a hydrogen atom is surrounded by three pairs of point charges, as shown in Figure 7.14. (Spin is irrelevant to this problem, so ignore it.)
(a) Assuming that r << d₁, r << d₂, and r << d₃, show that

where

(b) Find the lowest-order correction to the ground state energy.
(c) Calculate the first-order corrections to the energy of the first excited states (n = 2). Into how many levels does this four-fold degenerate system split, (i) in the case of cubic symmetry, β₁ = β₂ = β₃; (ii) in the case of tetragonal symmetry, β₁ = β₂ ≠ β₃;; (iii) in the general case of orthorhombic symmetry (all three different)?

Figure 7.14: Hydrogen atom surrounded by six point charges (crude model for a crystal lattice); Problem 7.48.

Transcribed Image Text:

H' = V₁ + 3 (P₁x². + 3 (B₁x² + B2y² + B32²) - (B₁ + B₂ + B3),²²,