The exchange operator is defined as chipsi1psi2 psi2psi1

Question

Problem -

Rydberg atoms are atoms in states with n > > 1. Their energy levels, even for complex atoms, obey a hydrogenlike formula E_nl = -1/2(n- δ_l)² in atomic units. Our (fictional) atom has δ_s = 3.3, δ_p = 2.8, δ_d = 1.5, δ_l>2 = 0. We also assume that the dipole matrix elements obey the approximate relation

We are interested in the energy of attraction or repulsion of two 97s atoms due to the dipole-dipole interaction. We will therefore focus on states close in energy to the 197s; 97 s) states. Fine-structure effects are negligible.

a) Identify the states that are of primary importance for dipole-dipole interactions between two 97s atoms. Will the dipole-dipole interaction be attractive or repulsive for 97s+97s states?

b) There are at least 18 possible states in your list. Use [H, L_1z+ L_2z] = 0 to reduce your list.

c) The exchange operator is defined as χ|ψ₁ψ₂) = |ψ₂ψ₁). Show that {H, χ] = 0, find the eigen-states and eigenvalues of χ and use that to further reduce your list of states.

d) You should now be down to just 4 states. Calculate the matrix elements and find the energy shifts in terms of the relevant unperturbed energy difference ΔE and U = e²ρ²/R⁵.

e) Plot your results and identify under what conditions the 97s97s energy shift can be written as C₅/R₅ or C₆/R₆. Give numerical values for C₃ and C₆.