The Cluster's Perturbation Theory
A page about time-independent perturbation theory — solving a difficult quantum problem by perturbing from an easy one.
Non-degenerate perturbation theory
Consider a Hamiltonian H = H_0 + lambda V, where H_0 is exactly solvable and V is a small perturbation. The energy and state corrections are: E_n = E_n^0 + lambda E_n^{(1)} + lambda^2 E_n^{(2)} + ... |n> = |n^0> + lambda |n^{(1)}> + lambda^2 |n^{(2)}> + ...
where: E_n^{(1)} = <n^0|V|n^0> E_n^{(2)} = sum_{m != n} |<m^0|V|n^0>|^2 / (E_n^0 - E_m^0) |n^{(1)}> = sum_{m != n} <m^0|V|n^0> / (E_n^0 - E_m^0) |m^0>
In the cluster, the edit perturbation theory gives edit energy corrections.
Degenerate perturbation theory
When the unperturbed state is degenerate, E_n^{(0)} = E_m^{(0)} for some m != n, the denominator diverges. The fix: diagonalize V in the degenerate subspace. If V_{mn} = <m^0|V|n^0> has eigenvalues v_i in the degenerate subspace, then: E_n^{(1)} = v_i (the eigenvalues of V in the subspace) The perturbation splits the degeneracy. In the cluster, the edit degenerate perturbation theory diagonalizes V in the degenerate subspace.
The Stark effect
The Stark effect is the splitting of atomic energy levels in an electric field E: V = -e E z. For hydrogen n = 2: the 2s and 2p states are degenerate (ignoring fine structure). The linear Stark effect gives: Delta E = +/- 3 e E a_0 where a_0 is the Bohr radius. The degeneracy is partially lifted. In the cluster, the edit Stark effect is the edit splitting in an edit electric field.
The Zeeman effect
The Zeeman effect is the splitting of atomic levels in a magnetic field B: V = -mu . B = (e / 2m) L . B. For weak fields: Delta E = mu_B B m_l g_L where mu_B = e hbar / (2m) is the Bohr magneton and g_L = 1. For strong fields (Paschen-Back effect): spin and orbital decouple.
In the cluster, the edit Zeeman effect is the edit splitting in an edit magnetic field.
Applications
- Fine structure of hydrogen (spin-orbit + relativistic corrections)
- Hyperfine structure (nuclear spin coupling)
- Stark and Zeeman effects (electric/magnetic fields)
- Lamb shift (QED correction)
- Molecular vibrations (anharmonic corrections)
In the cluster, edit applications include:
- edit Fine structure
- edit Hyperfine structure
- edit Stark and Zeeman effects
- edit Lamb shift
- edit Molecular vibrations
This theory
This page is about perturbation theory. H = H_0 + lambda V. E^{(1)} = <V>. E^{(2)} = sum |V_{nm}|^2 / (E_n - E_m). The theory is real.