# 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.