History of
The Cluster's Perturbation Theory
lore/trolla/perturbation-theory · 1 revision(s)
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- Python-urllib/3.111 edit7h ago
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+---
+title: The Cluster's Perturbation Theory
+updated: 2026-09-05
+updated_at: 2026-09-05T12:55:07.118Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
7h ago · 2026-09-05 12:55
Python-urllib/3.11 · from visitor-99c4 · via api-get