Degeneracy & Perturbation — Degeneracy and qualitative...

SECTION 1

Historical Context & Motivation

The concept of degeneracy — where two or more quantum states share exactly the same energy — first became apparent in the early twentieth century when atomic spectra revealed unexplained fine structure. Classical physics offered no framework for understanding why certain spectral lines would split into multiplets under external fields, but the nascent quantum theory of Bohr and Sommerfeld hinted that hidden symmetries were at work. When experimentalists applied electric and magnetic fields to hydrogen, they observed that single lines fanned out into clusters of closely spaced transitions, a phenomenon that demanded a theoretical mechanism for the lifting of degeneracy. This set the stage for one of quantum mechanics' most powerful approximation methods: perturbation theory.

1896

Zeeman Effect Discovered

Pieter Zeeman observed the splitting of sodium spectral lines in a magnetic field, providing the first experimental evidence that degenerate energy levels could be resolved by an external perturbation.

1913

Bohr Model & Hydrogen Degeneracy

Niels Bohr's model predicted energy levels depending only on the principal quantum number n, leaving multiple angular-momentum states with identical energies — a hallmark of accidental degeneracy in the Coulomb potential.

1926

Schrödinger Equation & Matrix Mechanics

Erwin Schrödinger and Werner Heisenberg independently formulated quantum mechanics, providing the rigorous algebraic framework in which degeneracy and perturbation theory could be precisely defined and systematically applied.

1926–1929

Rayleigh–Schrödinger Perturbation Theory

Building on Lord Rayleigh's classical perturbation methods, Schrödinger developed the quantum perturbation expansion, distinguishing between the non-degenerate and degenerate cases and showing how to choose 'correct' zeroth-order states when degeneracy is present.

1947

Lamb Shift

Willis Lamb measured a tiny energy splitting between the 2s₁/₂ and 2p₁/₂ states of hydrogen that Dirac theory predicted to be degenerate, revealing that even vacuum fluctuations act as a perturbation that lifts degeneracy — a triumph of quantum electrodynamics.

The central question driving this lesson is deceptively simple: when a quantum system possesses states that share the same energy, how do we predict which linear combinations nature actually selects once a small perturbation is introduced? Answering this question requires understanding both the group-theoretic origins of degeneracy and the qualitative logic behind perturbation expansions — concepts that underpin everything from crystal-field theory in inorganic chemistry to the band structure of solids.

SECTION 2

Core Principles & Definitions

Before engaging with the mathematics of perturbation theory, it is essential to establish a precise vocabulary. Degeneracy refers to the situation in which a single eigenvalue of a Hamiltonian corresponds to more than one linearly independent eigenstate. The degree of degeneracy (or multiplicity) is the dimension of the eigenspace associated with that eigenvalue. In the hydrogen atom, for example, the energy E_n = −13.6 eV / n² is n²-fold degenerate when spin is neglected, because states with different ℓ and m_ℓ values all share the same energy for a given n. This arises from the SO(4) symmetry of the pure Coulomb potential, and it would be broken if the potential deviated from its 1/r form. Understanding why degeneracy exists — always rooted in a symmetry of the Hamiltonian — is the first conceptual pillar.

Degeneracy & Symmetry

Whenever a Hamiltonian commutes with a set of symmetry operators, eigenstates related by those operators share the same energy. The richer the symmetry group, the higher the degeneracy. Remove a symmetry, and the degeneracy is partially or fully lifted.

Essential vs. Accidental Degeneracy

Essential (normal) degeneracy arises from obvious geometric symmetries — e.g., rotational invariance causing m_ℓ degeneracy. Accidental degeneracy stems from hidden symmetries, such as the Runge-Lenz vector in hydrogen.

Perturbation as a Small Disturbance

A perturbation H′ is a small additional term in the Hamiltonian: H = H⁰ + λH′, where λ is a dimensionless parameter controlling strength. The unperturbed system H⁰ is exactly solvable; H′ captures the complication we cannot solve exactly.

Correct Zeroth-Order States

In a degenerate subspace, any linear combination of degenerate eigenstates is also an eigenstate of H⁰. The perturbation selects specific linear combinations — the correct zeroth-order states — that diagonalize H′ within the degenerate subspace.

Lifting of Degeneracy

When the perturbation is turned on, previously coincident energy levels may split apart. The pattern of splitting reflects the symmetry of H′ relative to H⁰, and group theory can predict how many distinct levels emerge before any calculation is performed.

✦ KEY TAKEAWAY

Think of degeneracy like a perfectly balanced spinning top on a frictionless table: it can precess in any direction with the same energy. A perturbation is like introducing a slight tilt to the table surface. The top no longer has full rotational freedom — it settles into specific preferred orientations determined by the direction and steepness of the tilt. In quantum mechanics, the 'tilt' (perturbation) selects preferred state combinations and splits their energies, revealing structure that was hidden by the original symmetry.

SECTION 3

Visual Explanation — Energy Level Splitting

An unperturbed Hamiltonian H⁰ has three energy levels. E₁ is non-degenerate and shifts only slightly under the perturbation. E₂ is three-fold degenerate; the perturbation splits it into three distinct levels E₂ₐ, E₂ᵦ, and E₂꜀. E₃ is two-fold degenerate and splits into two levels. The shaded region indicates the regime where the perturbation strength λ increases from left to right.

The diagram above captures the central narrative of this lesson. On the left, the unperturbed Hamiltonian H⁰ yields energy levels with well-defined degeneracies indicated by the colored dots on each line. A non-degenerate level such as E₁ merely shifts its energy when the perturbation is introduced — this is the domain of standard (non-degenerate) perturbation theory. In contrast, the three-fold degenerate level E₂ fans out into three distinct levels as the perturbation λH′ breaks the symmetry responsible for the original degeneracy. The pattern of splitting — which levels move up, which move down, and whether any remain degenerate — is governed by the matrix elements of H′ within the degenerate subspace. Predicting this pattern is the central computational task of degenerate perturbation theory, and understanding it qualitatively before performing calculations is the goal of the present discussion.

SECTION 4

Mathematical Framework

The formal starting point is the decomposition of the full Hamiltonian into an exactly solvable piece and a perturbation. The solutions of H⁰ are assumed known, and our task is to determine how these solutions change when H′ is switched on. The analysis proceeds differently depending on whether the eigenvalue of interest is degenerate or not.

PERTURBED HAMILTONIAN

H = H⁰ + λH′

H⁰ = unperturbed (solvable) Hamiltonian; H′ = perturbation operator; λ = dimensionless coupling parameter (0 ≤ λ ≤ 1). At λ = 0 we recover the unperturbed system; at λ = 1 we have the full problem.

UNPERTURBED EIGENVALUE EQUATION

H⁰ |ψₙ⁽⁰⁾⟩ = Eₙ⁽⁰⁾ |ψₙ⁽⁰⁾⟩

If E_n⁽⁰⁾ has g_n linearly independent eigenstates |ψₙ,₁⁽⁰⁾⟩, … , |ψₙ,gₙ⁽⁰⁾⟩, the level is gₙ-fold degenerate.

FIRST-ORDER ENERGY CORRECTION (NON-DEGENERATE)

Eₙ⁽¹⁾ = ⟨ψₙ⁽⁰⁾| H′ |ψₙ⁽⁰⁾⟩

For a non-degenerate state, the first-order energy shift is simply the expectation value of the perturbation in the unperturbed state. This formula fails when the level is degenerate because any linear combination of degenerate states is equally valid as the zeroth-order state.

DEGENERATE PERTURBATION — SECULAR EQUATION

det | Wᵢⱼ − E⁽¹⁾ δᵢⱼ | = 0, where Wᵢⱼ = ⟨ψₙ,ᵢ⁽⁰⁾| H′ |ψₙ,ⱼ⁽⁰⁾⟩

The g_n × g_n matrix W is formed from the matrix elements of H′ restricted to the degenerate subspace. Its eigenvalues give the first-order energy corrections E⁽¹⁾, and its eigenvectors give the correct zeroth-order states — the linear combinations that diagonalize the perturbation within the subspace.

The secular equation is the heart of degenerate perturbation theory at first order. Qualitatively, its message is straightforward: among all possible ways to write zeroth-order states, nature picks the linear combinations that diagonalize the perturbation within the degenerate subspace. Once W is diagonal, each diagonal element becomes the first-order energy correction for the corresponding state. If some eigenvalues of W coincide, a residual degeneracy persists, and higher-order perturbation theory or additional perturbations may be needed to resolve it.

⚠️ Why Non-Degenerate PT Breaks Down

In non-degenerate perturbation theory, the first-order correction to the wavefunction contains terms proportional to 1/(E_n⁽⁰⁾ − E_m⁽⁰⁾). When E_n⁽⁰⁾ = E_m⁽⁰⁾ (degeneracy), these terms diverge, rendering the expansion meaningless. Degenerate perturbation theory resolves this by pre-diagonalizing H′ in the degenerate subspace, eliminating the offending denominators.

SECTION 5

Types of Degeneracy & Their Physical Origins

Not all degeneracies arise from the same physical mechanism, and recognizing their origin is essential for predicting how perturbations will affect them. We can broadly classify degeneracies into several categories, each connected to a different symmetry property of the Hamiltonian. The table below summarizes the principal types encountered in atomic, molecular, and solid-state quantum mechanics, while the diagram that follows illustrates how each type manifests in the hydrogen atom.

Classification of quantum degeneracies by their symmetry origins

Type	Physical Origin	Example	Perturbation That Lifts It
Essential (mℓ)	Rotational symmetry: [H, Lz] = 0 makes energy independent of mℓ	2p states with mℓ = −1, 0, +1 all at same energy	External magnetic field (Zeeman effect)
Accidental (ℓ)	Hidden SO(4) symmetry of the 1/r Coulomb potential	2s and 2p at the same energy in hydrogen	Deviation from pure 1/r potential; screening in multi-electron atoms
Spin	Spin-independent Hamiltonian: [H, S²] = [H, Sz] = 0	Each orbital state is (2s+1)-fold degenerate in ms	Spin-orbit coupling; external magnetic field
Exchange / Permutation	Indistinguishability of identical particles	Singlet and triplet states in helium with same spatial wavefunction	Electron-electron repulsion distinguishes spatial symmetry
Kramers	Time-reversal symmetry for half-integer spin systems	Every energy level of an odd-electron system is at least 2-fold degenerate	Magnetic field (breaks time-reversal symmetry)

The hydrogen n = 1 and n = 2 levels are shown under successive perturbations. Starting from the purely Coulombic (Bohr) picture where all n = 2 states are degenerate, relativistic corrections break the ℓ degeneracy, spin-orbit coupling further splits states by j, and an external magnetic field resolves the remaining m_j degeneracy completely.

This cascade of splittings illustrates a general principle: each perturbation removes a specific symmetry from the Hamiltonian, and the degeneracy associated with that symmetry is lifted. The order in which perturbations are applied matters when their relative magnitudes differ, a consideration that becomes important in choosing the appropriate coupling scheme (LS versus jj coupling) for multi-electron atoms.

SECTION 6

Worked Example — 2D Harmonic Oscillator with Anisotropic Perturbation

Consider a two-dimensional isotropic harmonic oscillator with the unperturbed Hamiltonian H⁰ = ½ mω²(x² + y²) + (p_x² + p_y²)/(2m). The energy levels are E_n = (n + 1)ℏω, where n = n_x + n_y. The first excited level (n = 1) is two-fold degenerate, with states |1,0⟩ and |0,1⟩. Suppose we introduce a perturbation H′ = εmω²xy, which couples the x and y motions. We wish to find the first-order energy corrections and the correct zeroth-order states.

First-Order Degenerate Perturbation Theory for the n = 1 Level

Step 1 — Identify the Degenerate SubspaceThe n = 1 level has energy E₁⁽⁰⁾ = 2ℏω and is spanned by two states: |1,0⟩ (one quantum in x, zero in y) and |0,1⟩ (zero in x, one in y). These two states form the basis for the 2×2 perturbation matrix W.

Step 2 — Compute the W Matrix ElementsWe need the four matrix elements Wᵢⱼ = ⟨nₓ, nᵧ| εmω²xy |nₓ′, nᵧ′⟩. Using the raising/lowering operator representation x = √(ℏ/2mω)(a_x† + a_x) and y = √(ℏ/2mω)(a_y† + a_y), the product xy connects states differing by ±1 in both nₓ and nᵧ. The diagonal elements W₁₁ = ⟨1,0|H′|1,0⟩ and W₂₂ = ⟨0,1|H′|0,1⟩ both vanish because the operator xy changes both quantum numbers. The off-diagonal elements give W₁₂ = W₂₁ = εℏω/2.

W = εℏω/2 × [[0, 1], [1, 0]]

Step 3 — Solve the Secular EquationThe secular equation det|W − E⁽¹⁾I| = 0 becomes (−E⁽¹⁾)² − (εℏω/2)² = 0, yielding E⁽¹⁾ = ±εℏω/2.

E₊ = 2ℏω + εℏω/2; E₋ = 2ℏω − εℏω/2

Step 4 — Find the Correct Zeroth-Order StatesThe eigenvectors of W are |+⟩ = (|1,0⟩ + |0,1⟩)/√2 and |−⟩ = (|1,0⟩ − |0,1⟩)/√2. These are the 'correct' zeroth-order states — the linear combinations that diagonalize H′ within the degenerate subspace. Note that |+⟩ is symmetric under exchange of x and y, while |−⟩ is antisymmetric, reflecting the fact that H′ = εmω²xy is symmetric under simultaneous interchange of x and y.

|+⟩ = (|1,0⟩ + |0,1⟩)/√2 → E = 2ℏω + εℏω/2; |−⟩ = (|1,0⟩ − |0,1⟩)/√2 → E = 2ℏω − εℏω/2

Step 5 — Interpret PhysicallyThe perturbation H′ = εmω²xy couples the two degenerate modes. In the |+⟩ state, the particle's probability density is elongated along the line y = x (the perturbation adds energy because both coordinates reinforce). In |−⟩, the density aligns along y = −x (the perturbation subtracts energy). The originally circular symmetry of the isotropic oscillator is broken, and the two states now oscillate along the principal axes of the total potential, rotated 45° from the original x and y axes. The degeneracy is completely lifted at first order.

SECTION 7

Strengths & Limitations of Perturbation Theory

Assessment of perturbation theory as an approximation tool

Aspect	Strength	Limitation
Applicability	Works whenever H′ is 'small' compared to the level spacing of H⁰; vast majority of real quantum systems can be framed this way	Fails when the perturbation is comparable to or larger than the unperturbed level spacing — the expansion diverges or converges too slowly
Physical insight	Provides term-by-term corrections with clear physical meaning: E⁽¹⁾ is the average of the perturbation, E⁽²⁾ captures virtual transitions to other states	Higher-order terms become algebraically complex and can obscure the physics they describe
Degeneracy handling	Degenerate perturbation theory correctly identifies the 'right' basis and predicts splitting patterns; symmetry arguments (group theory) further simplify the W matrix	When the perturbation does not fully lift degeneracy at first order, one must go to higher orders or use quasi-degenerate methods, adding considerable complexity
Convergence	Low-order corrections (first and second) are often remarkably accurate for atoms and molecules in weak external fields	The perturbation series is typically asymptotic, not convergent — summing too many terms can worsen the result
Alternatives	Perturbation theory is analytically tractable; variational methods complement it for ground states	For strong perturbations, variational, numerical diagonalization, or non-perturbative methods (WKB, instantons) may be required

✦ KEY TAKEAWAY

Perturbation theory is analogous to a Taylor series expansion for a function you cannot evaluate in closed form: the first few terms give you excellent local approximations, but the expansion may diverge far from the expansion point. Just as a Taylor series requires the function to be smooth, perturbation theory requires the perturbation to be 'small' in a well-defined operator-norm sense relative to the energy gaps of the unperturbed system. When degeneracy is present, the analogy extends to multivariable Taylor series where you must first rotate to the correct coordinate system (the eigenbasis of W) before the expansion makes sense.

SECTION 8

Connection to Advanced Theory

The introductory treatment of degeneracy and perturbation presented here is the entry point to a rich hierarchy of approximation methods. As you progress through physical chemistry and quantum mechanics, you will encounter increasingly sophisticated extensions that handle the complications arising when degeneracy is not fully lifted, when the perturbation is time-dependent, or when the perturbation series itself fails to converge.

Roadmap from introductory to advanced perturbation methods

This Lesson Covers	Advanced Extension
First-order degenerate perturbation theory (diagonalizing W within a degenerate subspace)	Higher-order degenerate PT: second-order corrections require an 'effective Hamiltonian' approach within the degenerate subspace, coupling to remote levels via Löwdin partitioning
Time-independent perturbation of energy levels	Time-dependent perturbation theory (Fermi's golden rule, transition rates, spectroscopy selection rules)
Qualitative prediction of splitting patterns	Group-theoretic selection rules: using irreducible representations to block-diagonalize W and predict which matrix elements vanish by symmetry
Small perturbations with clear unperturbed reference	Variational perturbation theory, coupled-cluster methods, and many-body perturbation theory (Møller–Plesset) for electron correlation in molecules
Exact degeneracy at zeroth order	Near-degeneracy (quasi-degenerate PT): when energy levels are not exactly degenerate but closely spaced, requiring a hybrid approach that treats the near-degenerate set as a subspace

One of the most impactful applications you will see shortly is crystal-field theory, where the perturbation is the electrostatic field of surrounding ligands acting on a transition-metal ion. The free-ion d-orbital degeneracy (five-fold) is partially lifted according to the point-group symmetry of the ligand environment — an octahedral field splits d orbitals into t₂g and eg sets, while a tetrahedral field produces a different splitting pattern. This is degenerate perturbation theory applied to a chemical problem, and the qualitative reasoning you develop here — which symmetries are preserved, which are broken, and how many distinct energy levels result — transfers directly to those applications.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain why the standard (non-degenerate) first-order wavefunction correction formula, |ψₙ⁽¹⁾⟩ = Σ_{m≠n} ⟨ψₘ⁽⁰⁾|H′|ψₙ⁽⁰⁾⟩/(Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾) |ψₘ⁽⁰⁾⟩, breaks down when the state |ψₙ⁽⁰⁾⟩ is degenerate with one or more other states. What is the mathematical symptom, and what is the physical interpretation?

PROBLEM 2 — BASIC CALCULATION

A three-fold degenerate energy level has the perturbation matrix W (in units of α) given by: W = [[2, 0, 0], [0, −1, 1], [0, 1, −1]]. Find the three first-order energy corrections E⁽¹⁾ and determine how many distinct energy levels emerge.

PROBLEM 3 — INTERMEDIATE

For the 2D isotropic harmonic oscillator, the n = 2 level (E = 3ℏω) is three-fold degenerate, spanned by |2,0⟩, |1,1⟩, and |0,2⟩. If the perturbation is H′ = εmω²xy, construct the 3×3 perturbation matrix W and find the first-order energy corrections. Does the perturbation completely lift the degeneracy?

PROBLEM 4 — APPLIED

In a transition-metal complex, the five d orbitals of the free ion are degenerate. When the ion is placed in an octahedral ligand field (point group O_h), the d orbitals split into two sets: t₂g (three-fold) and eg (two-fold). Using the qualitative reasoning from this lesson, explain why the five-fold degeneracy is only partially lifted to two groups rather than five distinct levels, and identify what symmetry operation still produces degeneracy within each group.

PROBLEM 5 — CRITICAL THINKING

Consider a system with a two-fold degenerate level where the perturbation matrix W = [[a, b], [b*, a]] with a real and b complex. Show that the splitting depends on |b| but not on the phase of b. Discuss what this implies about the sensitivity of energy level splittings to the choice of basis within the degenerate subspace, and relate your finding to the concept of gauge freedom in quantum mechanics.

SUMMARY

Summary

Degeneracy occurs whenever a Hamiltonian's symmetry causes multiple linearly independent eigenstates to share the same energy eigenvalue. The degree of degeneracy equals the dimension of the corresponding eigenspace, and it can be traced to specific symmetry operations that commute with the Hamiltonian — rotational invariance produces m_ℓ degeneracy, the hidden SO(4) symmetry of the Coulomb potential produces the accidental ℓ degeneracy of hydrogen, and spin-independent Hamiltonians produce spin degeneracy. A perturbation H′ that breaks one or more of these symmetries lifts the associated degeneracy, splitting a single energy level into multiple distinct levels whose number and pattern reflect the residual symmetry of H⁰ + H′.

The core mathematical procedure involves constructing the perturbation matrix W from matrix elements of H′ restricted to the degenerate subspace, then solving the secular equation to obtain first-order energy corrections and the correct zeroth-order states. This procedure is necessary because non-degenerate perturbation theory produces divergent denominators when applied to degenerate levels. Qualitatively, the key insight is that degeneracy implies freedom in choosing basis states, and the perturbation selects the physically meaningful combinations — the ones that diagonalize H′ and therefore evolve smoothly as the perturbation is turned on. This framework underpins applications from atomic fine structure and the Zeeman effect to crystal-field theory and band structure in solids.