Modern Physics · Upper Division

Perturbation Theory

Most quantum systems cannot be solved exactly. Perturbation theory gives corrections to known solutions when the Hamiltonian is close to a solvable one — and time-dependent perturbation theory gives transition rates for interactions with radiation.

PrerequisitesQuantum mechanics (Ch. 20) \cdot Spin & angular momentum (Ch. SP) \cdot Linear algebra
Learning Goals
  • Compute first- and second-order energy corrections using time-independent perturbation theory.
  • Diagonalize the perturbation within a degenerate subspace to find the "good" zero-order states.
  • AnalyzethelinearStarkeffectinhydrogenn=2usingdegenerateperturbationtheoryAnalyze the linear Stark effect in hydrogen n=2 using degenerate perturbation theory.
  • Apply Fermi's Golden Rule to calculate transition rates from an initial state to a continuum.
  • Estimate the helium ground-state energy using the variational method with a trial wavefunction.

PT.1 Time-Independent Perturbation Theory

Split the Hamiltonian H = H₀ + λH', where H₀ is solvable (eigenstates |n⟩, energies Eₙ⁰) and λH' is a small perturbation. Expand energies and states in powers of λ:

En=En0+\lambdaEn1+λ2En2+⋯∣n⟩=∣n0⟩+λ∣n1⟩+⋯E_{n} = E_{n}^{0} + \lambdaE_{n}^{1} + \lambda^{2}E_{n}^{2} + \cdots \qquad |n⟩ = |n^{0}⟩ + \lambda|n^{1}⟩ + \cdots(PT.1)

Substituting into H|n⟩ = Eₙ|n⟩ and collecting powers of λ:

En1=⟨n0∣H′∣n0⟩(first−orderenergycorrection)E_{n}^{1} = ⟨n^{0}|H'|n^{0}⟩ \qquad (first-order energy correction)(PT.2)
En2=∑(m≠n)∣⟨m0∣H′∣n0⟩∣2/(En0−Em0)(second−order)E_{n}^{2} = \sum(m≠n) |⟨m^{0}|H'|n^{0}⟩|^{2} / (E_{n}^{0} - Em^{0}) \qquad (second-order)(PT.3)

The second-order energy correction is always negative for the ground state (E₀² < 0), since the ground state energy Eₙ⁰ is the lowest and all denominator terms are negative. This means perturbations always lower the ground state energy (or leave it unchanged).

Example PT.1 — Anharmonic Oscillator

H=p2/2m+12mω2x2+\lambdax4(harmonicoscillatorwithquarticperturbation).Findthefirst−oH = p^{2}/2m + \frac{1}{2}m\omega^{2}x^{2} + \lambdax^{4} (harmonic oscillator with quartic perturbation). Find the first-order energy correction for the nth level.

Matrix element:En1=λ⟨n∣x4∣n⟩.Expressx=(ℏ/2mω)1/2(a+a†E_{n}^{1} = \lambda⟨n|x^{4}|n⟩. Express x = (\hbar/2m\omega)^{1/2}(a + a^\dagger.
x⁴:x4=(ℏ/2mω)2(a+a†)4.Expandandkeepnormal−orderedtermsx^{4} = (\hbar/2m\omega)^{2} (a+a^\dagger)^{4}. Expand and keep normal-ordered terms.
⟨n|x⁴|n⟩:Onlytermswithequalnumbersofaanda†contribute:⟨n∣x4∣n⟩=(ℏ/2mω)2(6n2+6n+3Only terms with equal numbers of a and a^\dagger contribute: ⟨n|x^{4}|n⟩ = (\hbar/2m\omega)^{2}(6n^{2}+6n+3.
Energy:En=ℏω(n+12)+λ(ℏ/2mω)2(6n2+6n+3E_{n} = \hbar\omega(n+\frac{1}{2}) + \lambda(\hbar/2m\omega)^{2}(6n^{2}+6n+3
Physical content:Theanharmoniccorrectiongrowsasn2—higherlevelsareshiftedmore.ThisexplainswhyThe anharmonic correction grows as n^{2} — higher levels are shifted more. This explains why real molecular vibrations (Morse potential) have levels that get closer together at high n.

PT.2 Degenerate Perturbation Theory

When multiple states share the same unperturbed energy Eₙ⁰, the first-order formula breaks down (zero denominators). Instead, diagonalize H' within the degenerate subspace:

det(Hα′β−E1δαβ)=0whereHα′β=⟨α0∣H′∣β0⟩det(H'_\alpha\beta - E^{1}\delta_\alpha\beta) = 0 \qquad where \qquad H'_\alpha\beta = ⟨\alpha^{0}|H'|\beta^{0}⟩(PT.4)

The eigenvalues E¹ are the first-order energy corrections; the eigenvectors are the "good" zero-order states (the correct linear combinations within the degenerate subspace). This procedure is essential for understanding the Stark effect (hydrogen in electric field), the Zeeman effect (in magnetic field), and crystal field splitting.

Example PT.2 — Linear Stark Effect in Hydrogen n=2

HydrogenatominelectricfieldE=Ez^.PerturbationH′=eEz=eErcosθ.Findthefirst−Hydrogen atom in electric field E = Eẑ. Perturbation H' = eEz = eEr cos \theta. Find the first-orderenergysplittingofthen=2levelorder energy splitting of the n=2 level.

n=2 states:|200⟩,∣210⟩,∣211⟩,∣21−1⟩—fourdegeneratestates(4−folddegeneracy200⟩, |210⟩, |211⟩, |21-1⟩ — four degenerate states (4-fold degeneracy
Selection rules:H′isoddunderparity(z→−z),so⟨nlm∣z∣nlm⟩=0.Off−diagonal:⟨200∣z∣210⟩≠0bypariH' is odd under parity (z \to -z), so ⟨nlm|z|nlm⟩ = 0. Off-diagonal: ⟨200|z|210⟩ ≠ 0 by parity (different l).
Only nonzero matrix element:⟨200∣eEz∣210⟩=eE⟨200∣rcosθ∣210⟩=−3eEa0200|eEz|210⟩ = eE ⟨200|r cos \theta|210⟩ = -3eEa_{0}
Diagonalize 4×4:The∣211⟩and∣21−1⟩statesdecouple(zdoesn′tchangem).The∣200⟩,∣210⟩blockhasmatrThe |211⟩ and |21-1⟩ states decouple (z doesn't change m). The |200⟩, |210⟩ block has matrix[[0,−3eEa0],[−3eEa0,0ix [[0, -3eEa_{0}],[-3eEa_{0}, 0.
Eigenvalues:E1=±3eEa0→levelssplitinto(atleast)three:E0±3eEa0andtwodegeneratemiddlelevelE^{1} = \pm3eEa_{0} \to levels split into (at least) three: E_{0}\pm3eEa_{0} and two degenerate middle levels.
Contrast to ground state:n=1hasnolinearStarkeffect(nomixingpartnerwithoppositeparityandsameenergy).In=1 has no linear Stark effect (no mixing partner with opposite parity and same energy). ItshowsonlyaquadraticStarkeffect(E2correctiont shows only a quadratic Stark effect (E^{2} correction.

PT.3 Time-Dependent Perturbation Theory

For a time-varying perturbation H'(t) turned on at t=0, the probability amplitude for a transition from |i⟩ to |f⟩ is (to first order):

cf(t)=(1/iℏ)∫(0tot)⟨f∣H′(t′)∣i⟩eiωfit′dt′(ωfi=(Ef−Ei)/ℏ)c_{f}(t) = (1/i\hbar) \int(0 to t) ⟨f|H'(t')|i⟩ e^{i\omega_fi t'} dt' \qquad (\omega_fi = (E_{f}-E_{i})/\hbar)(PT.5)

For a sinusoidal perturbation H' = V e^(−iωt) (e.g., a photon field):

∣cf(t)∣2≈(∣⟨f∣V∣i⟩∣2/ℏ2)×t×δ(ωfi−ω)×π|c_{f}(t)|^{2} \approx (|⟨f|V|i⟩|^{2}/\hbar^{2}) \times t \times \delta(\omega_fi - \omega) \times \pi(PT.6)

For a continuum of final states with density ρ(E_f), the transition rate (Fermi's Golden Rule) is:

Theorem PT.1 — Fermi's Golden Rule
The transition rate from initial state |i⟩ to a continuum of final states is:Γi\tof=(2π/ℏ)∣⟨f∣H′∣i⟩∣2ρ(EfΓ_i\tof = (2\pi/\hbar) |⟨f|H'|i⟩|^{2} \rho(E_{f}This is one of the most used formulas in all of physics — it governs radioactivedecay,photoionization,neutronscattering,nuclearreactions,andlaserphysics.Theδve decay, photoionization, neutron scattering, nuclear reactions, and laser physics. The \deltawhich transitions are fast and which are slow.
Definition PT.1 — Common Traps
  • Perturbation theory needs a small parameter: large corrections signal breakdown.
  • Degeneracy must be handled first: diagonalize the perturbation inside the degenerate subspace.
  • Energy denominators matter: nearly degenerate states can dominate corrections.
  • Series may be asymptotic: more terms do not always mean a better answer.
Exercises — PT.1–PT.3 Perturbation Theory
1.
AuniformperturbationH′=V0=3eVisaddedtotheentirebox.Findthefirst−ordereneA uniform perturbation H' = V_{0} = 3 eV is added to the entire box. Find the first-order energy correction to the ground state.
eV
Straightforward
2.
ForthelinearStarkeffectinhydrogenn=2,thekeymatrixelementis−3eEa0.WhatistheFor the linear Stark effect in hydrogen n=2, the key matrix element is -3eEa_{0}. What is the magnitude of the largest first-order energy shift?
eEa₀
Straightforward
3.Compute the first-order energycorrectiontothegroundstateofaparticleinaboxwhenasmallpotentialbumpV0nergy correction to the ground state of a particle in a box when a small potential bump V_{0}hird.
Straightforward
4.Find the quadratic Stark effect (second-order energy correction) for the hydrogen ground state in a uniform electric field. What is the polarizability?
Intermediate
5.Apply Fermi's Golden Rule to the photoelectric effect: absorption of a photon by a hydrogen atom in the ground state. What is the angular distribution of photoelectrons?
Intermediate
6.Use the variational method with hydrogenic trial wavefunctions to estimate the ground-state energy of helium. Why is Z_
Challenging
Key Takeaways
  • First−orderenergycorrection:En1=⟨n0∣H′∣n0⟩—justtheexpectationvalueoftheperturFirst-order energy correction: E_{n}^{1} = ⟨n^{0}|H'|n^{0}⟩ — just the expectation value of the perturbation.
  • Second−ordercorrection:En2=∑m≠n∣⟨m∣H′∣n⟩∣2/(En−Em).AlwayslowersgroundstateeneSecond-order correction: E_{n}^{2} = \sum_{m≠n} |⟨m|H'|n⟩|^{2}/(E_{n}-Em). Always lowers ground state energy.
  • Degenerate case: diagonalize H' within the degenerate subspace first.
  • Fermi′sGoldenRule:Γ=(2π/ℏ)∣⟨f∣H′∣i⟩∣2ρ(Ef)—governsalldecayandabsorptionratesFermi's Golden Rule: Γ = (2\pi/\hbar)|⟨f|H'|i⟩|^{2}\rho(Ef) — governs all decay and absorption rates.
  • Selection rules come from matrix element ⟨f|H'|i⟩ vanishing by symmetry (parity, angular momentum).
  • Variationalprinciple:⟨ψ∣H∣ψ⟩≥Eground—minimizeovertrialstatesforupperboundVariational principle: ⟨\psi|H|\psi⟩ \ge E_{ground} — minimize over trial states for upper bound.