Time-Independent Perturbation Theory

The catalogue of exactly solvable Hamiltonians is short: the free particle, the square well, the harmonic oscillator, the hydrogen atom, and a handful of others. Every real system deviates from these idealizations: a charged oscillator feels a stray field, hydrogen has relativistic and spin corrections, an atom sits in a laboratory magnet. Perturbation theory computes the response of a solved problem to a small addition without solving the new problem from scratch. The strategy is to write the Hamiltonian as a solvable piece plus a correction scaled by a bookkeeping parameter, expand the eigenvalues and eigenstates as power series in that parameter, and match order by order.

The perturbation expansion

Split the Hamiltonian into an unperturbed part with known spectrum and a perturbation,

\hat{H} = \hat{H}^{0} + λ \hat{H}^{'}, \hat{H}^{0} ∣ n^{0} ⟩ = E_{n}^{0} ∣ n^{0} ⟩,

where the $∣ n^{0} ⟩$ are orthonormal and complete, and $λ$ is a dimensionless parameter carried through the algebra to track orders; it is set to $1$ at the end, so $\hat{H}^{'}$ is the physical perturbation. Assume the exact eigenvalues and eigenstates deform smoothly from the unperturbed ones and admit power series in $λ$ ,

E_{n} = E_{n}^{0} + λ E_{n}^{1} + λ^{2} E_{n}^{2} + \dots, ∣ n ⟩ = ∣ n^{0} ⟩ + λ ∣ n^{1} ⟩ + λ^{2} ∣ n^{2} ⟩ + \dots .

The superscript counts the order in $λ$ , not a power. Substituting both series into $\hat{H} ∣ n ⟩ = E_{n} ∣ n ⟩$ and collecting terms of equal order produces one equation per power of $λ$ :

λ^{0} : λ^{1} : λ^{2} : \hat{H}^{0} ∣ n^{0} ⟩ = E_{n}^{0} ∣ n^{0} ⟩, \hat{H}^{0} ∣ n^{1} ⟩ + \hat{H}^{'} ∣ n^{0} ⟩ = E_{n}^{0} ∣ n^{1} ⟩ + E_{n}^{1} ∣ n^{0} ⟩, \hat{H}^{0} ∣ n^{2} ⟩ + \hat{H}^{'} ∣ n^{1} ⟩ = E_{n}^{0} ∣ n^{2} ⟩ + E_{n}^{1} ∣ n^{1} ⟩ + E_{n}^{2} ∣ n^{0} ⟩ .

The zeroth-order equation is the solved problem. The higher-order equations each determine one more term in the two series. A choice of phase and normalization removes the ambiguity in $∣ n ⟩$ : fix intermediate normalization $⟨ n^{0} ∣ n ⟩ = 1$ , which forces every correction to be orthogonal to the unperturbed state, $⟨ n^{0} ∣ n^{k} ⟩ = 0$ for $k \geq 1$ . The physical state is renormalized to unit length only at the end.

Each order of the expansion feeds the next: the solved zeroth-order level and state seed the first-order corrections, which seed the second, so the series is built one power of the coupling at a time.

First-order corrections

Project the first-order equation onto $⟨ n^{0} ∣$ . The term $⟨ n^{0} ∣ \hat{H}^{0} ∣ n^{1} ⟩$ equals $E_{n}^{0} ⟨ n^{0} ∣ n^{1} ⟩$ because $\hat{H}^{0}$ is Hermitian and acts to the left on its own eigenbra, and it cancels the matching term on the right. What survives is the first-order energy shift.

This is the single most-used result of the theory: to first order, the energy moves by the average of the perturbation over the unperturbed probability density. No new wavefunction is needed. For an oscillator nudged by a weak quartic term $\hat{H}^{'} = γ \overset{x}{^}^{4}$ , the ground-state shift is $E_{0}^{1} = γ ⟨ 0∣ \overset{x}{^}^{4} ∣0 ⟩ = 3 γ (ℏ/2 mω)^{2}$ , computed entirely from the known ground state.

The first-order state correction requires the components of $∣ n^{1} ⟩$ along the other unperturbed states. Expand $∣ n^{1} ⟩ = \sum_{m \neq = n} c_{m} ∣ m^{0} ⟩$ (the $n$ component vanishes by intermediate normalization) and project the first-order equation onto $⟨ m^{0} ∣$ with $m \neq = n$ :

E_{m}^{0} c_{m} + ⟨ m^{0} ∣ \hat{H}^{'} ∣ n^{0} ⟩ = E_{n}^{0} c_{m} ⟹ c_{m} = \frac{⟨ m ^{0} ∣ H ^ ^{'} ∣ n ^{0} ⟩}{E _{n}^{0} - E _{m}^{0}} .

The perturbation mixes into $∣ n ⟩$ a little of every other unperturbed state, weighted by the coupling matrix element $\hat{H}_{mn}^{'} = ⟨ m^{0} ∣ \hat{H}^{'} ∣ n^{0} ⟩$ and divided by the energy gap. States close in energy mix strongly; distant states barely contribute. The denominator $E_{n}^{0} - E_{m}^{0}$ is the seed of both the method's power and its failure: it is finite only when no other state shares the energy $E_{n}^{0}$ — the nondegeneracy assumption, restated.

The perturbation admixes other levels into the state with weight set by the coupling divided by the energy gap; the nearest levels dominate and the far ones are suppressed by the large denominator.

Second-order energy

The first-order shift can vanish by symmetry, and even when it does not, the next term measures how the admixed states pull on the energy. Project the second-order equation onto $⟨ n^{0} ∣$ . The $\hat{H}^{0}$ and $E_{n}^{1}$ terms drop by the same cancellation and orthogonality used before, leaving $E_{n}^{2} = ⟨ n^{0} ∣ \hat{H}^{'} ∣ n^{1} ⟩$ . Insert the first-order state:

Every term is a squared magnitude over a signed gap. Two structural facts follow at once.

The ground state always moves down. For $n$ the lowest level every denominator $E_{0}^{0} - E_{m}^{0}$ is negative, so $E_{0}^{2} \leq 0$ : second-order perturbation theory can only depress the ground-state energy, consistent with the variational bound.
Levels repel. A pair of coupled levels each push the other away. The lower of the two receives a negative shift (its partner sits above, negative denominator) and the upper a positive shift, so second order increases their separation. This level repulsion is why exact crossings of coupled levels are avoided.

Second order pushes coupled levels apart: the lower member is depressed and the upper raised by equal and opposite amounts set by the coupling squared over the gap, so the pair repels rather than crosses.

The full physical eigenvalue through second order, with $λ = 1$ , reads

E_{n} = E_{n}^{0} + \hat{H}_{nn}^{'} + m \neq = n \sum \frac{∣ H ^ _{mn}^{'} ∣ ^{2}}{E _{n}^{0} - E _{m}^{0}} + \dots .

Worked example. A charged oscillator in a uniform field tests the second-order formula against an exact answer. Take $\hat{H}^{0} = \overset{p}{^}^{2} /2 m + \frac{1}{2} m ω^{2} \overset{x}{^}^{2}$ and $\hat{H}^{'} = - q E \overset{x}{^}$ . The first-order shift vanishes for every level, since $\overset{x}{^}$ is odd and $⟨ n^{0} ∣ \overset{x}{^} ∣ n^{0} ⟩ = 0$ . The second-order shift needs the ladder matrix elements of $\overset{x}{^}$ , nonzero only for $m = n \pm 1$ , so with $E_{n}^{0} - E_{n \pm 1}^{0} = \mp ℏ ω$ ,

E_{n}^{2} = q^{2} E^{2} [\frac{∣ ⟨ n + 1∣ x ^ ∣ n ⟩ ∣ ^{2}}{- ℏ ω} + \frac{∣ ⟨ n - 1∣ x ^ ∣ n ⟩ ∣ ^{2}}{+ ℏ ω}] = q^{2} E^{2} \frac{ℏ}{2 mω} [\frac{n + 1}{- ℏ ω} + \frac{n}{ℏ ω}] = - \frac{q ^{2} E ^{2}}{2 m ω ^{2}} .

Every higher order vanishes. Completing the square, $\hat{H} = \overset{p}{^}^{2} /2 m + \frac{1}{2} m ω^{2} (\overset{x}{^} - q E / m ω^{2})^{2} - q^{2} E^{2} /2 m ω^{2}$ , confirms the exact spectrum is the unperturbed ladder shifted rigidly by $- q^{2} E^{2} /2 m ω^{2}$ . Perturbation theory is exact to second order here, a rare case where the series terminates.

The validity condition

The expansion is an approximation, useful when successive terms shrink. The ratio of the first-order state correction's typical coefficient to unity, or equivalently of the second-order energy to the first-order gap, gives the criterion.

The perturbation need not be small in absolute terms; it must be small compared to the gaps it bridges. A weak coupling between two nearly degenerate levels can still be large in this sense, and the series then fails. The remedy for the extreme case of exact degeneracy is the subject of the rest of the lesson.

Degeneracy breaks the formula

Suppose $E_{n}^{0}$ is $g$ -fold degenerate, with an orthonormal basis $∣ n_{1}^{0} ⟩, \dots, ∣ n_{g}^{0} ⟩$ of the degenerate subspace. The nondegenerate formulas contain denominators $E_{n}^{0} - E_{m}^{0}$ that vanish whenever $∣ m^{0} ⟩$ is another state of the same energy. Unless the offending numerator $⟨ m^{0} ∣ \hat{H}^{'} ∣ n^{0} ⟩$ also vanishes, the first-order state correction is infinite and the derivation collapses.

The failure is not physical but a symptom of a bad starting point. When a level is degenerate, the unperturbed problem does not single out a preferred basis inside the degenerate subspace: any orthonormal combination diagonalizes $\hat{H}^{0}$ equally well. The perturbation does select a preferred basis, and starting the expansion in the wrong one produces the singular denominators. The task is to identify the good states, the particular zeroth-order combinations toward which the exact eigenstates collapse as $λ \to 0$ .

A degenerate level admits any basis for its subspace under the unperturbed Hamiltonian; the perturbation lifts the degeneracy and picks out the unique good combinations that the exact states approach as the coupling is switched off.

Degenerate perturbation theory

Work inside the degenerate subspace. A good state is a combination $∣ ϕ ⟩ = \sum_{j = 1}^{g} c_{j} ∣ n_{j}^{0} ⟩$ whose first-order energy shift is well defined. Repeat the first-order derivation, but now project the first-order equation onto each degenerate basis bra $⟨ n_{i}^{0} ∣$ . The $\hat{H}^{0}$ terms still cancel, and because the exact state approaches $∣ ϕ ⟩$ the equation becomes an eigenvalue problem for the coefficients:

j = 1 \sum g W_{ij} c_{j} = E^{1} c_{i}, W_{ij} \equiv ⟨ n_{i}^{0} ∣ \hat{H}^{'} ∣ n_{j}^{0} ⟩ .

Diagonalizing $W$ replaces the singular sum with a finite linear-algebra problem. The perturbation is projected onto the subspace, the resulting matrix is diagonalized, and its eigenvalues are the split levels. Degeneracy is lifted completely when the eigenvalues are distinct; a repeated eigenvalue signals a residual degeneracy that a higher order, or a second perturbation, may still break.

Degenerate perturbation theory restricts the perturbation to the degenerate subspace, forming the matrix of its matrix elements; the eigenvalues of that block are the first-order splittings.

For the two-fold case the eigenvalues have a closed form. With diagonal elements $W_{aa}, W_{bb}$ and off-diagonal coupling $W_{ba} = W_{ab}^{*}$ ,

E_{\pm}^{1} = \frac{1}{2} (W_{aa} + W_{bb}) \pm \frac{1}{2} (W_{aa} - W_{bb})^{2} + 4 ∣ W_{ab} ∣^{2} .

The splitting $E_{+}^{1} - E_{-}^{1} = (W_{aa} - W_{bb})^{2} + 4 ∣ W_{ab} ∣^{2}$ never closes when the off-diagonal coupling $W_{ab}$ is nonzero: even states with equal diagonal elements split by $2 ∣ W_{ab} ∣$ . This is the two-level avoided crossing in its simplest algebraic form, the same structure that governs the Zeeman and Stark effects and molecular bonding.

Finding good states from symmetry

Diagonalizing $W$ is unnecessary when a symmetry already labels the degenerate states. Suppose an observable $\hat{A}$ commutes with both $\hat{H}^{0}$ and $\hat{H}^{'}$ , and the degenerate basis states are chosen as eigenstates of $\hat{A}$ with distinct eigenvalues $a_{i} \neq = a_{j}$ . Then

(a_{i} - a_{j}) W_{ij} = ⟨ n_{i}^{0} ∣ [\hat{A}, \hat{H}^{'}] ∣ n_{j}^{0} ⟩ = 0,

so $W_{ij} = 0$ for $i \neq = j$ : the matrix is already diagonal in the eigenbasis of $\hat{A}$ , and those states are the good ones.

This converts a diagonalization into an inspection. In hydrogen the perturbing interactions of the next lesson commute with the total angular momentum, so the coupled $∣ n, ℓ, j, m_{j} ⟩$ basis is good and the fine-structure shifts are read off as diagonal expectation values, no $g \times g$ matrix required. Choosing the right symmetry-adapted basis before perturbing is the practical heart of the degenerate theory.

When a symmetry operator commutes with both the unperturbed Hamiltonian and the perturbation, its distinct eigenvalues label the degenerate states and force the perturbation matrix diagonal, so those eigenstates are already the good ones.

Second order and the structure of the series

Beyond first order the degenerate and nondegenerate theories rejoin. Once the good states diagonalize $W$ and split the level, the remaining states are nondegenerate (assuming distinct eigenvalues) and the second-order formula applies with the good states as the unperturbed basis, the sum running over all states outside the original subspace. When a degeneracy survives to first order, second-order degenerate perturbation theory diagonalizes an effective operator built from couplings to the external states, but the same principle governs: find the basis the perturbation prefers, then expand.

The perturbation series is asymptotic, not convergent, for most physical Hamiltonians: the terms shrink at first and eventually grow, so truncating at low order is accurate while summing to all orders diverges. For the systems of this module the first two orders already capture fine structure, the Zeeman and Stark shifts, and molecular binding to spectroscopic accuracy. The variational method supplies a complementary tool bounding the ground state from above, and the WKB approximation handles the semiclassical regime where no small parameter multiplies a clean perturbation.