Time-Independent Perturbation Theory

Time-independent perturbation theory predicts how a static change to a Hamiltonian shifts its energies and mixes its states. The starting point is a solved problem $H_0$ and a perturbation $\lambda V$. The main choice is whether to follow one isolated level or treat several nearby levels together.

Helpful background. Spectral decomposition explains eigenstates and spectral subspaces; projectors separate the states kept together. The error-estimate guide and notation guide introduce small parameters and signed energy gaps.

Enter this chapter

Begin with nondegenerate perturbation theory, which derives the isolated-level formulas. Then use the focused pages to understand their different physical meanings:

Quantity	What to learn
First-order energy correction	How a diagonal expectation value gives the linear response of an energy
First-order state correction	How off-diagonal matrix elements mix states and change other observables
Second-order energy correction	How virtual mixing produces quadratic shifts and why the ground-state shift is nonpositive for a linear perturbation

These effects need not begin at the same order. In particular, a vanishing first-order energy does not mean the state is unchanged.

A displaced oscillator makes the distinction visible

Add a linear potential to the oscillator,

$$H=\frac{p^2}{2m}+\frac12m\omega^2x^2+Fx.$$

Here $F$ is the coefficient of the potential; the applied force is $-F$. Parity makes $\langle n\rvert x\lvert n\rangle=0$, so the first-order energy shift vanishes. Nevertheless, the oscillator’s center moves. Completing the square gives

$$\frac12m\omega^2x^2+Fx =\frac12m\omega^2\left(x+\frac{F}{m\omega^2}\right)^2 -\frac{F^2}{2m\omega^2}.$$

Every level is shifted by $-F^2/(2m\omega^2)$, while each wavefunction is translated by $-F/(m\omega^2)$. The energy change begins quadratically; the position change begins linearly. The oscillator perturbation examples work out the state mixing and compare linear, quadratic, cubic, and quartic perturbations.

The exact energy here stops at second order, but the displaced state generally needs infinitely many states of the original oscillator basis. An exact low-order energy is therefore not a guarantee of an exact low-order state.

Is the level isolated enough?

The characteristic mixing amplitudes are

$$\frac{\lambda V_{mn}}{E_n^{(0)}-E_m^{(0)}}.$$

Check their combined size as well as each individual ratio. Small energy gaps, a large number of coupled states, or a nearby continuum can defeat an apparently weak perturbation. The error-estimate guide explains these checks. For unbounded operators, the formulas also require suitable operator or quadratic-form domains; finite matrices hide this issue.

For an exactly degenerate eigenspace, degenerate perturbation theory diagonalizes $PVP$ within that eigenspace. For nearby but unequal reference energies, quasi-degenerate perturbation theory keeps the small splittings in $PH_0P$ as well. In both cases, the retained group must be sufficiently separated from omitted states, or it must be enlarged further.

Apply symmetry before summing. It can forbid mixing and protect a crossing, or make a first-order correction vanish while allowing higher orders. The Stark example and Zeeman example show how degeneracy and the strength of a field determine the useful basis. Atomic states in a static electric field may be resonances rather than strict bound states; the Stark page explains that limitation.

Organize higher orders and check them

Higher-order structure develops the recursion and its reduced resolvent. Rayleigh–Schrödinger theory expands consistently in the reference energy denominators. Brillouin–Wigner theory retains energy dependence in a projected equation. Solving that equation self-consistently can reorganize terms, but does not by itself prove a truncated calculation accurate.

Two tools often shorten or check the work. The Hellmann–Feynman theorem relates a differentiable eigenvalue’s parameter derivative to an expectation value. Sum rules and completeness constrain intermediate-state sums; energy denominators usually prevent replacing an entire weighted sum by a simple completeness identity.

Compare with the anharmonic oscillator to see a case whose weak-coupling series is asymptotic rather than terminating. The numerical benchmarks test both perturbative order and basis convergence. Zwiebach’s notes give a detailed first course through these methods; Kato treats the operator and analyticity questions at a more mathematical level.

Exercises

Do transition frequencies change? In the displaced oscillator above, determine the change in $E_{n+1}-E_n$. Explain why the state can nevertheless have a different mean position.

Solution

All levels receive the same energy shift, so the spacing remains $\hbar\omega$. Each wavefunction is translated, giving $\langle x\rangle=-F/(m\omega^2)$. The spectrum of energy differences does not specify every observable of the state.

Check the position from the energy. Use the Hellmann–Feynman relation and $\partial H/\partial F=x$ to recover the displaced oscillator’s mean position.

Solution

Differentiating $E_n(F)=\hbar\omega(n+1/2)-F^2/(2m\omega^2)$ gives $dE_n/dF=-F/(m\omega^2)=\langle x\rangle$. The sign agrees with the minimum of the shifted potential.

References

• T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer, 1995. Isolated spectral branches and operator assumptions.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020. Static perturbation theory and symmetry.
• B. Zwiebach, “Non-degenerate and Degenerate Perturbation Theory”, Quantum Physics III, MIT OpenCourseWare 8.06, 2018.