Perturbation Theory Automation

Perturbation theory can be automated by solving an ordered sequence of projected linear problems. The automation removes repeated algebra; it does not establish that a series converges or that a chosen unperturbed level remains isolated. This page treats a nondegenerate reference state and separates coefficient generation from validation at a finite coupling.

Required background. Nondegenerate perturbation theory develops the physical expansion and its assumptions; symbolic operator algebra supplies ordered products and exact simplification.

A recurrence with fixed normalization

Let $H(\lambda)=H_0+\lambda V$ and $H_0\lvert0\rangle=E_0\lvert0\rangle$, with an isolated simple eigenvalue. Use intermediate normalization,

$$\lvert\psi(\lambda)\rangle =\lvert0\rangle+\sum_{k\ge1}\lambda^k\lvert\psi_k\rangle, \qquad \langle0\vert\psi_k\rangle=0,$$

and $E(\lambda)=E_0+\sum_{k\ge1}\lambda^kE_k$. Matching powers of $\lambda$ gives

$$(H_0-E_0)\lvert\psi_k\rangle =-V\lvert\psi_{k-1}\rangle +\sum_{j=1}^k E_j\lvert\psi_{k-j}\rangle,$$

Here $\lvert\psi_0\rangle=\lvert0\rangle$ denotes the reference ket, not the zero vector. Projecting onto the reference state yields

$$E_k=\langle0\vert V\vert\psi_{k-1}\rangle.$$

Define $Q=I-\lvert0\rangle\langle0\rvert$ and the reduced inverse $R=Q(E_0-H_0)^{-1}Q$, with inversion only on the complementary subspace. Then

$$\lvert\psi_k\rangle =R\left[V\lvert\psi_{k-1}\rangle -\sum_{j=1}^{k-1}E_j\lvert\psi_{k-j}\rangle\right].$$

Compute $E_k$ before $\psi_k$, store earlier orders, and reuse a factorization of the projected operator when possible. Do not invert $H_0-E_0$ on the full space: it is singular by construction. In an eigenbasis, the reduced inverse is diagonal with entries $(E_0-E_n)^{-1}$ for $n\ne0$.

A two-level exact check

Take $\Delta>0$ and real $g$,

$$H(\lambda)=\begin{pmatrix}0&\lambda g\\\lambda g&\Delta\end{pmatrix}.$$

For the reference state $(1,0)^T$, the recurrence gives $E_1=0$, $\psi_1=-(g/\Delta)(0,1)^T$, and $E_2=-g^2/\Delta$. Continuing gives $E_3=0$ and $E_4=g^4/\Delta^3$. The exact lower eigenvalue is

$$E_-(\lambda)=\frac{\Delta-\sqrt{\Delta^2+4\lambda^2g^2}}2 =-\frac{\lambda^2g^2}{\Delta} +\frac{\lambda^4g^4}{\Delta^3}+O(\lambda^6).$$

This example checks signs of energy denominators, order bookkeeping, and symmetry under $\lambda\mapsto-\lambda$. At small coupling, evaluate the exact expression in a cancellation-resistant form, $-2\lambda^2g^2/(\Delta+\sqrt{\Delta^2+4\lambda^2g^2})$.

The nearest complex branch points are $\lambda=\pm i\Delta/(2g)$ for $g\ne0$, so the Taylor radius is $\Delta/(2|g|)$. Smooth behavior on the real axis does not imply convergence at arbitrarily large real coupling.

Checking generated coefficients

Substitute the truncated energy and state series back into $(H-E)\psi$. Every coefficient through the retained order should vanish exactly or within a stated arithmetic tolerance. Verify the reference overlap separately. A residual at finite $\lambda$ should scale with the first omitted power when the asymptotic regime is resolved.

Intermediate normalization is not unit normalization. Before computing a physical expectation value, use

$$\langle A\rangle= \frac{\langle\psi\vert A\vert\psi\rangle} {\langle\psi\vert\psi\rangle}$$

and expand numerator and denominator consistently to the same target order. Normalizing a truncated state numerically is useful for comparison, but silently mixes higher-order terms and is not identical to an order-by-order symbolic expansion.

For an infinite basis, coefficient generation introduces a second truncation in intermediate states. Increase that cutoff independently of perturbative order. Polynomial oscillator perturbations often connect only a finite band at each fixed order; using that selection rule is more reliable than assuming a small fixed matrix is always sufficient.

Degeneracies and divergent expansions

Small denominators warn that the chosen reference decomposition is inappropriate. For exact degeneracy, construct and diagonalize the effective perturbation in the degenerate subspace before applying a reduced inverse. See degenerate perturbation theory. An arbitrary small regulator in the denominator does not perform this reorganization.

Some physically useful perturbation series are asymptotic rather than convergent. Increasing order can initially improve and eventually worsen an answer. Compare successive truncations and an independent calculation; a symbolic system’s ability to generate a coefficient does not imply that summing more coefficients improves the prediction.

Exercises

For the two-level example, take $\Delta=2$, $g=1$. Find the energy through fourth order and the Taylor radius.

Solution

$E_-=-\lambda^2/2+\lambda^4/8+O(\lambda^6)$ and the radius is one. At $\lambda=1$ the series is on its convergence boundary; an interior asymptotic accuracy argument cannot be assumed there.

References

• Kato, T., Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976 — isolated spectral subspaces and analytic perturbation.
• Meurer, A., et al., “SymPy: Symbolic Computing in Python,” PeerJ Computer Science 3, e103 (2017), article — symbolic series and algebraic verification.
• Sakurai, J. J., and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2021 — nondegenerate and degenerate quantum perturbation theory.