Nondegenerate Perturbation Theory

Nondegenerate time-independent perturbation theory computes how an isolated eigenvalue and its eigenstate change when a Hamiltonian is changed slightly.

Write

$$H(\lambda) = H_0+\lambda V,$$

where $H_0$ is exactly solved and $\lambda$ is a bookkeeping parameter. The unperturbed eigenvalue equation is

$$H_0\lvert n^{(0)}\rangle = E_n^{(0)} \lvert n^{(0)}\rangle.$$

The level must be simple and separated by a positive gap from the rest of the spectrum, including any continuum. For self-adjoint operators, the perturbation must also define a suitable common-domain or quadratic-form family; an isolated level alone does not justify a power series for an arbitrary unbounded perturbation. The formulas below apply to a sufficiently smooth eigenvalue branch, or as a formal asymptotic expansion when convergence has not been established.

Required background. Use eigenvalues and eigenstates to project the eigenvalue equation and spectral decomposition to resolve the identity.

Expanding an isolated eigenvalue

Seek an eigenvalue and eigenstate of

$$H(\lambda)\lvert n(\lambda)\rangle = E_n(\lambda)\lvert n(\lambda)\rangle$$

as formal expansions:

$$E_n(\lambda) = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2E_n^{(2)} + \cdots,$$

and

$$\lvert n(\lambda)\rangle = \lvert n^{(0)}\rangle + \lambda\lvert n^{(1)}\rangle + \lambda^2\lvert n^{(2)}\rangle + \cdots.$$

The physical perturbation may be obtained by setting $\lambda=1$ after the order counting is clear, but this is safe only if the corresponding dimensionless ratios are small.

Intermediate Normalization

A common convention is intermediate normalization:

$$\langle n^{(0)}|n(\lambda)\rangle=1.$$

This implies

$$\langle n^{(0)}|n^{(1)}\rangle=0.$$

This condition fixes the component of $\lvert n^{(1)}\rangle$ parallel to $\lvert n^{(0)}\rangle$. Other normalization conventions are possible. Energy corrections do not depend on this convention, but state-correction formulas do.

Order-by-Order Equations

Substituting the expansions into the eigenvalue equation and matching powers of $\lambda$ gives, at first order,

$$(H_0-E_n^{(0)})\lvert n^{(1)}\rangle = (E_n^{(1)}-V) \lvert n^{(0)}\rangle.$$

Projecting with $\langle n^{(0)}|$ gives the first-order energy correction:

$$E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle.$$

Using the shorthand

$$V_{mn} = \langle m^{(0)}|V|n^{(0)}\rangle,$$

this is

$$E_n^{(1)}=V_{nn}.$$

For symmetry tests, sign checks, parameter-response interpretation, and focused examples, see First-Order Energy Corrections.

First-Order State Correction

Project the first-order equation onto $\langle m^{(0)}|$ with $m\ne n$:

$$(E_m^{(0)}-E_n^{(0)}) \langle m^{(0)}|n^{(1)}\rangle = -V_{mn}.$$

Thus

$$\lvert n^{(1)}\rangle = \sum_{m\ne n} \frac{V_{mn}} {E_n^{(0)}-E_m^{(0)}} \lvert m^{(0)}\rangle.$$

This formula is the first place energy-gap denominators appear. Small denominators are a warning sign, not a detail to ignore.

For normalization and phase conventions, observable corrections, selection-rule diagnostics, exact checks, and the geometric interpretation of this vector, see First-Order State Corrections.

Second-Order Energy Correction

The second-order energy correction is

$$E_n^{(2)} = \sum_{m\ne n} \frac{\lvert V_{mn}\rvert^2} {E_n^{(0)}-E_m^{(0)}}.$$

This formula is one of the main workhorses of bound-state perturbation theory. Its sign depends on whether the coupled states lie above or below the state being corrected.

For a ground state, all other unperturbed energies satisfy $E_m^{(0)}\gt E_0^{(0)}$, so every term in $E_0^{(2)}$ is nonpositive.

For the derivation, reduced-resolvent form, sign theorem, static-polarizability connection, continuum terms, and convergence diagnostics, see Second-Order Energy Corrections.

For the all-order recursion, third-order subtraction structure, convergence radius, asymptotic truncation, and the switch to subspace diagonalization, see Higher-Order Structure.

For the historical identity, normalization conventions, repartitioning, comparison with Brillouin–Wigner theory, and atomic and molecular uses of the order-by-order method, see Rayleigh–Schrödinger Perturbation Theory.

Validity Criterion

A useful diagnostic is

$$\epsilon_{mn} = \left| \frac{\lambda V_{mn}} {E_n^{(0)}-E_m^{(0)}} \right|.$$

Small individual $\epsilon_{mn}$ are necessary for weak mixing in each channel. Also check their sum: many small amplitudes can produce a large state correction,

$$\|\lambda n^{(1)}\|^2 =\sum_{m\ne n}\epsilon_{mn}^2\ll1.$$

Continuum contributions require spectral integrals. This first-order diagnostic does not prove convergence of the full series. For bounded $V$, a conservative sufficient condition for the isolated spectral subspace to remain separated is $|\lambda|\|V\|<g/2$, where $g$ is the unperturbed gap.

If the perturbation has nonzero matrix elements connecting nearly degenerate levels, diagonalize the coupled subspace instead.

Worked Two-Level Check

Consider

$$H = \begin{pmatrix} E_1&\lambda v\\ \lambda v^*&E_2 \end{pmatrix},$$

with $E_1\lt E_2$ and $\Delta=E_2-E_1\gt 0$. For the lower level, $V_{11}=0$ and

$$E_1^{(2)} = \frac{\lvert v\rvert^2}{E_1-E_2} = -\frac{\lvert v\rvert^2}{\Delta}.$$

The exact lower eigenvalue is

$$\frac{E_1+E_2}{2} - \sqrt{ \left(\frac{\Delta}{2}\right)^2 + \lambda^2\lvert v\rvert^2 }.$$

Expanding for $\lambda\lvert v\rvert\ll\Delta$ gives

$$E_1 - \lambda^2 \frac{\lvert v\rvert^2}{\Delta} + O(\lambda^4),$$

matching the perturbative result.

What the Method Computes Well

Nondegenerate perturbation theory is excellent for small shifts of isolated levels, weak mixing between well-separated states, and expectation-value corrections when state corrections are included consistently.

It is not automatically reliable for wavefunction details, near-degenerate levels, tunneling splittings, or observables sensitive to high-energy tails.

Exercises

1. Show that if $V_{nn}=0$, the first nonzero energy correction may be second order.

Solution

The first-order correction is

$$E_n^{(1)}=V_{nn}.$$

If $V_{nn}=0$, this term vanishes. The second-order correction is

$$E_n^{(2)} = \sum_{m\ne n} \frac{\lvert V_{mn}\rvert^2} {E_n^{(0)}-E_m^{(0)}}.$$

This can be nonzero if the perturbation couples $\lvert n^{(0)}\rangle$ to other unperturbed states.

2. For the two-level Hamiltonian above, explain why the perturbation expansion fails when $\Delta\to0$.

Solution

The second-order correction contains

$$-\frac{\lvert v\rvert^2}{\Delta}.$$

As $\Delta\to0$, this correction diverges, indicating that the assumed unperturbed basis is no longer a good starting point. One must diagonalize the coupled two-state subspace, which is degenerate perturbation theory in the $\Delta=0$ limit.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.