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 method assumes the level $E_n^{(0)}$ is isolated: no other unperturbed state has the same energy, and the perturbation does not mix it too strongly with nearby levels.

Expansion

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

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

For nondegenerate perturbation theory to be reliable, the relevant $\epsilon_{mn}$ should be much smaller than $1$. This is not a rigorous universal bound, but it is the basic physical test.

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.

Common Mistakes

• Using the formula when $E_m^{(0)}=E_n^{(0)}$ or nearly so.
• Forgetting that the component of $\lvert n^{(1)}\rangle$ along $\lvert n^{(0)}\rangle$ depends on the normalization convention.
• Treating state corrections themselves as observables.
• Setting $\lambda=1$ before identifying the actual small dimensionless ratio.
• Ignoring symmetry, which can force $V_{nn}$ or entire blocks of $V$ to vanish.

Cross-Links

• Time-Independent Perturbation Theory
• First-Order Energy Corrections
• First-Order State Corrections
• Second-Order Energy Corrections
• Higher-Order Structure
• Rayleigh–Schrödinger Perturbation Theory
• Small Parameters and Error Estimates
• Degenerate Perturbation Theory
• Anharmonic Oscillator
• Eigenvalues and Eigenstates
• Spectral Decomposition
• Symmetry Constraints on Hamiltonians
• Approximate Symmetry

References

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

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.