Sum Rules and Completeness Tricks

Perturbation theory often produces sums over intermediate states. Sum rules and completeness tricks simplify those sums, check calculations, and sometimes avoid explicit summation altogether. The general many-body hierarchy of spectral moments, nested commutators, high-energy response, and density $f$-sums is developed on Sum Rules; this page owns perturbative closure and state-by-state rearrangements.

The central idea is that a complete set of exact or unperturbed states resolves the identity:

$$\sum_m \lvert m\rangle\langle m\rvert=I,$$

with integrals over continuum states included when needed.

Completeness Relations

If $\{\lvert m\rangle\}$ is a complete orthonormal basis, then for any operator $A$,

$$\sum_m |\langle m|A|n\rangle|^2 = \langle n|A^\dagger A|n\rangle.$$

If the state $\lvert n\rangle$ should be excluded from the sum, then

$$\sum_{m\ne n} |\langle m|A|n\rangle|^2 = \langle n|A^\dagger A|n\rangle - |\langle n|A|n\rangle|^2.$$

This identity is exact when the basis is complete. In truncated numerical bases, it becomes a convergence test.

Second-Order Sums

A typical second-order energy shift is

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

The canonical derivation, sign analysis, polarizability interpretation, and continuum form are on Second-Order Energy Corrections. This page focuses on methods for reorganizing or checking the resulting spectral sums.

Completeness alone does not remove the energy denominator. Sum rules become useful when the numerator and denominator can be related by commutators, when the denominator is approximated by an average energy, or when an auxiliary equation is solved instead of summing explicitly.

Closure Approximation

If a denominator varies slowly over the important intermediate states, one may replace it by an effective average $\Delta$:

$$\frac{1}{E_n-E_m} \approx \frac{1}{\Delta}.$$

Then

$$\sum_{m\ne n} \frac{|\langle m|A|n\rangle|^2}{E_n-E_m} \approx \frac{1}{\Delta} \left( \langle n|A^\dagger A|n\rangle - |\langle n|A|n\rangle|^2 \right).$$

This is an approximation, not an identity. It is useful only when the intermediate-state energy distribution is narrow enough for the observable being estimated.

Commutator Sum Rules

Let $H\lvert n\rangle=E_n\lvert n\rangle$ and let $A$ be Hermitian. Then

$$\frac12 \langle n|[A,[H,A]]|n\rangle = \sum_m (E_m-E_n) |\langle m|A|n\rangle|^2.$$

This identity is exact under the usual domain and completeness assumptions. It turns an energy-weighted sum over states into an expectation value of a double commutator.

For a one-dimensional Hamiltonian

$$H=\frac{P^2}{2m}+V(X),$$

using $A=X$ gives

$$[X,[H,X]] = \frac{\hbar^2}{m}.$$

Therefore

$$\sum_m (E_m-E_n) |\langle m|X|n\rangle|^2 = \frac{\hbar^2}{2m}.$$

This is the basic energy-weighted position sum rule for one particle in one dimension.

Oscillator Strength Preview

In atomic physics, dipole matrix elements and excitation energies are often packaged into oscillator strengths. The Thomas–Reiche–Kuhn sum rule is the many-state statement that the total oscillator strength is fixed by commutation relations and particle number. Oscillator Strengths develops the spectroscopic definitions, degeneracy conventions, continuum budget, and relation to integrated absorption.

The detailed atomic and spectroscopic applications live elsewhere. The perturbation-theory lesson is simpler: not every sum over unknown excited states is unconstrained. Commutators can enforce exact global checks.

Dalgarno-Lewis Method Preview

Some second-order sums can be avoided by solving an inhomogeneous equation. Suppose one needs

$$\sum_{m\ne n} \frac{\langle n|A|m\rangle\langle m|A|n\rangle} {E_n-E_m}.$$

Define an auxiliary state $\lvert F_n\rangle$ by

$$(H_0-E_n) \lvert F_n\rangle = - (A-\langle A\rangle_n) \lvert n\rangle,$$

with the component along $\lvert n\rangle$ fixed by a normalization convention. Then the desired sum can be extracted from

$$\langle n|A|F_n\rangle.$$

This method is especially useful for polarizabilities, where explicit sums over infinitely many excited and continuum states are inconvenient.

Exact Versus Approximate

Completeness relations are exact only when the basis is complete and includes all discrete and continuum states with the correct normalization.

Commutator sum rules are exact only when the operators and states lie in the domains required for the commutators and spectral resolution.

Closure approximations are not exact. They replace detailed spectral information by an average denominator. They are estimates whose accuracy must be justified by the energy distribution and observable.

Numerical Use

In a truncated basis, sum rules are powerful diagnostics. For example, compute

$$S_N = \sum_{m=0}^{N} (E_m-E_n) |\langle m|X|n\rangle|^2$$

and compare it with $\hbar^2/(2m)$. Failure to approach the sum rule can indicate that the basis is too small, the continuum is missing, or matrix elements are inconsistent.

The same warning applies to perturbative sums: a partial sum may look converged for low states but still violate an exact sum rule sensitive to high-energy tails.

Common Mistakes

• Using completeness while forgetting continuum states.
• Treating closure as exact without checking denominator variation.
• Applying commutator sum rules outside the operator domain where the commutators are valid.
• Dropping the excluded state term in sums over $m\ne n$.
• Trusting a truncated basis that badly violates an exact sum rule.
• Confusing energy-weighted and inverse-energy-weighted sums.

Cross-Links

• Many-Body Sum Rules
• Oscillator Strengths
• Time-Independent Perturbation Theory
• Nondegenerate Perturbation Theory
• Second-Order Energy Corrections
• Spectral Decomposition
• Commutators
• Small Parameters and Error Estimates
• Perturbation Theory for the Harmonic Oscillator
• Anharmonic Oscillator by Perturbation Theory
• Quantum Harmonic Oscillator

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• A. Dalgarno and J. T. Lewis, “The exact calculation of long-range forces between atoms by perturbation theory,” Proceedings of the Royal Society A 233, 70-74, 1955.
• W. Thomas, “Uber die Zahl der Dispersionselektronen, die einem stationaren Zustand zugeordnet sind,” Naturwissenschaften 13, 627, 1925.

Exercises

1. Prove the simple completeness identity

$$\sum_m |\langle m|A|n\rangle|^2 = \langle n|A^\dagger A|n\rangle.$$

Solution

Insert the identity between $A^\dagger$ and $A$:

$$\langle n|A^\dagger A|n\rangle = \langle n|A^\dagger \left( \sum_m\lvert m\rangle\langle m\rvert \right) A|n\rangle.$$

This becomes

$$\sum_m \langle n|A^\dagger|m\rangle \langle m|A|n\rangle = \sum_m |\langle m|A|n\rangle|^2.$$

2. For $H=P^2/(2m)+V(X)$, show that $[X,[H,X]]=\hbar^2/m$.

Solution

Since $X$ commutes with $V(X)$,

$$[H,X] = \frac{1}{2m}[P^2,X].$$

Using $[P,X]=-i\hbar$,

$$[P^2,X] = P[P,X]+[P,X]P = -2i\hbar P.$$

Thus

$$[H,X] = - \frac{i\hbar}{m}P.$$

Then

$$[X,[H,X]] = - \frac{i\hbar}{m}[X,P] = \frac{\hbar^2}{m}.$$