Completeness and Orthonormal Bases

A complete orthonormal basis is an orthonormal family large enough to approximate every Hilbert-space vector in norm. It is the infinite-dimensional version of the finite-dimensional statement

$$I = \sum_i \lvert e_i\rangle\langle e_i\rvert,$$

but the infinite-dimensional word “complete” carries convergence content that must not be suppressed.

The finite-dimensional algebra is Orthonormal Bases. This page explains what changes in Hilbert spaces such as $L^2(\mathbb R)$.

For why standard quantum Hilbert spaces usually admit countable complete orthonormal bases, see Separable Hilbert Spaces.

Orthonormal Systems

Let $\mathcal H$ be a Hilbert space. A sequence or countable family

$$\{e_1,e_2,\ldots\}$$

is orthonormal if

$$\langle e_m\vert e_n\rangle = \delta_{mn}.$$

For any finite linear combination

$$f = \sum_{n=1}^{N} c_n e_n,$$

orthonormality gives

$$\lVert f\rVert^2 = \sum_{n=1}^{N}\lvert c_n\rvert^2.$$

This part is exactly like finite-dimensional linear algebra.

Completeness

The orthonormal family is complete if its closed linear span is the whole Hilbert space:

$$\overline{\operatorname{span}\{e_n:n\ge1\}} = \mathcal H.$$

Equivalently, the only vector orthogonal to every $e_n$ is the zero vector:

$$\langle e_n\vert\psi\rangle=0 \quad \text{for all }n \quad \Longrightarrow \quad \psi=0.$$

The closure is essential. Infinite sums usually converge as limits of partial sums, not as finite combinations.

Expansion Coefficients

If $\{e_n\}$ is complete and orthonormal, then every $\psi\in\mathcal H$ has coefficients

$$c_n = \langle e_n\vert\psi\rangle.$$

The expansion is

$$\psi = \sum_{n=1}^{\infty} c_n e_n,$$

meaning

$$\lim_{N\to\infty} \left\lVert \psi - \sum_{n=1}^{N}c_n e_n \right\rVert = 0.$$

This is convergence in Hilbert-space norm. It does not automatically mean pointwise convergence of representative functions.

Parseval and Inner Products

For a complete orthonormal basis, Parseval’s identity says

$$\lVert\psi\rVert^2 = \sum_{n=1}^{\infty} \lvert c_n\rvert^2, \qquad c_n=\langle e_n\vert\psi\rangle.$$

If

$$\phi = \sum_n d_n e_n, \qquad \psi = \sum_n c_n e_n,$$

then

$$\langle\phi\vert\psi\rangle = \sum_n d_n^*c_n.$$

This is the infinite-dimensional analogue of taking dot products of coordinate columns.

Partial-Sum Projectors

Define

$$P_N = \sum_{n=1}^{N} \lvert e_n\rangle\langle e_n\rvert.$$

Then $P_N\psi$ is the best approximation to $\psi$ inside the finite-dimensional subspace spanned by $e_1,\ldots,e_N$:

$$P_N\psi = \sum_{n=1}^{N} \langle e_n\vert\psi\rangle e_n.$$

Completeness means

$$P_N\psi\to\psi$$

for every $\psi\in\mathcal H$, with convergence in norm. This is sometimes called strong convergence of $P_N$ to the identity. It is weaker than saying the operators converge in operator norm.

Norm Convergence Versus Pointwise Convergence

For $L^2$ spaces, norm convergence means

$$\int \lvert f_N(x)-f(x)\rvert^2\,dx \to0.$$

It does not say that $f_N(x)\to f(x)$ at every point. Nor does it say that the series can be differentiated term by term.

This distinction matters in quantum mechanics because probabilities depend on integrals of squared moduli. A basis expansion can converge perfectly well as a state expansion even when the pointwise behavior of the representative functions is delicate.

Fourier Example

On an interval of length $L$ with periodic boundary conditions, the functions

$$e_n(x) = \frac{1}{\sqrt L} e^{2\pi i n x/L}, \qquad n\in\mathbb Z,$$

form a complete orthonormal basis of $L^2([0,L])$ with periodic identification.

For $f\in L^2([0,L])$,

$$f(x) \sim \sum_{n\in\mathbb Z} c_n e_n(x), \qquad c_n = \int_0^L e_n(x)^*f(x)\,dx.$$

The symbol $\sim$ is a useful reminder: the expansion converges to $f$ in $L^2$ norm. Extra hypotheses are needed for stronger pointwise statements. The finite-interval expansion is developed in Periodic Functions and Fourier Series, while full-line transform conventions are collected in Fourier Transform.

Bound-State Example

For the infinite square well on $0\lt x\lt L$, the normalized eigenfunctions

$$u_n(x) = \sqrt{\frac{2}{L}} \sin\frac{n\pi x}{L}, \qquad n=1,2,\ldots,$$

form a complete orthonormal system in $L^2(0,L)$ for functions satisfying the corresponding Dirichlet boundary behavior in the standard Sturm–Liouville setting.

A normalizable state in the well can be expanded as

$$\psi(x) = \sum_{n=1}^{\infty} c_n u_n(x), \qquad c_n = \int_0^L u_n(x)^*\psi(x)\,dx,$$

with convergence in $L^2$ norm. The physical model is Infinite Square Well, and the mathematical background is Sturm–Liouville Theory.

Discrete Bases Versus Generalized Bases

A countable complete orthonormal basis consists of Hilbert-space vectors. Position kets $\lvert x\rangle$ and momentum kets $\lvert p\rangle$ are not normalizable Hilbert-space vectors, so they are not orthonormal bases in this strict sense.

Physicists still write formal resolution relations such as

$$I = \int \lvert x\rangle\langle x\rvert\,dx,$$

but these belong to the language of generalized eigenvectors and distributions. The practical position-momentum representation is introduced in Position and Momentum Representations.

Common Mistakes

• Treating “complete” as a decorative word rather than a statement about closed spans.
• Assuming norm convergence implies pointwise convergence.
• Differentiating or multiplying an infinite expansion without checking convergence and domains.
• Confusing a countable orthonormal basis with a continuum of generalized eigenvectors.
• Forgetting boundary conditions when choosing basis functions for a differential Hamiltonian.
• Assuming that a set of mutually orthogonal functions is complete without proof or a theorem.
• Treating a truncated expansion as exact without estimating the discarded norm.

Cross-Links

• Hilbert Spaces
• L2 Spaces
• Separable Hilbert Spaces
• Orthonormal Bases
• Periodic Functions and Fourier Series
• Fourier Transform
• Position and Momentum Representations
• Sturm–Liouville Theory
• Bases and Representations
• Infinite Square Well

References

• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1980.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Let $\{e_n\}$ be a complete orthonormal basis and let $c_n=\langle e_n\vert\psi\rangle$. What does

$$\psi = \sum_{n=1}^{\infty}c_n e_n$$

mean?

Solution

It means convergence in Hilbert-space norm:

$$\lim_{N\to\infty} \left\lVert \psi - \sum_{n=1}^{N}c_n e_n \right\rVert = 0.$$

It does not automatically assert pointwise convergence of functions.

2. Suppose $\{e_n\}$ is complete orthonormal and $\langle e_n\vert\psi\rangle=0$ for every $n$. What is $\psi$?

Solution

Completeness implies that the only vector orthogonal to every basis vector is the zero vector. Therefore $\psi=0$.

3. If $\psi=\sum_n c_n e_n$ in a complete orthonormal basis and $\lVert\psi\rVert=1$, what does Parseval’s identity give?

Solution

Parseval’s identity gives

$$1 = \lVert\psi\rVert^2 = \sum_n \lvert c_n\rvert^2.$$

In quantum mechanics, these squared moduli become probabilities when the basis corresponds to a projective measurement context.

4. Why are position kets not a countable orthonormal basis of $L^2(\mathbb R)$?

Solution

The objects $\lvert x\rangle$ are generalized eigenvectors, not normalizable Hilbert-space vectors. They are labeled by a continuum and satisfy delta-normalization formally. A countable orthonormal basis of a Hilbert space consists of actual Hilbert-space vectors with finite norm.