Orthonormal Bases

An orthonormal basis is a basis made of unit vectors that are mutually orthogonal. It is the coordinate system in which inner products, norms, projections, probabilities, and matrix elements take their cleanest finite-dimensional form.

The general idea of a basis is developed in Bases and Coordinates. This page focuses on the extra structure supplied by an inner product.

For infinite-dimensional Hilbert spaces, where expansions converge as limits and completeness must be stated carefully, see Completeness and Orthonormal Bases.

Definition

Let $\mathcal H$ be a finite-dimensional complex inner-product space. A basis

$$\mathcal B = \{\lvert e_1\rangle,\ldots,\lvert e_n\rangle\}$$

is orthonormal if

$$\langle e_i\vert e_j\rangle = \delta_{ij}.$$

This equation combines two statements:

$$\lVert e_i\rVert=1$$

for every $i$, and

$$\langle e_i\vert e_j\rangle=0 \qquad \text{when } i\ne j.$$

The word “orthogonal” means mutually perpendicular with respect to the inner product. The word “normal” means unit norm. “Orthonormal” means both.

Expansion Coefficients

If $\mathcal B$ is orthonormal, every vector $\lvert\psi\rangle\in\mathcal H$ has a unique expansion

$$\lvert\psi\rangle = \sum_i c_i\lvert e_i\rangle.$$

Taking the inner product with $\lvert e_j\rangle$ gives

$$\langle e_j\vert\psi\rangle = \sum_i c_i\langle e_j\vert e_i\rangle = \sum_i c_i\delta_{ji} = c_j.$$

Thus the coordinates in an orthonormal basis are

$$c_i = \langle e_i\vert\psi\rangle.$$

This formula is one of the main reasons orthonormal bases are so useful: no inverse Gram matrix is needed.

Completeness Relation

The identity operator can be written

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

This is the finite-dimensional completeness relation. Applying it to a vector gives

$$I\lvert\psi\rangle = \sum_i \lvert e_i\rangle \langle e_i\vert\psi\rangle = \lvert\psi\rangle.$$

Each term $\lvert e_i\rangle\langle e_i\rvert$ is the rank-one projector onto the direction $\lvert e_i\rangle$. The sum of all these mutually orthogonal projectors is the identity.

Norms and Inner Products in Components

If

$$\lvert\psi\rangle = \sum_i c_i\lvert e_i\rangle, \qquad \lvert\phi\rangle = \sum_i d_i\lvert e_i\rangle,$$

then

$$\langle\phi\vert\psi\rangle = \sum_i d_i^*c_i.$$

The norm becomes

$$\lVert\psi\rVert^2 = \sum_i \lvert c_i\rvert^2.$$

For a normalized quantum state,

$$\sum_i \lvert c_i\rvert^2=1.$$

This simple squared-modulus formula assumes an orthonormal basis. In a nonorthonormal basis, the Gram matrix appears.

Projection Onto a Subspace

If $\{\lvert u_1\rangle,\ldots,\lvert u_k\rangle\}$ is an orthonormal basis for a subspace $U\subset\mathcal H$, the orthogonal projector onto $U$ is

$$P_U = \sum_{a=1}^k \lvert u_a\rangle\langle u_a\rvert.$$

For any vector $\lvert\psi\rangle$, the projected vector is

$$P_U\lvert\psi\rangle = \sum_{a=1}^k \lvert u_a\rangle \langle u_a\vert\psi\rangle.$$

The squared norm $\lVert P_U\psi\rVert^2$ is the total weight of $\psi$ in the subspace $U$. This is the linear algebra behind degenerate projective measurements.

Change Between Orthonormal Bases

Two orthonormal bases of the same finite-dimensional Hilbert space are related by a unitary matrix. If

$$S_{ai} = \langle f_a\vert e_i\rangle$$

is the overlap matrix between old basis $\{\lvert e_i\rangle\}$ and new basis $\{\lvert f_a\rangle\}$, then

$$\sum_i S_{ai}S_{bi}^* = \delta_{ab}.$$

Thus

$$SS^\dagger=I.$$

The mathematical transformation law is developed in Change of Basis. The physics-facing change-of-basis workflow is developed in Change of Basis, while Unitary Operators explains norm and inner-product preservation.

Physical Meaning of Basis Choice

An orthonormal basis can represent a measurement context. If a state is expanded as

$$\lvert\psi\rangle = \sum_i c_i\lvert e_i\rangle,$$

and the measurement projects onto the basis vectors $\lvert e_i\rangle$, then

$$p(i)=\lvert c_i\rvert^2.$$

But “being a superposition” is basis-relative. A vector can be a single basis vector in one orthonormal basis and a superposition in another. The physical content lies in the state, the measurement, and their inner products, not in a component list alone.

For the Core Formalism interpretation, see Bases and Representations and Probability in Different Bases.

Worked Example

In $\mathbb C^2$, define

$$\lvert +\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ 1 \end{pmatrix}, \qquad \lvert -\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ -1 \end{pmatrix}.$$

They are normalized:

$$\langle+\vert+\rangle = 1, \qquad \langle-\vert-\rangle = 1,$$

and orthogonal:

$$\langle+\vert-\rangle = \frac12(1-1) = 0.$$

Thus $\{\lvert+\rangle,\lvert-\rangle\}$ is an orthonormal basis.

For

$$\lvert\psi\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix},$$

the expansion coefficients in this basis are

$$c_+ = \langle+\vert\psi\rangle = \frac{1}{\sqrt2}, \qquad c_- = \langle-\vert\psi\rangle = \frac{1}{\sqrt2}.$$

Therefore

$$\lvert\psi\rangle = \frac{1}{\sqrt2}\lvert+\rangle + \frac{1}{\sqrt2}\lvert-\rangle.$$

Common Mistakes

• Using $c_i=\langle e_i\vert\psi\rangle$ in a basis that is not orthonormal.
• Forgetting to check unit norms as well as orthogonality.
• Treating an orthonormal basis label as a physical outcome before the observable or measurement is specified.
• Confusing a passive basis change with a physical transformation of the state.
• Assuming that a continuous generalized basis works exactly like a finite orthonormal basis.
• Calling a state a superposition without saying relative to which basis.

Cross-Links

• Bases and Coordinates
• Dirac Notation as Linear Algebra
• Index Notation and Summation Conventions
• Finite-Dimensional Hilbert Spaces
• Change of Basis
• Projectors
• Unitary Operators
• Bases and Representations
• Change of Basis
• Probability in Different Bases

References

• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• G. Strang, Linear Algebra and Its Applications, 4th ed., Brooks/Cole, 2006.
• P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed., Springer, 1974.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Show that the vectors $(1,i)^T/\sqrt2$ and $(1,-i)^T/\sqrt2$ form an orthonormal basis of $\mathbb C^2$.

Solution

Each vector has squared norm

$$\frac{\lvert1\rvert^2+\lvert i\rvert^2}{2} = 1.$$

Their inner product is

$$\frac12 \begin{pmatrix} 1 & -i \end{pmatrix} \begin{pmatrix} 1\\ -i \end{pmatrix} = \frac12(1-1) = 0.$$

Two orthonormal vectors in $\mathbb C^2$ form an orthonormal basis.

2. Let $\{\lvert e_1\rangle,\lvert e_2\rangle,\lvert e_3\rangle\}$ be orthonormal and let $\lvert\psi\rangle=2\lvert e_1\rangle-i\lvert e_3\rangle$. Find $\lVert\psi\rVert^2$.

Solution

In an orthonormal basis, square the moduli of the coefficients:

$$\lVert\psi\rVert^2 = \lvert2\rvert^2+\lvert-i\rvert^2 = 5.$$

3. If $P=\lvert e_1\rangle\langle e_1\rvert+\lvert e_2\rangle\langle e_2\rvert$ for two orthonormal vectors, what subspace does $P$ project onto?

Solution

It projects onto the span of $\lvert e_1\rangle$ and $\lvert e_2\rangle$. Components orthogonal to both basis vectors are removed.