Orthonormality

Given a set of functions {fn}n∈N\{ f_{n} \}_{n\in \mathbb{N}} or vectors {vn}n∈N\{ v_{n} \}_{n\in \mathbb{N}} from a space equipped with a Scalar product, the set is said to be orthonormal if all of the elements are individually normalized and also orthogonal to each other. In other words, the scalar product between any two elements mm and nn must be the Kronecker delta:

(fm,fn)=δmn={1if m=n0if m≠n(f_{m},f_{n})=\delta_{mn}=\begin{cases} 1&\text{if }m=n \\ 0&\text{if }m\neq n \end{cases}

This definition applies to discrete sets, but it can be extended to also provide a similar form of "continuous orthonormality". Given a continuous set {fz}z∈R\{ f_{z} \}_{z\in \mathbb{R}} of functions or vectors, the scalar product between any two elements zz and ww must be the Delta di Dirac:

(fz,fw)=δ(z−w)={∞if z=w0if z≠w(f_{z},f_{w})=\delta(z-w)=\begin{cases} \infty&\text{if }z=w \\ 0&\text{if }z\neq w \end{cases}

This form of orthonormality is sometimes refereed to with a different name, such as Dirac orthonormality1, to distinguish it from the discrete case.

Footnotes

  1. This name is from Griffiths' Introduction to Quantum Mechanics, 2nd ed.. ↩