Editing Operator (physics) (section)

===Operators in matrix mechanics ===

An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a [[linear transformation]] (aka transition matrix) between bases. Each basis element <math>\phi_j </math> can be connected to another,<ref name="Quantum Mechanics Demystified 2006"/> by the expression:

:<math>A_{ij} = \left\langle \phi_i \left| \hat{A} \right| \phi_j \right\rangle,</math>

which is a matrix element:

:<math>\hat{A} = \begin{pmatrix}
  A_{11} & A_{12} & \cdots & A_{1n} \\
  A_{21} & A_{22} & \cdots & A_{2n} \\
  \vdots & \vdots & \ddots & \vdots \\
  A_{n1} & A_{n2} & \cdots & A_{nn} \\
\end{pmatrix}
</math>

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.<ref name="QUANTUM CHEMISRTY 1977"/> In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the [[characteristic polynomial]]:

:<math> \det\left( \hat{A} - a \hat{I} \right) = 0 ,</math>

where ''I'' is the ''n'' × ''n'' [[identity matrix]], as an operator it corresponds to the identity operator. For a discrete basis:

:<math> \hat{I} = \sum_i |\phi_i\rangle\langle\phi_i|</math>

while for a continuous basis:

:<math> \hat{I} = \int |\phi\rangle\langle\phi| \mathrm{d}\phi</math>