Matrix elements of an operator

The matrix elements of an operator are the entries of the matrix representation of the operator.

Consider a linear map from a vector space \vert\psi\rangle to the same vector space, i.e. \vert\phi\rangle=\hat{O}\vert\psi\rangle, where and the orthonormal basis states \vert\varphi_n\rangle span . The matrix representation of the equation is

\begin{pmatrix} \phi_1\\\phi_2 \\ \vdots \end{pmatrix}=\begin{pmatrix} O_{11} &O_{12} &\cdots \\ O_{21} &O_{22} &\cdots \\ \vdots & \vdots &\ddots \end{pmatrix}\begin{pmatrix} \psi_1\\\psi_2 \\ \vdots \end{pmatrix}

where \phi_m and \psi_n are the coefficients of the vectors \vert\phi\rangle and \vert\psi\rangle respectively.

The matrix elements of \vert\phi\rangle are given by

\phi_m=\sum_{n}O_{mn}\psi_n\; \; \; \; \; \; \; \; 32a

Since the orthonormal basis states \vert\varphi_n\rangle span , we have . So,

\vert\phi\rangle=\hat{O} \sum_{n}\vert\varphi_n\rangle\langle\varphi_n\vert\psi\rangle =\sum_{n}\hat{O}\vert\varphi_n\rangle\langle\varphi_n\vert\psi\rangle


Similarly, and so

Comparing eq33 with eq32a, O_{mn}=\langle\varphi_m\vert\hat{O}\vert\varphi_n\rangle. Therefore, O_{mn} are the matrix elements of \hat{O} with respect to the basis states of \vert\varphi_n\rangle.


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