An operator matrix element

The matrix element of the operator Aαβ is defined as & # x27E8; α | A ^ | β & # x27E9; {\displaystyle \langle \alpha |{\hat {A}}|\beta \rangle } . The above can be interpreted in two ways:

Vector scalar product & # x27E8; α | {\displaystyle \langle \alpha |} and vector A ^ | β & # x27E9; {\displaystyle {\hat {A}}|\beta \rangle }

The integral of all n parameters of rk from which vectors depend, ie:

& # x27E8; α | A ^ | β & # x27E9; = & # x222B; & # x2212; & # x221E; & # x221E; α ( r ¯ ) A ^ β ( r ¯ ) d n r {\displaystyle \langle \alpha |{\hat {A}}|\beta \rangle =\int \limits _{-\infty }^{\infty }\alpha ({\bar {r}}){\hat {A}}\beta ({\bar {r}})d^{n}r}

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