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Chemistry LibreTexts

Physical Observables

Physical observables are represented by linear, hermitian operators that act on the vectors of the Hilbert space. If \(A\) is such an operator, and \(\vert\phi\rangle\) is an arbitrary vector in the Hilbert space, then \(A\) might act on \(\vert\phi\rangle\) to produce a vector \(\vert\phi ' \rangle\), which we express as

\[A\vert\phi\rangle = \vert\phi'\rangle\]

Since \(\vert\phi\rangle\)   is representable as a column vector, \(A\) is representable as a matrix with components

\[A = \left(\matrix{A_{11} & A_{12} & A_{13} & \cdots \cr A_{21} & A_{22} & A_{23} & \cdots \cr\cdot & \cdot & \cdot & \cdots }\right)\]

The condition that \(A\) must be hermitian means that

\[A^{\dagger} = A\]


\[A_{ij} = A_{ji}^*\]