9.2: 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

$A\vert\phi\rangle = \vert\phi'\rangle \nonumber$

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

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

The condition that $$A$$ must be hermitian means that

$A^{\dagger} = A \nonumber$

or

$A_{ij} = A_{ji}^* \nonumber$

This page titled 9.2: Physical Observables is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark Tuckerman.