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

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

The condition that $A$ must be hermitian means that

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

or

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