Matrix Representation of Operators and States

We may define the components of a state vector $\psi$ as the projections of the state on a complete, orthonormal set of states, like the eigenfunctions of a Hermitian operator.

$$\begin{eqnarray*} \psi_i & \equiv & \langle u_i\vert\psi\rangle \\ \vert\psi\rangle & = & \sum\limits_i\psi_i\vert u_i\rangle \end{eqnarray*}$$

Similarly, we may define the matrix element of an operator in terms of a pair of those orthonormal basis states

$$\bgroup\color{black}O_{ij}\equiv \langle u_i\vert O\vert u_j\rangle.\egroup$$

With these definitions, Quantum Mechanics problems can be solved using the matrix representation operators and states. An operator acting on a state is a matrix times a vector.

$$\bgroup\color{black} \left(\matrix{(O\psi)_1\cr (O\psi)_2\cr ... ...matrix{\psi_1\cr \psi_2\cr ...\cr \psi_j\cr ...}\right) \egroup$$

The product of operators is the product of matrices. Operators which don't commute are represented by matrices that don't commute.