Matrix Representation of Operators and States

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

\psi_i & \equiv & \langle u_i\vert\psi\rangle \\
\vert\psi\rangle & = & \sum\limits_i\psi_i\vert u_i\rangle

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

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

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.

\begin{displaymath}\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\end{displaymath}

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

Jim Branson 2013-04-22