## Matrix Representation of Operators and States

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

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

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.

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