We will define our vectors and matrices using a
complete set of, orthonormal basis states
, usually the set of eigenfunctions of a Hermitian operator.
These basis states are analogous to the orthonormal unit vectors in Euclidean space
Similarly, we can look at the product of two operators
(using the identity
So, wave functions are represented by vectors and operators by matrices, all in the space of orthonormal functions.
The Harmonic Oscillator Hamiltonian Matrix.*
* Example: The harmonic oscillator raising operator.*
* Example: The harmonic oscillator lowering operator.*
Now compute the matrix for the
of an operator.
Check that this is true for and .
We know that there is a difference between a bra vector and a ket vector.
This becomes explicit in the matrix representation.
then, the dot product is
Jim Branson 2013-04-22