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.
* Example:
The Harmonic Oscillator Hamiltonian Matrix.*
* Example:
The harmonic oscillator raising operator.*
* Example:
The harmonic oscillator lowering operator.*
Now compute the matrix for the
Hermitian Conjugate
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.
If
and
then, the dot product is
Jim Branson 2013-04-22