##

The Matrix Representation of Operators and Wavefunctions

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
.

Define the **components of a state vector**
(analogous to
).

The wavefunctions are therefore represented as **vectors.**Define the **matrix element**

We know that an **operator acting on a wavefunction** gives a wavefunction.

If we dot
into this equation from the left, we get

This is exactly the formula for a state vector equals a **matrix operator** times a state vector.

Similarly, we can look at the **product of two operators**
(using the identity
).

This is exactly the formula for the product of two matrices.

**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.

The Hermitian Conjugate matrix is the (complex) **conjugate transpose.**
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

We can write this in **dot product in matrix notation** as

The bra vector is the conjugate transpose of the ket vector.
The both represent the same state but are different mathematical objects.

Jim Branson
2013-04-22