### Review of Operators

First, a little review. Recall that the square integrable functions form a vector space, much like the familiar 3D vector space.

in 3D space becomes

The scalar product is defined as

and many of its properties can be easily deduced from the integral.

As in 3D space,

the magnitude of the dot product is limited by the magnitude of the vectors.

This is called the Schwartz inequality.

Operators are associative but not commutative.

An operator transforms one vector into another vector.

Eigenfunctions of Hermitian operators

form an orthonormal

complete set

Note that we can simply describe the eigenstate at .

Expanding the vectors and ,

we can take the dot product by multiplying the components, as in 3D space.

The expansion in energy eigenfunctions is a very nice way to do the time development of a wave function.

The basis of definite momentum states is not in the vector space, yet we can use this basis to form any state in the vector space.

Any of these amplitudes can be used to define the state.

Jim Branson 2013-04-22