We can represent a state with either or with . We can (Fourier) transform from one to the other.

We have the symmetric Fourier Transform.

When we change variable from to , we get the

These formulas are worth a little study.
If we define
to be the **state with definite momentum**
, (in position space)
our formula for it is

Similarly, the state (in momentum space) with

These states cannot be normalized to 1 but they do have a normalization convention which is satisfied due to the constant shown.

Our Fourier Transform can now be read to say that we **add up states of definite momentum to get**

and we add up states of definite position to get .

There is a more abstract way to write these states.
Using the notation of Dirac, the state with definite momentum
,
might be written as

and the state with definite position , might be written

The arbitrary state represented by either
or
,
might be written simple as

The actual wave function
would be written as

This gives us the amplitude to be at for any value of .

We will find that there are other ways to represent Quantum states. This was a preview. We will spend more time on Dirac Bra-ket notation later.

Jim Branson 2013-04-22