We can represent a state with either or with . We can (Fourier) transform from one to the other.
We have the symmetric Fourier Transform.
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
Our Fourier Transform can now be read to say that we add up states of definite momentum 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
The arbitrary state represented by either
or
,
might be written simple as
The actual wave function
would be written as
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