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
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