## Position Space and Momentum Space

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 Fourier Transforms in terms of and .  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 definite position is 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