In quantum mechanics you can denote the wave-function in the position or the momentum basis. Written in the momentum basis, it would look something like:
But momentum is a continuous variable and it varies from to .
Therefore changing to the integral representation we get that:
But is just the projection of the momentum vector on the wavefunction:
We are also aware from our knowledge of Fourier Transform* that the wave function written in momentum space is given as :
Comparing both the above equations if we take the momentum basis as , then:
We can perform a similar analysis by expanding the wavefunction about the position basis and get
** Where does the in the Fourier Transform come from ?
We know from Fourier transform is defined as follows:
Plugging in and rewriting the above equation we get,
We find that from
that the normalization constant is not but . Therefore,