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,

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