Quantum Mechanics for Mathematicians

Heuristically (ignoring problems of interchange of integrals that don’t make
sense), the Fourier inversion formula can be written as follows

ψ(q) =

1

√

2 π

∫+∞

−∞

eikqψ ̃(k)dk

=

1

√

2 π

∫+∞

−∞

eikq

(

1

√

2 π

∫+∞

−∞

e−ikq

′

ψ(q′)dq′

)

dk

=

1

2 π

∫+∞

−∞

(∫+∞

−∞

eik(q−q

′)

ψ(q′)dk

)

dq′

=

∫+∞

−∞

δ(q−q′)ψ(q′)dq′

Physicists interpret the above calculation as justifying the formula

δ(q−q′) =

1

2 π

∫+∞

−∞

eik(q−q

′)

dk

and then go on to consider the eigenvectors

|k〉=

1

√

2 π

eikq

of the momentum operator as satisfying a replacement for the Fourier series
orthonormality relation (equation 11.1), with theδ-function replacing theδnm:

〈k′|k〉=

∫+∞

−∞

(

1

√

2 π

eik′q

)(

1

√

2 π

eikq

)

dq=

1

2 π

∫+∞

−∞

ei(k−k

′)q

dq=δ(k−k′)

11.4 Linear transformations and distributions

The definition of distributions as linear functionals on the vector spaceS(R)
means that for any linear transformationAacting onS(R), we can get a linear
transformation onS′(R) as the transpose ofA(see equation 4.1), which takes
Tto
AtT:f∈S(R)→(AtT)[f] =T[Af]∈C

This gives a definition of the Fourier transform onS′(R) as

FtT[f]≡T[Ff]

and one can show that, as forS(R) andL^2 (R), the Fourier transform provides
an isomorphism ofS′(R) with itself. Identifying functionsψwith distributions