Quantum Mechanics for Mathematicians

Tψ, one has

FtTψ[f]≡Tψ[Ff]

=Tψ

[

1

√

2 π

∫+∞

−∞

e−ikqf(q)dq

]

=

1

√

2 π

∫+∞

−∞

ψ(k)

(∫+∞

−∞

e−ikqf(q)dq

)

dk

=

∫+∞

−∞

(

1

√

2 π

∫+∞

−∞

e−ikqψ(k)dk

)

f(q)dq

=TFψ[f]

showing that the Fourier transform is compatible with this identification.
As an example, the Fourier transform of the distribution √^12 πeikais the

δ-functionδ(q−a) since

FtT√^1

2 πe

ika[f] =

1

√

2 π

∫+∞

−∞

eika

(

1

√

2 π

∫+∞

−∞

e−ikqf(q)dq

)

dk

=

∫+∞

−∞

(

1

2 π

∫+∞

−∞

e−ik(q−a)dk

)

f(q)dq

=

∫+∞

−∞

δ(q−a)f(q)dq

=Tδ(q−a)[f]
For another example of a linear transformation acting onS(R), consider the
translation action on functionsf→Aaf, where

(Aaf)(q) =f(q−a)

The transpose action on distributions is

AtaTψ(q)=Tψ(q+a)

since

AtaTψ(q)[f] =Tψ(q)[f(q−a)] =

∫+∞

−∞

ψ(q)f(q−a)dq=

∫+∞

−∞

ψ(q′+a)f(q′)dq′

The derivative is an infinitesimal version of this, and one sees (using inte-
gration by parts), that
(
d
dq

)t

Tψ(q)[f] =Tψ(q)

[

d

dq

f

]

=

∫+∞

−∞

ψ(q)

d

dq

f(q)dq

=

∫+∞

−∞

(

−

d

dq

ψ(q)

)

f(q)dq

=T−dqdψ(q)[f]