Quantum Mechanics for Mathematicians

In order to have the standard derivative when one identifies functions and dis-
tributions, one defines the derivative on distributions by

d

dq

T[f] =T

[

−

d

dq

f

]

This allows one to define derivatives of aδ-function, with for instance the first
derivativeδ′(q) ofδ(q) satisfying

Tδ′(q)[f] =−f′(0)

11.5 Solutions of the Schr ̈odinger equation in momentum space

Equation 11.7 shows that under Fourier transformation the derivative opera-
tordqd becomes the multiplication operatorik, and this property will extend
to distributions. The Fourier transform takes constant coefficient differential
equations inqto polynomial equations ink, which can often much more readily
be solved, including the possibility of solutions that are distributions. The free
particle Schr ̈odinger equation

i

∂

∂t

ψ(q,t) =−

1

2 m

∂^2

∂q^2

ψ(q,t)

becomes after Fourier transformation in theqvariable the simple ordinary dif-
ferential equation

i

d

dt

ψ ̃(k,t) =^1

2 m

k^2 ψ ̃(k,t)

with solutions
ψ ̃(k,t) =e−i^21 mk^2 tψ ̃(k,0)
Solutions that are momentum and energy eigenstates will be distributions,
with initial value
ψ ̃(k,0) =δ(k−k′)

These will have momentumk′and energyE=k

′ 2
2 m. The space of solutions can
be identified with the space of initial value dataψ ̃(k,0), which can be taken to
be inS(R),L^2 (R) orS′(R).
Instead of working with time-dependent momentum space solutionsψ ̃(k,t),
one can Fourier transform in the time variable, defining

ψ̂(k,ω) =√^1

2 π

∫∞

−∞

eiωtψ ̃(k,t)dt

Just as the Fourier transform inqtakes dqd to multiplication byik, here the

Fourier transform inttakesdtd to multiplication by−iω. Note the opposite
sign convention in the phase factor from the spatial Fourier transform, chosen