Quantum Mechanics for Mathematicians

• The linear functional that takes a function to its Fourier transform atk:

f∈S(R)→f ̃(k)

• The linear functional that takes a function to its value atq:

f∈S(R)→f(q)

• The linear functional that takes a function to the value of its derivative
  atq:
  f∈S(R)→f′(q)

We would like to think of these as “generalized functions”, corresponding toTψ
given by the integral in equation 11.8, for someψwhich is a generalization of a
function.
From the formula 11.3 for the Fourier transform we have

f ̃(k) =T√ 1

2 πe−ikq

[f]

so the first of the above linear functionals corresponds to

ψ(q) =

1

√

2 π

e−ikq

which is a function, but “generalized” in the sense that it is not inS(R) (or
even inL^2 (R)). This is an eigenfunction for the operatorP, and we see that
such eigenfunctions, while not inS(R) orL^2 (R), do have a meaning as elements
ofS′(R).
The second linear functional described above can be written asTδwith the
corresponding generalized function the “δ-function”, denoted by the symbol
δ(q−q′), which is taken to have the property that

∫+∞

−∞

δ(q−q′)f(q′)dq′=f(q)

δ(q−q′) is manipulated in some ways like a function, although such a function
does not exist. It can however be made sense of as a limit of actual functions.
Consider the limit as→0 of functions

g=

1

√

2 π

e−

(q− 2 q′)^2

These satisfy ∫
+∞

−∞

g(q′)dq′= 1

for all >0 (using equation 11.6).