Quantum Mechanics for Mathematicians

the momentum space representation, as opposed to the previous position space
representation. By the Plancherel theorem (11.2) these are unitarily equivalent
representations of the groupR, which acts in the position space case by transla-
tion byain the position variable, in the momentum space case by multiplication
by a phase factoreika.
In the momentum space representation, the eigenfunctions ofPare the dis-
tributionsδ(k−k′), with eigenvaluek′, and the expansion of a state in terms
of eigenvectors is

ψ ̃(k) =

∫+∞

−∞

δ(k−k′)ψ ̃(k′)dk′ (12.2)

The position operator is

Q=i

d

dk

which has eigenfunctions
1
√
2 π

e−ikq

′

and the expansion of a state in terms of eigenvectors ofQis just the Fourier
transform formula 11.3.

12.3 Dirac notation

In the Dirac bra-ket formalism, position and momentum eigenstates will be
denoted|q〉and|k〉respectively, with

Q|q〉=q|q〉, P|k〉=k|k〉

Arbitrary states|ψ〉can be thought of as determined by coefficients

〈q|ψ〉=ψ(q), 〈k|ψ〉=ψ ̃(k) (12.3)

with respect to either the|q〉or|k〉basis. The use of the bra-ket formalism
requires some care however since states like|q〉or|k〉are elements ofS′(R) that
do not correspond to any element ofS(R). Given elements|ψ〉inS(R), they
can be paired with elements ofS′(R) like〈q|and〈k|as in equation 12.3 to get
numbers. When working with states like|q′〉and|k′〉, one has to invoke and
properly interpret distributional relations such as

〈q|q′〉=δ(q−q′), 〈k|k′〉=δ(k−k′)

Equation 12.1 is written in Dirac notation as

|ψ〉=

∫∞

−∞

|q〉〈q|ψ〉dq

and 12.2 as

|ψ〉=

∫∞

−∞

|k〉〈k|ψ〉dk