From Classical Mechanics to Quantum Field Theory

10 From Classical Mechanics to Quantum Field Theory. A Tutorial

More generally, one can define the mean value (and variance) ofOˆover a mixed
state by taking averages over the statistical mixture, i.e. by:

〈

Oˆ

〉

≡

∑N

j=1

pjTr

[

ρψjOˆ

]

=Tr

[

ρOˆ

]

. (1.13)

Example 1.2.2. Position and momentum.
Let us considerH=L^2 (R)={ψ(x)}. On this space we can define two impor-
tant operators, the position ˆxand the momentum ˆprespectively given by the
multiplication by the coordinate and the derivative with respect to it position:

xψˆ (x)=xψ(x), (1.14)

pψˆ (x)=−ıd

dx

ψ(x), (1.15)

which satisfy the so called canonical commutation relation (CCR):

[ˆx,pˆ]⊆ıI, (1.16)

where the inclusion comes from the fact that, as explained previously, the operators
q,pare unbounded. Notice that Eq. (1.16) looks the same as in the classical case,
if we replace the Poisson brackets with the Lie commutator. It is said that Eqs.
(1.14, 1.15) realize the CCR in thecoordinate representation.
Relation (1.16) implies that neither ˆxnor ˆpcan be realized as bounded opera-
tors: thus one needs to specify their domainsDx,Dp⊂H, on which they are (at
least essentially) self-adjoint. Usually, one choosesDx=Dp=S(R), the space of
Schwartz functions. The spectrum of both these operators is continuous and equal
toR, while the corresponding generalized eigenfunctions are respectively given by:

|x 0 〉=δ(x−x 0 ) with eigenvaluex 0 ∈R, (1.17)

|p 0 〉=eıp^0 x with eigenvaluep 0 ∈R. (1.18)

Let us recall that the Fourier transform:

ψ(p)≡√^1

2 π

∫

dx eipxψ(x) (1.19)

is a unitary operator onL^2 (R). Thus, we may work as well onH ̃={ψ(p)}(∼H),
where the ˆxand the ˆpoperators are now given by:

xψˆ (p)=−ıd

dp

ψ(p), (1.20)

pψˆ (p)=pψ(p). (1.21)

Eqs. (1.20, 1.21) define the so-calledmomentum representationof the CCR.