102 From Classical Mechanics to Quantum Field Theory. A Tutorial
define thek-axis position operatorXminL^2 (Rn,dnx) with domain
D(Xm):={
ψ∈L^2 (Rn,dnx)∣∣
∣∣
∫
Rn|xmψ(x)|^2 dkn}
and
(Xmψ)(x):=xmψ(x),x∈Rn. (2.35)Just by applying the definition of adjoint, one sees thatXm†=Xmso thatXmis
selfdjoint[ 5 ]. Again applying the definition of adjoint, one sees (see below) that
Xm′†=Xm′′†=Xmwhere we know that the last one is selfadjoint. By definition,
Xm′ andXm′′are therefore essentially selfadjoint. By (b) in proposition 2.2.28Xm′
andXm′′admit a unique selfadjoint extension which must coincide withXmitself.
We conclude thatC∞ 0 (Rn;C)andS(Rn)arecores(Def. 2.2.20) for them-axis
position operator.
Let us prove that forXm′†=Xm, the proof forXm′′†is identitcal. By direct
inspection, one easily sees thatXm′†⊂Xm. Let us prove the converse inclusion.
We have thatφ∈D(Xm′†) if and only if there existsηφ∈L^2 (Rn,dnx) such that
∫
φ(x)xmψ(x)dx=
∫
ηφ(x)ψ(x)dx,thatis∫
φ(x)xm−ηφ(x)ψ(x)dx= 0, for every
ψ ∈C 0 ∞(Rn). Fix a compactK ⊂Rn, obviouslyK x→φ(x)xm−ηφ(x)
isL^2 (K, dx). Since we canL^2 (K)-approximate that function with a sequence of
ψn∈C∞ 0 (Rn) such thatsupp(ψn)⊂K, we conclude thatKx→φ(x)xm−ηφ(x)
is zero a.e.. SinceKwas arbitrary, we conclude thatRnx→φ(x)xm=ηφ(x)
a.e. In particular, bothφandRnx→xmφ(x)areL^2 (Rn,dx) (the latter because
it is a.e. identical toηφ∈L^2 (Rn,dx)), namely,D(Xm′†)φimpliesφ∈D(Xm).
This proves thatD(Xm′†)⊂D(Xm) and consequentlyXm′†⊂Xmas wanted.
(2)Form∈{ 1 , 2 ,...,n},thek-axis momentum operator,Pm, is obtained
from the position operator using the Fourier-Plancherel unitary operatorFˆintro-
duced in example 2.2.32.
D(Pm):={
ψ∈L^2 (Rn,dnx)∣∣
∣∣
∫
Rn|km(Fˆψ)(k)|^2 dnk}
and
(Pmψ)(x):=(Fˆ†KmFˆψ)(x),x∈Rn. (2.36)Above Km is the m-axis position operator just written for functions (in
L^2 (Rn,dnk)) whose variable, for pure convenience, is denoted bykinstead of
x.SinceKmis selfadjoint,Pmis selfadjoint as well, as established in Proposition
2.2.30 as a consequence of the fact thatFˆis unitary.
It is possible to give a more explicit form toPmif restricting its domain. Taking
ψ∈C∞ 0 (Rn;C)⊂S(Rn) or directlyψ∈S(Rn),Fˆreduces to the standard integral