From Classical Mechanics to Quantum Field Theory

82 From Classical Mechanics to Quantum Field Theory. A Tutorial

section). We just mention the fact that all these operators admitS(Rn) as common
invariant subspace included in their domains. Thereon

(Xkψ)(x)=xkψ(x), (Pkψ)(x)=−i

∂ψ(x)

∂xk ,ψ∈S(R

n) (2.19)

and so

〈Xkψ,φ〉=〈ψ,Xkφ〉, 〈Pkψ,φ〉=〈ψ,Pkφ〉 for allψ,φ∈S(Rn) , (2.20)

By direct inspection, one easily proves that thecanonical commutation relations
(CCR) hold when all the operators in the subsequent formulas are supposed to be
restricted toS(Rn)

[Xh,Pk]=iδhkI, [Xh,Xk]=0, [Ph,Pk]=0. (2.21)

We have introduced thecommutator[A, B]:=AB−BAof the operatorsAandB
generally with different domains, defined on a subspace where both compositions
ABandBAmakes sense,S(Rn) in the considered case. Assuming that (2.5) and
(2.8) are still valid forXkandPkreferring toψ∈S(Rn), (2.21) easily leads to
theHeisenberg uncertainty relations,

ΔXkψΔPkψ≥

2

, forψ∈S(Rn), ||ψ||=1. (2.22)

Exercise 2.1.14.Prove inequality (2.22) assuming (2.5) and (2.8).

Solution. Using (2.5), (2.8) and the Cauchy-Schwarz inequality, one easily
finds (we omit the indexkfor simplicity),

ΔXψΔPψ=||X′ψ||||P′ψ||≥|〈X′ψ,P′ψ〉|.

whereX′:=X−〈X〉ψIandP′:=X−〈X〉ψI. Next, notice that

|〈X′ψ,P′ψ〉|≥|Im〈X′ψ,P′ψ〉|=^1

2

|〈X′ψ,P′ψ〉−〈P′ψ,X′ψ〉|

Taking advantage from (2.20) and the definitions ofX′ andP′and exploiting
(2.21),

|〈X′ψ,P′ψ〉−〈P′ψ,X′ψ〉|=|〈ψ,(X′P′−P′X′)ψ〉|

=|〈ψ,(XP−PX)ψ〉|=|〈ψ,ψ〉|.