10.1. THEFREEPARTICLE,ANDMOMENTUMCONSERVATION 155
Ifwerelabelthecoordinatesx→x′,
H ̃[−i ̄h ∂
∂x′,x′]φα(x′)=Eαφα(x′) (10.9)andthenusetheinvarianceoftheHamiltonianunderx′=f(x),wefind
H ̃[−i ̄h∂
∂x,x]φα(f(x))=Eαφα(f(x)) (10.10)whichprovesthatthetransformedwavefunction
φ′(x)=φα(f(x)) (10.11)isalsoanenergyeigenstate,withenergyEα.
Thereasonthatthesymmetries oftheHamiltonian areimportantisthatthey
areinvariablyassociatedwithconservationlaws;i.e. theexpectationvalueofsome
observableQ(differentfromtheenergy)isconstantintime
d
dt= 0 (10.12)
For example, symmetryofthe Hamiltonian withrespect to arbitrarytranslations
(10.2)isassociatedwiththeconservationofmomentum;andsymmetryoftheHamil-
tonianwithrespecttorotationsbyanarbitraryangle(10.5)isassociatedwiththe
conservationofangularmomentum. Itisalwayspossibletochoosethesetofenergy
eigenstates {φα}to beeigenstates notonly ofthe Hamiltonian,but alsoof some
subsetoftheconservedobservables,andthevaluesoftheseobservablescanbeused
todistinguishbetweendifferentenergyeigenstateswhichmayhavethesameenergy
eigenvalue.ThislecturewillbedevotedtosomeexamplesofsymmetricHamiltonians,
andtheircorrespondingconservationlaws.
10.1 TheFreeParticle,andMomentumConserva-
tion
ThefreeparticleHamiltonianinonedimensionisinvariantunderarbitrarytransla-
tions
x′=x+a (10.13)
becauseV(x′)=V(x)= 0 everywhere,andbecause
∂
∂x′=
∂
∂x(10.14)
LetusdefinealinearoperatorT,whichactsonfunctionsbytransformingthecoor-
dinate
TF(x)=F(x′)=F(x+a) (10.15)