QMGreensite_merged

340 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

1. Beginwithasystemwithasetofcoordinatesandmomenta{qa,pa},whosedy-
   namicsisdescribedbyaHamiltonianH=H(p,q).


2. Physicalstatesbecomevectors|ψ>inaHilbertspace.


3. AllobservablesareassociatedwithHermitianoperatorsactingonvectorsinHilbert
   space. Inparticular,thecoordinatesqaandmomentapacorrespondtoopera-
   tors
   qa→Qa pa→Pa (21.217)

whichsatisycommutationrelations

[Qa,Qb] = 0

[Pa,Pb] = 0

[Qa,Pb] = ih ̄δabI (21.218)

4. PhysicalstatesevolveintimeaccordingtotheSchrodingerequation

ih ̄|ψ>=H[P,Q]|ψ> (21.219)

5. TheexpectationvalueofanobservableO(p,q),forasysteminstate|ψ>,
   isgivenby=<ψ|O[P,Q]|ψ> (21.220)
   

Expressedinthisway,thepassagefromclassicaltoquantummechanicsissimple:
itsjustaquestion ofimposingcommutationrelationsbetweenQ andP, andpos-
tulatingtheSchrodingerequation. Innoneofthesestepsisthereanycommittment
toanyparticularbasisinHilbertspace. Practicalcalculations,however,areusually
carriedoutintheSchrodingerrepresentation. Thereasonissimple:themomentum
operatorP appearsonlyquadraticallyinthe Hamiltonian,but thecoordinatesQ
generallyappear(inthepotentialV(Q))inmorecomplicatedways. Itthenmakes
sensetoworkinabasiswhichareeigenstatesofthecomplicatedpart(V(Q))ofthe
Hamiltonian.TheSchrodingerrepresentation(orQ-representation)isdefinedbythe
basisofQ-eigenstates

Qa|{q}>=qa|{q}> (21.221)

ThewavefunctioninSchrodingerrepresentationis

ψ(q)=<{q}|ψ> (21.222)