QMGreensite_merged

7.2. OPERATORSANDOBSERVABLES 99

Intermsofφx 0 , φp 0 , φn,theprobabilitiescanberewrittenasfollows:

Pdx(x 0 ) = |<φx 0 |ψ>|^2 dx

Pdp(p 0 ) = |<φp 0 |ψ>|^2 dp

P(En) = |<φn|ψ>|^2 (7.8)

The(generalized)functionsφx 0 , φp 0 , φnallsharetwothingsincommon.First,they
allsatisfyEigenvalueEquations:

x ̃φx 0 (x) = x 0 φx 0 (x)

p ̃φp 0 (x) = p 0 φp 0 (x)

H ̃φn(x) = Enφn(x) (7.9)

Thefunctionsφx 0 , φp 0 ,φnareknownas”Eigenfunctions”or”Eigenstates”(the
termsaresynonymous),oftheoperatorsx ̃, p, ̃ H ̃respectively,andthecorresponding
numbersx 0 , p 0 , Enareknownas”Eigenvalues”. Secondly,theinnerproductsof
theseeigenstatesareeitherdelta-functionsorKroneckerdeltas,dependingonwhether
theeigenvalues belongtoacontinuous range(x 0 ,p 0 ∈[−∞,∞]),or adiscreteset
(En∈{Ek}):

<φx 1 |φx 2 > =

∫

dxδ(x−x 1 )δ(x−x 2 )

= δ(x 1 −x 2 )

<φp 1 |φp 2 > =

∫

dx

1

2 π ̄h

ei(p^2 −p^1 )x/ ̄h

= δ(p 1 −p 2 )

<φn|φm> = δnm (7.10)

Therelationshipthatexistsbetweenobservablesx,p,Handoperatorsx ̃,p, ̃H ̃,and
betweenprobabilitesandinnerproducts,generalizestoallotherobservablequantities
ofasystem.

7.2 Operators and Observables

AnObservableisanypropertyof asystemthat canbemeasured; e.g. position,
momentum,energy,angularmomentum,magneticmoment,andsoon. Inclassical
mechanics,allobservablesarefunctionsofthegeneralizedcoordinatesandmomenta
{qa,pa}, so aknowledge of the physical state impliesaknowledge of all possible
observablesofthesystem.
AnOperatorisaruleforchangingfunctionsintootherfunctions. Givenany
function|ψ >as input, an operatorspecifiesaunique function|ψ′ >as output.
Symbolically,
|ψ′>=O|ψ> or |Oψ> (7.11)