QMGreensite_merged

92 CHAPTER6. ENERGYANDUNCERTAINTY

6.3 The Energy of Energy Eigenstates

Wehaveseenthattheexpectationvalueofenergyisgivenby

<E>=

∫

dxψ∗(x,t)H ̃ψ(x,t) (6.37)

butofcourseaseriesofexperimentscangivemuchmoreinformationabouttheenergy
distributionthanjust.AccordingtoBohr’smodeloftheHydrogenatom,an
orbitingelectroncanbefoundtohaveonlycertaindefiniteenergies

En=−

(

me^4

2 ̄h^2

)

1

n^2

(6.38)

andtheremustbesometruthtothisidea,becauseitexplainsatomicspectrasowell.
WewouldliketousetheSchrodingerequationtocomputewhichenergiesarepossible
foranorbitingelectronorforanyothersystem,andtheprobabilitiesoffindingthese
energiesinameasurementprocess.
Afirstsignthat theenergies of boundstates maybe discretecomesfrom the
particleinatube,whereitwasfoundthattheenergyeigenvaluesare

En=n^2

π^2 ̄h^2

2 mL^2

(6.39)

Thequestion is: Inwhat way aretheenergy eigenvaluesof the time-independent
Schrodingerequationrelatedtotheenergiesthatmightactuallybefoundinapartic-
ularmeasurement? Wenowshow,usingeq. (6.37),andtheorthogonalityproperty
(seebelow)oftheenergyeigenstates

<φn|φm>= 0 if En+=Em (6.40)

thatthesetofenergyeigenvaluescoincideswiththesetofenergiesthatcanbefound
bymeasurement,andalsoshowhowtocomputetheprobabilityassociatedwitheach
energy.
Forsimplicity,assumethattheeigenvaluesarediscreteandnon-degenerate,which
meansthatnotwoeigenvaluesareequal.Toprovetheorthogonalityproperty(6.40),
beginbyconsideringthequantity

Hmn=

∫

dxφ∗m(x)H ̃φn(x) (6.41)

Fromthetime-independentSchrodingerequation

H ̃φk(x)=Ekφk(x) (6.42)

thisbecomes

Hmn = En

∫

dxφ∗m(x)φn(x)

= En<φm|φn> (6.43)