QMGreensite_merged

94 CHAPTER6. ENERGYANDUNCERTAINTY

Nowrecalltheexpression(4.67),fromprobabilitytheory,fortheexpectationvalue
ofanyquantityQ
=

∑

n

QnP(Qn) (6.54)

where{Qn}arethesetofpossiblevaluesofthequantity,andP(Qn)istheprobability
thatthevalueQnwillbefoundbymeasurement.Comparing(6.53)to(6.54),wefind
that:

I.Thesetofenergiesthatcanbefoundbymeasurementisthesetofenergyeigenvalues
{En}ofthetimeindependentSchrodingerequation;and

II.Theprobabilitythatastateψ(x,t)willbefoundtohaveanenergyEnisgiven
bythesquaredmodulusofcoefficients

P(En)=|an|^2 (6.55)

Tocomputean intermsof thewavefunction ψ(x,t),we multiply bothsidesof
(6.52)byφ∗m(x),andintegrateoverx,andusetheorthogonalityproperty(6.51)
∫
dxφ∗m(x)ψ(x,t) =

∫

dxφ∗m(x)

∑

n

anφne−iEnt/ ̄h

<φm|ψ> =

∑

n

ane−iEnt/ ̄h<φm|φn>

=

∑

n

ane−iEnt/ ̄hδmn

= ame−iEmt/ ̄h (6.56)

Soweseethat
an(t)≡ane−iEnt/ ̄h=<φn|ψ> (6.57)

andtheprobabilityforfindingtheenergyEn,whenthesystemisinthephysicalstate
|ψ>,isgivenby
P(En)=|<φn|ψ>|^2 (6.58)
Inparticular,fortheparticleinatube,thelowestpossibleenergyfortheparticle
isthelowesteigenvalue

E 1 =

π^2 ̄h^2

2 mL^2

(6.59)

andthecorrespondingeigenstate

φ 1 (x)=

√

2

L

sin(

πx

L

) (6.60)

isknownasthe”groundstate.”Ingeneral,thegroundstateisthelowestenergy
stateofaquantumsystem(inquantumfieldtheory,itisalsoknownasthe”vacuum