QMGreensite_merged

66 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

Finally, weneedtochecktheconsistency oftheSchrodingerequationwiththe
BornInterpretation. AccordingtotheBornInterpretation,atanyinstantoftimet,

<ψ|ψ>=

∫

dxψ∗(x,t)ψ(x,t)= 1 (5.22)

foraparticlemovinginonedimension. Nowsupposethisnormalizationconditionis
trueatsometimet 0 ,andandthewavefunctionevolvesaccordingtotheSchrodinger
equation. Willthisconditionalsobevalidatlatertimest>t 0? Tocheckthis,we
differentiate<ψ|ψ>withrespecttotime

∂t<ψ|ψ> =

∫

dx{(∂tψ∗)ψ+ψ∗(∂tψ)}

=

∫

dx

{(

−

i ̄h

2 m

∂x^2 ψ∗+

i

̄h

Vψ∗

)

ψ+ψ∗

(

i ̄h

2 m

∂^2 xψ−

i

̄h

Vψ

)}

=

i ̄h

2 m

∫

dx

{

−(∂x^2 ψ∗)ψ+ψ∗∂x^2 ψ

}

= 0 (5.23)

wherewehaveonceagainusedtheintegrationbypartsformula(5.7). Thisresult
meansthat the normofthe wavefunction isconstantintime, which isknownas
”ConservationofProbability”. Theinnerproduct<ψ|ψ>isthetotalprobabil-
itytofindtheparticleanywhereinspace. Ifthenormisconstant,and<ψ|ψ>= 1
atsometimet=t 0 ,then<ψ|ψ>= 1 atanyanylatertime.
TheSchrodingerequationformotionin3-dimensionsisastraightforwardgener-
alizationoftheone-dimensionalcase:

i ̄h

∂ψ

∂t

=

− ̄h^2

2 m

∇^2 ψ+V(x,y,z)ψ (5.24)

Inthiscase,thegeneralizedcoordinatesareq^1 =x, q^2 =y, q^3 =z,andthefirstof
theEhrenfestequations(5.2)tellsus

<px>=

∫

d^3 xψ∗(x,y,z,t)p ̃xψ(x,y,z,t)

<py>=

∫

d^3 xψ∗(x,y,z,t)p ̃yψ(x,y,z,t)

<pz>=

∫

d^3 xψ∗(x,y,z,t)p ̃zψ(x,y,z,t) (5.25)

where

p ̃x ≡ −i ̄h

∂

∂x

p ̃y ≡ −i ̄h

∂

∂y

p ̃z ≡ −i ̄h

∂

∂z

(5.26)