5.2. THESCHRODINGERWAVEEQUATION 63
Equation(5.8) shouldbeunderstoodas aprediction, giventhequantumstate
ψ(x,t),fortheexpectationvalueofmomentum. Next,takingthetime-derivativeof
,soastocomparetothesecondequationof(5.3),
∂t<p> = ∂t∫
dxψ∗(
−i ̄h∂
∂x)
ψ=
(
ih ̄
2 m)
(−i ̄h)∫
dx{−(∂^2
∂x^2ψ∗)∂
∂xψ+ψ∗∂
∂x∂^2
∂x^2ψ}=
(
ih ̄
2 m)
(−i ̄h)∫
dxψ∗{−∂^2
∂x^2∂
∂xψ+∂
∂x∂^2
∂x^2ψ}= 0 (5.9)weconcludethat∂t
=0,whichagreeswithEhrenfest’sprincipleforthecase
thatthepotentialV =0.
InthecasethatV +=0, theequationfordeBrogliewavesdoesnotsatisfythe
secondof equations(5.3). SothedeBrogliewaveequationmustbegeneralizedso
astoagreewithEhrenfest’sprincipleforanypotentialV(x).Thisgeneralizedwave
equationwasfoundbySchrodinger;itisascentraltoquantummechanicsasF=ma
istoclassicalmechanics.
5.2 The Schrodinger Wave Equation
ThequantummechanicallawofmotionfoundbySchrodinger,describingaparticle
ofmassmmovinginonedimensioninanarbitrarypotentialV(x),isknownas
TheSchrodingerWaveEquation
ih ̄
∂ψ
∂t
= −
h ̄
2
2 m
∂^2
∂x^2
ψ+V(x)ψ (5.10)
Itiseasytoverifythatthisequationdoes,infact,satisfyEhrenfest’sprinciple. Once
again,startingwiththefirstoftheEhrenfestequations(5.3),
<p> = m∂t<x>= m∫
dx{
∂ψ
∂t∗
xψ+ψ∗x∂ψ
∂t}
(5.11)