5.3. THETIME-INDEPENDENTSCHRODINGEREQUATION 67
andonecanthenreadilyverifythe2ndEhrenfestequation
∂t<%p>=<−∇V > (5.27)
Intermsofthep ̃-operators,theSchrodingerequationin3-dimensionscanbewritten
ih ̄
∂ψ
∂t
=
[ 1
2 m
(p ̃x^2 +p ̃^2 y+p ̃z^2 )+V(x,y,z)
]
ψ
= H ̃ψ (5.28)
Problem: Verify conservation of probability andthe Ehrenfest equations inthe
three-dimensionalcase.
5.3 The Time-Independent Schrodinger Equation
WhenthepotentialV(x)istime-independentwecansimplifytheSchrodingerequa-
tionbythemethodofseparationofvariables.Write
ψ(x,t)=φ(x)T(t) (5.29)
andsubstituteintothe(one-dimensional)Schrodingerequation:
ih ̄φ(x)
∂T
∂t
=TH ̃φ (5.30)
dividebothsidesbyφT
i ̄h
1
T(t)
∂
∂t
T(t)=
1
φ(x)
H ̃φ(x) (5.31)
Sincethelhsdependsonlyont,andtherhsonlyonx,theonlywaythisequation
canbetrueisifbothsidesequalaconstant,callitE:
i ̄h
1
T(t)
∂
∂t
T(t) = E
1
φ(x)
H ̃φ(x) = E (5.32)
Thefirstofthesetwodifferentialequationscanbesolvedimmediately:
T(t)=e−iEt/ ̄h (5.33)
whilethesecondequation
H ̃φ=Eφ (5.34)