QMGreensite_merged

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)