82 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE
wherewehavedefined
Xmn ≡∫L0φ∗m(x)xφn(x)=
2
L
∫L0dxxsin[mπx
L]sin[nπx
L]
=
1
2 L m=n
0 m−neven
2 L
π^2 [(m+n)− (^2) −(m−n)− (^2) ] m−nodd
(5.127)
Similarly,formomentum
<p> =∫
dxψ∗(x,t)p ̃ψ(x,t)=
∫
dx{∞
∑
i=1aiφi(x)e−iEit/ ̄h}∗(
−i ̄h∂
∂x)∞
∑
j=1ajφj(x)e−iEjt/ ̄h=
∑∞i=1∑∞j=1a∗iajei(Ei−Ej)t/h ̄Pij (5.128)where
Pmn ≡∫L0φ∗m(x)(
−i ̄h∂
∂x)
φn(x)= −i ̄h2 πn
L^2∫L0dxsin[mπx
L]cos[nπx
L]
=
{
−iL ̄hm^42 mn−n 2 m−nodd
0 m−neven(5.129)
Example: TheStepFunctionWavepacket
Asanexampleoftheuseoftheseformulas,supposethataparticleisinitiallyin
thephysicalstate
ψ(x,0) =1
√
2 a{
eip^0 x/ ̄h x 0 −a<x<x 0 +a
0 otherwise
= Θ[a^2 −(x−x 0 )^2 ]eip^0 x/ ̄h (5.130)whereΘ(x)isthestepfunction
Θ(x)={
1 x≥ 0
0 x< 0(5.131)
Problem:Showthatthisstateisnormalized,andthatthepositionandexpectation
valuesattimet= 0 are
<x>=x 0 <p>=p 0 (5.132)