5.7. THEPARTICLEINACLOSEDTUBE 83
Therealpartof ψ(x,0) issketched inFig. [5.4]. Then thecoefficientsanare
easilycalculated:
an =√
2
L∫x 0 +ax 0 −adxeipx/ ̄hsin[nπx
L]
=
√
2
L
expi[p
̄h−nπ
L]
x 0
p
̄h−nπ
Lcos[
p
̄h−
nπ
L]
a−
expi[
p
̄h+nπ
L]
x 0
p
̄h+nπ
Lcos[p̄h+
nπ
L]
a
(5.133)Nowcollectingformulas:
ψ(x,t) =√
2
L∑∞n=1ansin[nπx
L]e−iEnt/ ̄h<x> =∑∞i=1∑∞j=1a∗iajei(Ei−Ej)t/ ̄hXij<p> =∑∞i=1∑∞j=1a∗iajei(Ei−Ej)t/ ̄hPijEn = n^2π^2 ̄h^2
2 mL^2(5.134)
Fromthissetofequationswecancalculateψ(x,t), atanytimetto andalsochoose Thecorrespondingwavefunction,calculatedatvariousinstants,isshowninFig.[5.5].
anydesiredaccuracy,bykeepingasufficientlylargenumberoftermsinthesums.
Nowletspluginsome numbers. Supposetheparticle isanelectron, p 0 isthe
momentumcorrespondingtoanenergyof 10 eV,
p 0 = 1. 7 × 10 −^24 kg-m/s (5.135)
a = 10 −^8 m
L = 10 −^7 m (5.136)
Theexpectationvaluesofpositionandmomentumareplottedasafunctionoftime
inFig.[5.6].Theseshouldbecomparedtothecorrespondingfiguresforx(t)andp(t)
foraclassicalpointlikeelectronofmomentump 0 ,initiallylocatedatthepointx 0.
Notethatthewavefunctionfortheelectron”bounces”fromoneendofthewalltothe
otherwhileslowlyspreadingout.Theexpectationvaluesofpositionandmomentum,
asexpected,closelyparalleltheclassicalbehavior.