QMGreensite_merged

80 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

Itisusefultonormalizeφn(x),sothatthesolutiontothetime-dependentSchrodinger
equation
ψn(x,t)=φn(x)e−iEnt/ ̄h (5.114)

isaphysicalstate. Thenormalizationconditionis

1 =

∫∞

−∞

dxψ∗n(x,t)ψn(x,t)

= |N|^2

∫L

0

dxsin^2

[

nπx

L

]

= |N|^2

L

2

(5.115)

Thisdeterminesthenormalizationconstant

N=

√

2

L

(5.116)

sothat


φn(x)=

√

2

L

sin[

nπx

L

], En=

n^2 π^2 h ̄^2

2 mL^2





, n=^1 ,^2 ,^3 ,... (5.117)

isthe complete set of energyeigenvaluesand energy eigenfunctions (inthe inter-
val[0,L]). Thenthegeneral solutionofthetime-dependentSchrodingerequation,
accordingtoeq. (5.38)is

ψ(x,t) =

∑∞

n=1

anφn(x)e−iEnt/ ̄h

=

√

2

L

∑∞

n=1

ansin[

nπx

L

]e−iEnt/ ̄h (5.118)

Nowsupposethatthewavefunctionψ(x,t)isspecifiedataninitialtimet=0.
Howdowefind thewavefunctionatanylatertimet>0,usingeq. (5.118)? The
problemistofindthesetofconstants{an}givenψ(x,0).
Themethodforfindingthean,fortheparticleinatube,iscloselyanalogousto
solvingforf(p)forthefreeparticleineq.(5.44),givenψ(x,0).Beginwith

ψ(x,0)=

∑∞

n=1

anφn(x) (5.119)

andmultiplybothsidesoftheequationbyφ∗k(x)^3

φ∗k(x)ψ(x,0)=

∑∞

n=1

anφ∗k(x)φn(x) (5.120)

(^3) Sinceφnisarealfunction,complexconjugationdoesnothing. Itisindicatedherebecausein
general,whentheenergyeigenstatesarecomplex,thisoperationisnecessary.