QMGreensite_merged

5.4. THEFREEPARTICLE 69

5.4 The Free Particle

WhenthepotentialV(x)vanisheseverywhere,theSchrodingerequationinonedi-
mensionreducestotheequationfordeBrogliewaves

i ̄h

∂ψ

∂t

=−

̄h^2

2 m

∂^2 ψ

∂x^2

(5.41)

andthisequationwasdeducedfromthewavefunctionforplanewaves,corresponding
toaparticlewithdefinitemomentumpandenergyEp=p^2 / 2 m

ψp(x,t) = Nexp[i(px−Ept)/ ̄h]

= Nexp[i(px−

p^2

2 m

t/ ̄h] (5.42)

TheSchrodingerequation(whetherV = 0 ornot)isalinearequation,andanylinear
equationhasthepropertythat,ifψp 1 , ψp 1 , ψp 3 ,...areallsolutionsoftheequation,
thensoisanylinearcombination

ψ=cp 1 ψp 1 +cp 2 ψp 2 +cp 3 ψp 3 +... (5.43)

Inparticular,

ψ(x,t)=

∫∞

−∞

dp

2 π ̄h

f(p)exp[i(px−

p^2

2 m

t)/ ̄h] (5.44)

isalinearcombinationofplanewavesolutions(5.42). Thismeansthat(5.44)isa
solutionofthefree-particleSchrodingerequation(5.41),foranychoiceoffunction
f(p)whatsoever.

Problem:Showthatthewavefunctionineq. (5.44)isasolutionofthefreeparticle
Schrodingerequationbysubstituting(5.44)directlyinto(5.41).

Equation(5.44)isanexampleofthegeneralsolution(5.39)ofatime-dependent
Schrodingerequation. Thetime-independentSchrodingerequationinthiscaseis

−

h ̄^2

2 m

∂^2 φ

∂x^2

=Eφ (5.45)

ForeachpositiveeigenvalueE∈[0,∞]therearetwolinearlyindependentsolutions

φp(x)=eipx/ ̄h p=±

√

2 mE (5.46)

suchthatanyothersolutionforagivenEcanbeexpressedasalinearcombination

φ(x)=c 1 ei

√

2 mEx/ ̄h+c

2 e

−i

√

2 mEx/ ̄h (5.47)