36 CHAPTER3. THEWAVE-LIKEBEHAVIOROFELECTRONS
3.1 Wave Equation for de Broglie Waves
Wherethereisawave,thereisawavefunction. Thewavefunctionψ(x,t)ofaplane
waveoffrequencyfandwavelengthλmaybegiven,e.g.,byasinwave
sin(kx−ωt) (3.1)
wherekisthewavenumber
k=
2 π
λ
=
p
̄h
(3.2)
andωistheangularfrequency
ω= 2 πf=
E
̄h
(3.3)
andwhere we have used thedeBroglie relationsto express λandf intermsof
electronmomentumpandenergyE. Thentheelectronwavefunctioncouldhavethe
formofasinwave
ψ(x,t)=sin(kx−ωt)=sin
(px−Et
̄h
)
(3.4)
oracosine
ψ(x,t)=cos(kx−ωt)=cos
(
px−Et
̄h
)
(3.5)
oranylinearcombinationofsinandcosine,includingthecomplexfunction
ψ(x,t)=ei(px−Et)/ ̄h (3.6)
Normallywewouldruleoutacomplexwavefunctionoftheform(3.6),onthegrounds
that, e.g., the displacement of avibrating string,or thestrength of electric and
magneticfieldsinaradiowave,orthepressurevariationinasoundwave,arestrictly
realquantitites.
Giventhewavefunction, whatis thewave equation? Waveson strings, sound
waves,andlightwaves,allsatisfywaveequationsoftheform
∂^2 ψ
∂t^2
=α
∂^2 ψ
∂x^2
(3.7)
whereα= 1 /v^2 isaconstant. However,ifweplugthesin-waveform(3.4)intothis
expressionwefind
E^2
̄h^2
sin
px−Et
̄h
=α
p^2
̄h^2
sin
px−Et
̄h
(3.8)
whichimplies
E^2 =αp^2 (3.9)