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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)