(a)
If particles can exhibit wave nature, they should be described by a function that satisfies a ”wave equation”.
For classical waves
∂x2∂2y(x,t)=v21∂t2∂2y(x,t)
f(x,t)=Acos(kx−ωt)+Bsin(kx−ωt) ω=vk
For free-particle waves
E=2mp2 E=hf=ℏω p=λh=ℏk
ℏω=2mℏ2k2
The Schrodinger Equation for a free particle
−2mℏ∂x2∂2Ψ(x,t)=iℏ∂t∂Ψ(x,t)
The Wave Function
Ψ(x,t)=Acos(kx−ωt)+iAsin(kx−ωt)=Aei(kx−ωt)
(b)
Ψ(x,t)=A[cos(k1x−ω1t)+sin(k1x−ω1t)]+A[cos(k2x−ω2t)+sin(k2x−ω2t)]
Ψ(x,t)=2∣A∣2(1+cos((k1−k2)x−(ω1−ω2)t))
Ψ(x,0)=2∣A∣2(1+cos((k1−k2)x))
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