Answer to Question #192888 in Quantum Mechanics for Krishnendu

Question #192888

The wave function Ψ(x, t) = A exp [i(k1x − ω1t)] + A exp [i(k2x − ω2t)] is a superposition of two free-

particle wave functions. Both k1 and k2 are positive.

(a) Show that this wave function satisfies the Schodinger equation for a free particle of mass m.

(b) Find and plot the probability distribution function for Ψ(x, 0).


1
Expert's answer
2021-05-14T09:42:35-0400

(a)

If particles can exhibit wave nature, they should be described by a function that satisfies a ”wave equation”.


For classical waves


2y(x,t)x2=1v22y(x,t)t2\dfrac{\partial^2y(x,t)}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial ^2y(x,t)}{\partial t^2}

f(x,t)=Acos(kxωt)+Bsin(kxωt)            ω=vkf(x,t)=A cos (kx − ωt) + B sin (kx − ωt)\space\space\space\space\space\space\space\space\space\space\space\space ω = vk


For free-particle waves


E=p22m               E=hf=ω             p=hλ=kE =\dfrac{p^2}{2m}\space\space \space \space \space \space \space \space \space \space \space \space \space \space \space E = hf = \hbar ω \space \space \space \space \space \space \space \space \space \space \space \space \space p=\dfrac hλ= \hbar k

ω=2k22m\hbar\omega=\dfrac{\hbar^2k^2}{2m}


The Schrodinger Equation for a free particle


2m2Ψ(x,t)x2=iΨ(x,t)t-\dfrac{\hbar}{2m}\dfrac{\partial^2\Psi(x,t)}{\partial x^2}=i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}


The Wave Function


Ψ(x,t)=Acos(kxωt)+iAsin(kxωt)=Aei(kxωt)Ψ(x, t) = A cos (kx − ωt) + iA sin (kx − ωt) = Ae^{i(kx−ωt)}


(b)

Ψ(x,t)=A[cos(k1xω1t)+sin(k1xω1t)]+A[cos(k2xω2t)+sin(k2xω2t)]\Psi(x,t)=A[cos(k_1x-\omega_1t)+sin(k_1x-\omega_1 t)]+A[cos(k_2x-\omega_2t)+sin(k_2x-\omega_2 t)]

Ψ(x,t)=2A2(1+cos((k1k2)x(ω1ω2)t))\Psi(x,t)=2|A|^2(1+cos((k_1-k_2)x-(\omega_1-\omega_2)t))

Ψ(x,0)=2A2(1+cos((k1k2)x))\Psi(x,0)=2|A|^2(1+cos((k_1-k_2)x))


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