Answer to Question #192888 in Quantum Mechanics for Krishnendu

Question

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).

Answer 1

(a)

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

For classical waves

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

$f(x,t)=A cos (kx − ωt) + B sin (kx − ωt)\space\space\space\space\space\space\space\space\space\space\space\space ω = vk$

For free-particle waves

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

$\hbar\omega=\dfrac{\hbar^2k^2}{2m}$

The Schrodinger Equation for a free particle

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

The Wave Function

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

(b)

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

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

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