Question

I believe I have a right, but I’m not sure, and I’m not sure how to do b

(a) Show that the general solution in one dimension for Ψ(z,t) given a snapshot of the wavefunction Ψ(x, to) at t to is given by $(z,t) = 늚 J Ψ(z',to)eik(z-n-i.it-to) dr.dk This gives Ψ(z,t) at the price of two integrations. Reduce it to a single integral by carrying out the integral over k. (b)

Answer 1

(b)

$\psi(x,t)=\frac {1}{2\pi}\int \psi(x',t_0)e^{-i\omega_k(t-t_0)}\left ( \int e^{ik(x-x_0)}dk \right )dx'$

$\psi(x,t)=\frac {1}{2\pi}\int \psi(x',t_0)e^{-i\omega_k(t-t_0)}\cdot 2\pi\cdot \delta (x-x') dx'$