Problem set 1

Wave function of a particle moving in free space is given by, $\psi=e^{ikx}+2e^{-ikx}$. Find the energy of the particle.

one dimensional Schrödinger equation is $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V\psi=E\psi$
For a free particle $V=0$
$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi$
$\frac{d^2\psi}{dx^2}=-k^2\left(e^{ikx}+2e^{-ikx}\right)=-k^2\psi$
$E=\frac{\hbar^2k^2}{2m}$