Exploring Quantum Physics - Week 1 Question 3

Problem:

What is the time-dependent Schrödinger equation for a particle in a potential $V = \frac{1}{2}m\omega^2x^2$ ?

Solution:

The Schrödinger equation with potential energy is summarized as:

$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2 \nabla^2}{2m}\psi + \frac{1}{2}m\omega^2x^2\psi$.

Note the potential energy term also have to multiply the wave function.