Trying to write a code to analyze a PDE of the Schrodinger Equation

these are the equations

I want to solve the pde and plot the \Psi(x,t)^{2} on a 3d time vs x plot and then i wanted to plot the expectation value of x vs time. I have been trying for weeks and just can't get it to work figure it out. Where x0,m,w,h,and sigma are setable variables