# Numerically solving 2D “time-dependent Schrödinger equation” in MATLAB

I need to numerically solve the following second-order ODE in MATLAB

$$2ik\frac{\partial U(\vec{\rho},z)}{\partial z}+\left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right]U(\vec{\rho},z)=0$$

which looks a lot like a 2-D time-dependent Schrödinger equation but with time replaced by $$z$$. Here, $$\rho$$ is the planar position vector. Any recommendations on which numerical technique to use would be helpful.

• Your spatial domain completely determines the adopted numerical method. Describe it please. – HBR Dec 25 '18 at 12:32
• For my project's purpose, I was going to use a box with z ranging from 0 to 1000km, and x and y ranging from 0 to 100km. The initial conditions will be known for the entire base of the box, i.e. U(x, y, 0) is known. – Cédric Arseneau Dec 27 '18 at 13:55