# The Schrodinger equation is not a reaction-diffusion equation

Recently, a stackexchange answer claimed that the Schrodinger equation is effectively a reaction-diffusion equation. I’ll set aside semantic arguments about the meaning of “effectively”, and give a more obvious example to explain why I think this statement is misleading.

Consider the wave equation

$u_{tt} = u_{xx}$

Introducing a new variable $$v=u_t$$ we can rewrite the wave equation as

\begin{align*} v_t & = u_{xx} \\ u_t & = v. \end{align*}

Observe that the first of these equation is the diffusion equation, while the second is a reaction equation. Thus we have reaction-diffusion!

Right?

Wrong. We’ve disguised the true nature of this equation by applying our intuition (which is based on scalar PDEs) to a system of PDEs. In the same way, the “reaction-diffusion” label for Schrodinger is obtained by applying intuition based on PDEs with real coefficients to a PDE with complex coefficients.

Of course, in both cases you can use numerical methods that are appropriate for reaction-diffusion problems in order to solve a wave equation.
Here is a quick ipython notebook implementation of the obvious method for the system above.