The Schrodinger equation is not a reaction-diffusion equation

Note: this post was originally written by David Ketcheson.

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!


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.

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