Schrödinger's Equation

Question

I have two questions concerning the Schroedinger equation. The Schroedinger equation is of the form \begin{equation} i \partial_t \Psi = -\Delta \Psi + V\Psi. \end{equation} My first question is, why does the $i$ appears in the equation? Secondly, in the mathematical literature (up to my knowledge) most of the time equations of the type \begin{equation} \partial_t \Phi = \Delta \Phi + f(\Phi) \end{equation} are studied. The $i$ does not appear in this form. Can we reduce the Schroedinger equation to this case somehow (or maybe reduce it to the wave equation instead, since the Schroedinger equation behaves more like a wave equation)? Or else, why do mathematicians mostly study this equation?

My thoughts: My first approach to this question was to expand $\Psi = u +iv$ for two real-valued functions $u$ and $v$. In this case, the Schroedinger equation is equivalent with the system \begin{equation} \begin{cases} \partial_t u = -\Delta v + Vv\\ \partial_t v = \Delta u - Vu \end{cases} \end{equation} However, I don't see how to proceed from here.

Best, Luke

Answer 1

Consider $\dot{x} = i x$ and $\dot{x} = x$. They are similar, but the behavior of the solution $x(t)$, where $t$ is real, is different (a bounded oscillatory function VS an unbounded exponential). So the presence of the $i$ makes a big difference (oscillation VS growth/decay).

This is why the Schroedinger equation is different from, say, the heat equation. If you use an "imaginary time", then you can map the Shroedinger equation into the heat equation (this is known as "Wick rotation").

For extra clarity: you cannot map it into the wave equation because the wave equation is second order in time... the mapping provided by the Wick rotation is into the heat equation.

Answer 2

$$i \partial_t \Psi = H\Psi$$ has the formal solution $$\Psi(t) = e^{-iHt}\Psi_0 .$$

If $H$ is Hermitian, $e^{-iHt}$ is unitary so $\parallel \Psi(t) \parallel = \parallel \Psi_0 \parallel$, which is conservation of probality.