# Does Schrödinger equation have dual-property with Heat equation?

I have experimental data that Schödinger equation maintains high frequencies, while heat equation low.

Does Schrödinger equation have some duality property with heat equation?

• Are you referring to FT duality property? Oct 9 '13 at 11:58
• Both equations are of the type $\partial_tf=C\partial_x^2f$, but that's about where the similarities end. Oct 9 '13 at 14:08
• Heat equation is a diffusion equation so I dont get what you mean by "maintain frequency" Oct 11 '13 at 7:22
• The only thing which happen seems to be that heat equation and Schrödinger equation change to be each other, when time or frequency from real to imaginary or vice versa. Oct 11 '13 at 8:30
• @Masi the technical term for that is Wick rotation. (Unfortunately I don't know much about Wick rotation, and the Wikipedia page isn't very useful, but maybe someone with a bit more knowledge can post an answer detailing the correspondence between the two.) Oct 11 '13 at 9:04

The heat equation is of the form $\partial_t f = \nabla^2 f$ so eigenfunctions of $\nabla^2$ decay exponentially with time, and the higher the spatial frequency (eigenvalue), the faster the decay. Therefore low spatial frequencies decay more slowly.
There are lots of what seem to be deep connections between the two equations, however. These are best studied from the point of view of the path integral and the partition function. The Schödinger equation is what happens when a system samples all paths weighted by a phase factor $e^{-\frac{i}{\hbar} S}$. The heat equation is what happens when a system samples all microstates weighted by a factor that leads to maximum entropy.
Both equations are of the type $\partial_{t} f = C \partial_{x}^{2} f$. They are related by Wick rotation. However, there is a slight difference. Statistical mechanics n-point functions satisfy positivity whereas Wick-rotated quantum field theories satisfy reflection positivity. TODO more info about this is needed.