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

Does Schrödinger equation have some duality property with heat equation?

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  • $\begingroup$ Are you referring to FT duality property? $\endgroup$
    – mcodesmart
    Oct 9 '13 at 11:58
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    $\begingroup$ Both equations are of the type $\partial_tf=C\partial_x^2f$, but that's about where the similarities end. $\endgroup$
    – Kyle Kanos
    Oct 9 '13 at 14:08
  • $\begingroup$ Heat equation is a diffusion equation so I dont get what you mean by "maintain frequency" $\endgroup$
    – mcodesmart
    Oct 11 '13 at 7:22
  • $\begingroup$ 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. $\endgroup$ Oct 11 '13 at 8:30
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    $\begingroup$ @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.) $\endgroup$
    – Nathaniel
    Oct 11 '13 at 9:04
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"I have experimental data that Schödinger equation maintains high frequencies, while heat equation low."

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.

The Schödinger equation, as far as I understand, doesn't have quite this behaviour because all eigenfunctions of the Hamiltonian merely oscillate in time. It maintains both high and low frequencies, so this isn't exactly a duality.

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.

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    $\begingroup$ +1 Very pithy and important point in your last paragraph: so simple, yet I've never thought of it quite like this. $\endgroup$ Nov 22 '13 at 0:16
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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.

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