# A Schrödinger's equation: a diffusion or a wave equation?

#### fluidistic

Gold Member
From many sources (Internet, Landau & Lifshitz, etc.), it is claimed that the Schrödinger's equation is a wave equation. However I do not understand why for the following reasons:

• It is Galilean invariant, unlike the wave equation which is Lorentz invariant. Note that the diffusion/heat equation is also Galilean invariant.
• If one takes the free particle localized in a finite region at time t0, then at any instant afterwards, the wavefunction will have non zero values arbitrarily far away from that region. I.e. there is a diffusion without any speed limit, of the wavepacket. That's another point making the Schrödinger equation looking more like the heat equation than the wave equation.
• Mathematically its determinant is such that the Schrödinger's equation qualifies as a parabolic PDE, same as the heat equation and unlike the wave equation (hyperbolic).
On IRC someone said something about a Wick rotation in QFT (and there is an obscure Wikipedia sentence about it), and that apparently this makes the Schrödinger's equation a wave equation rather than a diffusion or heat one. Can someone shed some light?

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#### A. Neumaier

The standard criterion for parabolicity assumes real dynamical variables, hence your corresponding claim is wrong.

In a diffusion equation or any parabolic euation, wavelike excitations decay to an equilibrium state. But wavelike initial conditions in solutions of the Schrödinger equations persist. This makes it a wave equation, not parabolic.

Wick rotation removes the factor $i$ from the Schrödinger equation and turns it into a diffusion equation.

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#### PeterDonis

Mentor
It is Galilean invariant, unlike the wave equation which is Lorentz invariant
What is "the" wave equation? There are many wave equations. Some are Galilean invariant, some are Lorentz invariant.

#### fluidistic

Gold Member
The standard criterion for parabolicity assumes real dynamical variables, hence yuour corresponding claim is rong.)
Thanks a lot! I was unaware of this!

Arnold said:
In a diffusion equation or any parabolic euation, wavelike excitations decay to an equilibrium state. But wavelike initial conditions in solutions of the Schrödinger equations persist. This makes it a wave equation, not parabolic.
Hmm, let's take the free particle initially as a localized wave packet. The equilibrium state is the total uncertainty in spatial space. After an infinite amount of time, the information of where the particle was initially localized, is lost. It is exactly the same as in the case of say an infinite rod that had initially a heat source that was later removed. After an infinite (or extremely large) time, the information of where that perturbation was applied is lost.
Am I missing something here?

Arnold said:
Wick rtation removes the factor $i$ from the Schrödinger equation and turns it into a diffusion equation.
I see. I thought there was something much deeper than that.

#### fluidistic

Gold Member
What is "the" wave equation? There are many wave equations. Some are Galilean invariant, some are Lorentz invariant.
Whoops, I had in mind the one of the E and B fields that satisfy Maxwell equation.

"Schrödinger's equation: a diffusion or a wave equation?"

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