Re:Is the Schrödinger Equation True?
by gus_massa on Tuesday January 12, 2021 @10:32AM
(#25748130)
> The laplacian in there feels like it's a "diffusion" term (as in the heat equation, things tend to spread and smoothen over time). Why isn't there ever a intuitive explanation in text books about this?
In an equation like
df/dt = k Δf
the Laplacian create a diffusion-like effect only if k is a real positive number. If k is a real negative number, it creates the opposite effect that concentrates whatever f represent and make it more spiky.
When k is an imaginary number, it is not like diffusion o anti-diffusion (I don't know the technical name). The associated operator is unitary and the behavior is different.
Many books about the heat/diffusion equations and Schrodinger equation have this discussion. And the generalized versions where k is a functions that depends on x or t, or is a complex number instead of a real or complex number.
For an intuitive idea of what is happening, the idea is that you consider at time=0 the function
f(x,t=0)=A sin(ωx).
If k is real and positive, the amplitude of the function decrease with time. And when ω is big, it decrease faster, so the small features disappear very faster than the smooth parts.
If k is real and negative, the amplitude of the function increase with time, and you have the opposite effect.
If k is imaginary, the function just rotate in the complex plane but does not increase or decrease the amplitude. So you get a wavy effect instead of a diffusion effect.
The initial equation is linear, i.e. there is no f^2 or something like that. With some hand waving you can extend the simple cases where f is sin or cos to more complex cases. The technical details are in https://en.wikipedia.org/wiki/Fourier_transform
In an equation like
the Laplacian create a diffusion-like effect only if k is a real positive number. If k is a real negative number, it creates the opposite effect that concentrates whatever f represent and make it more spiky.When k is an imaginary number, it is not like diffusion o anti-diffusion (I don't know the technical name). The associated operator is unitary and the behavior is different.
Many books about the heat/diffusion equations and Schrodinger equation have this discussion. And the generalized versions where k is a functions that depends on x or t, or is a complex number instead of a real or complex number.
For an intuitive idea of what is happening, the idea is that you consider at time=0 the function
If k is real and positive, the amplitude of the function decrease with time. And when ω is big, it decrease faster, so the small features disappear very faster than the smooth parts.If k is real and negative, the amplitude of the function increase with time, and you have the opposite effect.
If k is imaginary, the function just rotate in the complex plane but does not increase or decrease the amplitude. So you get a wavy effect instead of a diffusion effect.
The initial equation is linear, i.e. there is no f^2 or something like that. With some hand waving you can extend the simple cases where f is sin or cos to more complex cases. The technical details are in https://en.wikipedia.org/wiki/Fourier_transform