https://www.youtube.com/watch?v=qtIsYbYdzCI
"But when you look at it, one thing to notice is that, if you just look at the probability densities of each of the energy eigenstates, they're actually stationary. But when you add the eigenstates, because you are adding the complex numbers and there's that complex interference going on, the subsequence probablity density of the sum of those states, the superposition of those states, is actually this thing that varies in time. So here we can see this cool kind of dynamics coming out of the machinery of complex numbers.
So it's that imaginary component that is the noncommutative nonlocal phase secret of quantum de Broglie-Bohm physics - as a "guiding" antigravity force.
"So for example here's a two-dimensional plane wave. It's defined in the plane of your screen, it's constant amplitude and the color represents the phase of the wave function at every point. I've also superimposed these little arrows and what the arrows represent is numbers in the complex plane. Now I want to use to illustrate a couple of points. First, when you look at a picture like this, it almost looks like a vector field and there's a temptation to think that the complex numbers are embedded in this two-dimensional space, and that their direction is sort of pointing in a direction in that space. This is a common misconception and I had this misconception for awhile when I was learning quantum mechanics, cuz one of the things that confused me about complex numbers was they're two-dimensional right? So why?
I mean if you have a three-dimensional wave function for example, shouldn't you have some three-dimensional thing? How do you stick a two-dimensional arrow at a point in space? How does that even make sense? But I hope based on everything you've seen so far you've realized that the two-dimensionality of the complex numbers actually is not about any direction in physical space. The fact that the complex numbers are two-dimensional is the fact is that a wave is up and down or left and right or back and forth or high pressure, low pressure. It's yin and yang. When you see that then the confusion goes away.
Richard Behiel
Super fascinating new de Broglie "Matter wave" nonlocal quantum physics vid on the role of the imaginary number on Schroedinger wave function.
This is the Dirac Dance....
in noncommutativity nonlocality the minus sign can't just be washed away.
"that actually is something that is relevant to the wavefunction of relativistic electron, for example....that is a very very deep mystery as to why that's the case." -
OK thanks!
AI: Yes, the anticommutation relations for 1/2 spin are a manifestation of noncommutativity,
Matrix imaginary exponential=90 degree rotation
the rate of a change of a certain state is perpendicular to that state and hence that the way things have to evolve over time will involve a kind of oscillation.
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