Complex numbers in physics

>>8592071
Complex numbers are useful in math modeling. You model math in physics.

Example: optics and qm.

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110KB, 850x565px

http://www.ams.org/notices/200902/rtx090200212p.pdf

>When I look at the history of mathematics, I see a succession of illogical jumps, improbable coincidences, jokes of nature. One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrödinger’s idea seemed to be going nowhere. But then came the surprise. Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom.

>It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.

>All through the nineteenth century, mathematicians from Abel to Riemann and Weierstrass had been creating a magnificent theory of functions of complex variables. They had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.

In optics the imaginary part of a wave describes the phase.

The imaginary part of a material's refractive index describes absorption.

Oscillation.

>>8592071
Complex numbers like to rotate, rotation is multidimensional oscilation, osxilations are waves, waves are physics.

>>8592317
>Complex numbers like to rotate

Can you expand?

>>8592411
The square root of a complex number can be interpreted as the rotation of a point in the complex plane represented by the complex number.

>>8592071
>be rational
>said by i
wut

>>8592468
He's saying it to pi you moron

>>8592071
They are used a lot in EE and optics

>>8592525
yea, but i is still contradicting itself.

>>8592071
phase differences

Complex numbers are used in Quantum mechanics to describe probability amplitudes and are core to QM so that's definitely worth looking at. Theyre used alot in optics when doing fourier transforms; so like in finding a diffraction pattern due to a given aperture. We just finished an optics experiment making holograms and fourier transforms (and so complex numbers) are most of the maths but describe something pretty neat. Also in non-linear physics imaginary eigenvalues can show that a fixed point is a centre/spiral/limiting cycle rather than just a saddle or a node. That's the examples I can think of off the top of my head

>>8592071
potential flow has a subset that makes use of the power of complex numbers. This is called complex potential

complex potential in 2-D flow is given
[math]
w=\phi + i\psi
[/math]
where phi is the velocity potential
and psi the stream function. defined as follows
[math]
\nabla \phi = \vec{u} \\
\nabla \times (\psi\vec{k}) = \vec{u}
[/math]
which leads to blasius's theorem to calculate force on objects immersed in the flow and other powerful results. The function w is analytic

>>8592085
Perhaps it is in what we are attempting to achieve that is the problem, that need not be.

Squares and square roots have had their place but in a sustainable environment there would be no need of anyone or anything calc'ing squares or square roots. I find it hard to imagine anyone would want problems in which to seek solutions rather than have all answers and to socialize, relax. What purpose if this not the goal.

However...

It seems our math notational system is corrupt. I was trying to define a new one based on x^y^z format: but exponential curves kept exponentiating, they would not stabilize.

It seems we see x^2 as x times x when it is, or should be, x to the second vector. It should not produce the x by x result.