E² = p² + m²
is "ugly". (We have set c = 1 in the equation.)
Since the equation is ugly, also its solutions probably are ugly. The ugliness would give rise to the electron spin.
A. O. Barut and A. J. Bracken (1981) explain the Schrödinger argument (1930) of the zitterbewegung.
Positive and negative energy solutions: the complex value rotates either clockwise or counter-clockwise with time
The zitterbewegung seems to be associated with the fact that the Dirac equation admits solutions both with a positive E > 0 and a negative E < 0. Let us call these positive and negative frequency solutions.
Let us make a wave packet which contains an identical amount of positive and negative frequencies.
--->
___
____ / \_____ positive
\___/
___
____ / \_____ negative
\___/
--->
Let the wave packet move to the right. The wave at the front of the packet may be something like
ψ(t) = exp(-i (E t - p x)) + exp(-i (-E t + p x)).
The wave alternates between destructive interference and constructive interference. Does that mean that the expectation value of the position x of the particle moves back and forth? Probably yes.
Constructive interference happens when t = 0 and x = 0. Where is there destructive interference? When
p x = π / 2,
that is 1/4 of the de Broglie wavelength
λ = 2 π / p.
The expected location of the particle jumps back and forth the distance λ / 4.
We want the location to jump back and forth by a fixed distance that does not depend on p. An obvious solution is to "mix less" of the second term in the formula of ψ if |p| is small:
ψ(t) = exp(-i (E t - p x)) + C |p| * exp(-i (-E t + p x)).
There C is a (small) constant.
Above we have standard plane wave solutions for the Dirac equation (by Jim Branson, 2013).
Let us assume that p in the formula is non-zero only to the x direction. If we sum the ψ(1) in the upper left corner to the ψ(4) in the lower right corner, then the the first and the fourth component of the spinor wave function look somewhat like what we derived above. Maybe we found a simple model which describes the zitterbewegung?
Above ψ(1) is an electron with the spin-z up and ψ(4) is a positron with the spin-z down. That matches nicely pair production.
Is an "electron" actually a mix of electron and positron solutions?
Then the electron would not be the solution ψ(1) or ψ(2). It would be a mix of positive frequency (E > 0) and negative frequency (E < 0) waves. If the momentum |p| is small, then there is only little negative in the mix.
Why is the electron spin 1/2 and not 1?
The "mix" model above gives us a heuristic explanation. The length of the jump path, or the zitterbewegung is
2 * 1/4 λ = 1/2 λ,
where λ is the de Broglie wavelength. For relatively large momenta |p|, the de Broglie wavelength is close to the electron Compton wavelength.
Thus, a nice value for the zitterbewegung pathlength is 1/2 of the Compton wavelength, which corresponds to the spin 1/2.
In this blog we have worked very hard trying to understand how the electron may return to its original state in zitterbewegung after moving just 1/2 of the Compton wavelength.
The mix model explains this: the path is formed by the interference of two waves. The state of both of these waves only returns to the original after two constructive interference events.
This is probably the origin of the strange 720 degree rotation rules for the electron spin.
What is the origin of the gyromagnetic ratio 2?
Above we were able to make the interference pattern to move back and forth. How can we make it to follow a circular path?
Negative frequencies in a chirp
In this blog we have studied hypothetical Unruh and Hawking radiation. We learned that a "chirp" contains both positive and negative frequencies.
If we have an electron under an accelerating motion, then its wave function presumably is a chirp.
It seems to be so that an electron wave under an interaction always contains both positive and negative frequencies. It is not possible to restrict us to just positive frequencies.
Some people have claimed that the electron wave function should only contain positive frequencies, but that seems to be impossible to implement.
The zitterbewegung model of David Hestenes
David Hestenes (1990) suggests that the phase of the electron wave function determines its location in a circular motion. The circular motion is the electron spin.
The Hestenes model may be too bold.
How to make the Schrödinger equation more precise about the energy-momentum relation?
If we try to improve the Schrödinger equation in such a way that it estimates the energy-momentum relation
E = sqrt( p² + m² )
more precisely, then we have to add more terms.
The square root has the Taylor series:
sqrt(1 + a) ≅ 1 + 1/2 a - 1/8 a² + 1/16 a³ ...
Let us assume that m = 1 and p² = a. Then we can use the series to approximate the energy-momentum relation.
How to add the term -1/8 a² to the Schrödinger equation? Could we use the fourth derivative
d⁴ / dx⁴
to keep the equation linear?
The Scrödinger equation "codes" the value of p² into the second spatial derivative of the wave function Ψ. Can we code the value of p⁴ into the fourth spatial derivative? No, that does not work. We cannot make sure that the fourth derivative stays as the square of the second derivative.
What about adding a term
(d² / dx² Ψ)² ?
That makes the equation nonlinear. We might try to solve the nonlinear equation by writing it as a linear equation plus a perturbation term. But a perturbation will scatter the wave. It is hard to maintain conservation of momentum, if the wave is scattered to various directions.
Make the electron to move at the speed of light and "bounce" in a pipe?
The energy-momentum relation is very simple for massless particles which move at the speed of light:
E = | p |.
If we make the electron to be massless and move at the speed of light, then we maybe can keep the wave equation linear.
pipe wall
------------------------------
/\/\/\/\/\/\/\/\/\ bouncing electron
------------------------------
pipe wall
The bouncing of the electron would be the zitterbewegung, and it would be responsible for the electron spin and the magnetic moment.
This is a method of simulating a massive particle with a massless particle. If we have a set of photons confined in a box, the photons, in a way, behave like a massive object.
We still have to find an explanation to why the spin-z of this bouncing has to be +- 1/2 ħ.
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