An electromagnetic quadrupole wave has the spin 2?
^
| rod
+ ● -------------------- ● + charge
|
v rotation
If the rod is microscopic, it returns to the original state after a 180 degree rotation. It may be reasonable to require that the wave function of the system returns to the original state after a 180 degree rotation.
If the wave function only does one full cycle in a full rotation, then the angular momentum is presumably ħ. Niels Bohr built his atomic model from the assumption that angular momentum is quantified in units of ħ.
If the wave function of the rotating rod does two cycles in a 360 degree rotation, then the angular momentum is 2 ħ.
The rotating rod produces an electromagnetic quadrupole wave. The spin (helicity) of that wave, in some sense, is 2.
Maybe it is possible to extract from the wave a photon with the spin 1? If that is true, then we can split a spin 2 photon into two photons of the spin 1.
In gravity, people usually assume that a rotating quadrupole produces spin 2 gravitons.
Spin 0 and spin 1
An electromagnetic spin 1 wave is generated by a rotating dipole. The dipole has to rotate 360 degrees to return to its original state.
Spin 0 waves are longitudinal waves, or pressure waves. The simplest way to generate them is to have a monopole whose charge varies with time. For example, in a drum skin we can generate waves by pressing it rhythmically with our finger.
Spin 1/2
If the rod system would return to its original state after two full rotations, then, presumably, it would be able to store angular momentum in units of 1/2 ħ.
Question. If we are able to construct a system of electric charges which has the property above, are we able to create photons whose spin or helicity is 1/2?
Above we argued that a quadrupole creates a spin 2 wave. A quadrupole is two dipoles rotating with a 180 degree phase shift.
x------● + dipole
x = axis of rotation
A dipole has to be rotated 360 degrees to bring it to the original state.
We can reduce the required rotation to 180 degrees by combining two dipoles:
+ ●------x------● + quadrupole
x = axis of rotation
An antenna, which has to be rotated 720 degrees to bring it to its original state, would produce spin 1/2 waves. If we combine two such antennas, then we get the familiar dipole antenna.
Does this cast light on the spin 1/2?
Suppose that we have a torus with a sphere inside. We twist and bend the torus so that it becomes a coil with two loops.
If the sphere has a charge, it can probably absorb angular momentum in units of 1/2 ħ.
loop 1 loop2
| |
| ● charge moving in the coil
| |
-----
bridge
In the diagram we schematically have the two loops. The "bridge" is the two tubes which connect the loops.
If the charge moves in the coil, it creates a main wave which is like from a rotating dipole. Also, it creates another, smaller, back-and-forth dipole wave whose frequency is 1/2 of the main wave.
Could it be that a spin 1/2 boson is this simple? Just two dipole waves superimposed?
Spin 1/2 fermions
The waves which we have studied above probably do not conserve the number of particles. We may be able to extract the energy in many different combinations from a wave of a random form. Also, there is no problem in stacking many identical waves on top of each other. The waves describe bosons.
Fermions, like electrons, contain a fixed charge which is conserved. Their wave model must strictly conserve the number of particles.
Erwin Schrödinger in 1925 discovered his famous equation which works for non-relativistic fermions. Paul Dirac in 1928 was able to make a relativistic equation for electrons and positrons.
What kind of a classical wave model strictly conserves the number of particles? An answer is a model of a gas. The number of molecules does not change in a gas, but a gas does exhibit wave phenomena in the form of sound.
A semiconductor might be a suitable classical wave model for fermions. If we have an extra electron somewhere, the associated excess density of electrons is the "wave". If we have a missing electron, then the missing electron density is the associated wave.
Conclusions
We found simple classical models for spin 0, 1, and 2 bosons. We may have a model for a spin 1/2 boson, too.
It remains unclear what the spin of a boson actually means. Is it a unit of what amount of angular momentum can an absorber harvest from the wave? Or is it always the same fixed amount for one type of a boson field? Can a photon transfer angular momentum which is not parallel to its path?
In our blog we claim that gravity is fundamentally analogous to electromagnetism, and that the spin of the graviton is 1. Our observation that a quadrupole radiates "spin 2" photons may clarify why some people say that the spin of the graviton is 2.
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