^ rotation
|
● ---------- ● symmetric device
|
v
Classically, the device outputs into the scalar field energy and angular momentum, but no linear momentum because the device is symmetric.
Definition of spin and orbital angular momentum
If angular momentum is input to a field, but the associated linear momentum is zero, then we may call it "spin" angular momentum. An example is when we rotate a disk. The disk spins but does not move in another way.
If we input angular momentum relative to a point x, but there is a point y relative to which the angular momentum is zero, then we may call it "orbital" angular momentum. An example is when someone riding in a carousel throws a grain of sand tangentially.
Circularly polarized photons: the frisbee model
Let us assume that the rotating device is an electric dipole and and analyze a single photon. The electromagnetic field is not a scalar field, but it is a good analogy.
If the frequency of the rotation is f, then the energy of a photon is
h f,
where h is the Planck constant.
Let us have a mass
m = h f / c²
moving in a circular path at the speed of light. Let it do one cycle in a time 1 / f. The radius is
r = c / (2 π f)
and the angular momentum is
J = c m r
= h / (2 π)
= ħ.
We can imagine that the field of the rotating dipole loses the energy h f and the angular momentum ħ to the free electromagnetic field from the distance r from the system.
The absorption process is this in the reverse.
The photon itself can be imagined as a "frisbee" which moves at the speed of light and carries the energy and the angular momentum lost by the rotating system.
The frisbee model does not properly explain the photon, though. Since the frisbee moves at the speed of light, it cannot rotate at all - if it would, then the edge would move at more than the speed of light. This is a paradox.
The classical limit of waves made in a scalar field
We finally found an argument which shows that a rotating device outputs into a scalar field orbital angular momentum. Thus, the spin is 0.
● ---------- ● absorbing device
--------->
absorbed orbital
angular momentum
^ rotation
|
● ---------- ● emitting device
|
v
Let us have a rotating device which creates pressure waves in a gas or a liquid. We use a similar, freely floating, device to absorb the waves.
Suppose that the absorbing device would start rotating around its axis when it absorbs a single quantum of the pressure wave.
The classical limit would then be that the absorbing device starts to rotate macroscopically under a strong wave.
But we have already argued that a classical pressure wave mostly moves any obstacle back and forth to the direction of the wave source.
The emitting device does output angular momentum to the field. Freely floating absorbing devices start to circle around the source of the wave. They receive orbital angular momentum with respect to the emitting device. But they do not receive spin angular momentum.
Conclusions
Waves in a solid can be transverse shear waves or longitudinal pressure waves. Transverse waves can easily make a small freely floating absorbing device to rotate around its axis. Transverse waves carry spin angular momentum.
Longitudinal pressure waves cannot easily turn a small absorbing device around its axis. They give up their angular momentum as orbital angular momentum. Pressure waves are scalar.
We found an explanation of the spin 0 and spin 1 in classical physics.
Let us next study the following two questions:
Can we find a classical model for the spin 1/2 of the electron?
Can the concept of a shear wave clarify how in photon absorption, the absorber harvests the entire energy and angular momentum of the photon from a circularly polarized wave? If the absorber is at an oblique angle from the source, how can it absorb the entire angular momentum?
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