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It's well known that classical mechanics is a crude approximation of reality, and that it can be derived from quantum mechanics. But if this is so, why is it not a linear theory, like quantum mechanics? And why does it not require complex numbers like quantum mechanics does?

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In short, because you can apply classical mechanics when you can average of a large number of quantum mechanical events. The average hides quantum mechanics.

In part, it is like air pressure. Classically it is a smooth force that presses on the walls. Microscopically, it is atoms bouncing off the walls. Each atom gives the wall a discrete kick. When you average the effect of a large number of atoms, the details of the kicks don't matter.

In part, a complex number describes an amplitude and a phase. Classically, this is replaced by a real amplitude and a real phase.

Instead of air, suppose we shine photons on the wall. Each photon interacts with an atom in the wall. Depending on the type of atom it may be absorbed and the energy reappears as heat. Or for phosphorus, it may excite the atom and another photon will be emitted a while later. Or for silver as in a mirror, it may interact with many electrons spread out in a conduction band, and the effect will be a reflected photon. Or for a dipole, it would give the dipole a kick that would affect its orientation.

Suppose we have phosphorus and a double slit in front of the wall. The photon is described by a complex wave function. For a free photon, the solution to the Schrodinger equation is $\psi = Ce^{ikr-\omega t}$. It propagates through both slits and interferes with itself. Since the two paths from each slit to a point on the wall have different lengths, the wave function may interfere destructively or constructively. The probability density of finding the photon a the point is $\psi^*\psi$. There is a greater probability of finding a bright spot where interference is constructive.

If you send many photons, there are bright and dark bands because intensity is proportional to the number of photons at the point, which is proportional to the probability.

But in this case you can use the classical explanation. Light is an oscillating electromagnetic field. Taking just the electric field, $E = Csin(kr-wt+\phi)$. Again, the electric field passes through both slits and interferes with itself. Again, interference may be constructive or destructive. The intensity of the light at a point is $I \propto E^2$. You get the same bright and dark bands.


This explained where photons are detected. The same thing explains forces.

Suppose you have a wall full of vertical wires, and light in the radio wavelengths, such that a photon has an equal probability of hitting any point on the wall. Some dipole will receive a kick each time a photon lands.

Suppose you have many vertically polarized photons in phase. $\psi = Ce^{ikx-\omega t}$. At some time $t_0$, the phase at the wires is just right to kick an electron upward. Half a period later, the electron would be kicked downward.

If you have many photons, the kicks create an alternating current in the wires. This is how a radio antenna works.

You can describe it with an alternating electric field where an electric force drives the electrons up and down.

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