There are three predominant interpretations of quantum mechanics: in the Copenhagen Interpretation the event (say, the path of a particle following diffraction) is determined probabilistically by the act of observation, the Multiverse interpretation has all probabilities eventuate somewhere, while the Pilot Wave has the event decided causally. One major failing with the Pilot Wave is that it becomes almost impossible, in its current form, to contribute to chemistry.
My interest in this problem started in my honours year, with a lecture on the hydrogen molecule. I stopped the lecture to point out that the Hamiltonian operator as presented led to the system becoming increasingly stable as the internuclear distance D diminished. The operator had no causal reason to diminish electron probability between the nuclei, and the lecturer could only agree that something was wrong. Shortly after, it occurred to me that the answer must lie in wave interference, so I tried a wave approach. Ten minutes later I had an analytical answer: 1/3 the energy of the hydrogen atom. (Look it up – it's not bad!)
Either the wave determines the motion of the particle or it does not. If it does not, what is it doing? Why do all quantum computations accidentally end up in agreement with the predictions that it does? With that thought, I decided that for me, the wave must determine the particle motion, but how? Conceptually, this is the Pilot Wave, but in detail, what I have come up with is somewhat different.
For the wave to act on the particle, they must interact. Now, assume the particle has a velocity
v. What is the wave doing? With a little algebra, it is easy to show that the phase velocity (the velocity of, say, a wave crest) is given by
E/p, where
p is the momentum of the particle (by derivation). The problem then is, what is
E? Some textbooks woodenly quote
E = m(c squared). This has the rather remarkable property that the phase velocity
always exceeds the speed of light, and the stationary particle gives off waves at infinite velocity. Some say that this does not violate relativity because the wave carries no information. That is absolute nonsense: it defines the velocity of the particle. Worse than that, it defines an absolute velocity, which requires a fixed background as a reference, and that also violates the most fundamental principle of relativity. Heisenberg objected, and put
E equal to the kinetic energy, which has the rather odd property of requiring the wave to travel at half the particle velocity, which makes it difficult to see how it can affect the particle.
Which gets to my version. I put
E =
m(v squared), twice the kinetic energy, which means the wave must contain energy, and hence is real. The wave now travels at exactly the same velocity as the particle, and hence can affect it. Now the wave function only works if it is in the form exp (2pi.
i S/h). (A sine wave does not give you quantum mechanics.) My guess is that puts the wave in an additional dimension, which is almost required for energy conservation, in which case quantum mechanics is the first actual evidence of the additional dimensions proposed by string theorists.
Of course, you won't believe that. So, how do you explain how the wave function makes the particles do what they do?
