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Why does the electron come in exact packages of mass, electrical charge, weak charge, and spin? I believe that if we can come up with a model which implies quantum mechanics and which also gives a pleasing interpretation to the wave-particle characteristics of say an electron then we will be much closer to answering parts of the above question.
How are we going to solve it, clues and thoughts on my quantum puzzle. How to come up with this pleasing interpretation? Learn, read, and study the relevant physics. What relevant physics, this I will try to answer in what follows.
One might still like to ask: "How does it work? What is the machinery behind the law?" No one has found any machinery behind the law. No one can "explain" any more than we have just "explained". No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced."
The preceding picture is also taken from Feynman . In this section of the book he talks about how probability amplitudes change with time. This section basically encapsulates de Broglie's idea on matter waves. The main idea in the de Broglie formulation of the electron can be understood by considering a bunch of synchronized pendulums. Pendulums which appear to be synchronized in my frame, will only appear synchronized in other frames which are at rest relative to my frame. If you move past me my pendulums they will no longer appear to be in phase. For small velocities the distance between pendulums with the same phase will be very large but they will still appear to have the same frequency. But if you move past my synchronized pendulums with great speed then the distance between in phase pendulums will be much shorter and the frequency of the pendulums will appear reduced. It was by this picture I was able to begin to understand de Broglie's work on matter waves. A particle in de Broglies picture is the combination of two things:
The picture above shows some lines of constant phase of the wave exp(-iwot) in the x-t frame, the frame in which the particle is at rest. Now consider how the wave exp(-iwot) looks in another frame x'-t'. Lines of constant phase will now depend on position as well as on time, and the frequency of the wave will now appear higher. In appendix A is a summary of de Broglie's idea on matter waves.
The picture below is adapted from Main's book . Given the dispersion curve for say the electron one can quickly determine the characteristics of both phase velocity and group velocity. 
Consider two nuclei at rest. Let one be a proton and let the other be a nucleus of deuterium. As the deuterium nucleus has roughly twice the mass of a proton, its' "clock" will be twice as fast as the protons clock, call these frequencies 2w and w.  But let us consider the clocks of the neutron and proton which make up deuterium nucleus, they each have a frequency of roughly w but they combine to act like a single object of frequency 2w. Now a puzzle is how to think of two things, each with a frequency w, that can be combined to yield one thing with a frequency 2w.
Along a similar line, consider several massless balls of radius R to which particles are attached around each equator. Consider three cases:
The last two configurations must surely exist in some form in nature. Now consider the states where the balls are spun around the polar axis and have one unit of angular momentum. If one considers the wavelengths of the individual particles of mass 4M, 2M, and M , they will have a wavelengths of 2pR, 4pR, and 8pR respectively. The individual particles in the 2 and 4 mass systems do not have the "proper" wavelength, 2pR, it is only when they act as a system does one get the "proper" wavelength. An interesting thought (doubtful). Take two waves of wavelength 2l and shift the phase of one wave so that the zero of one wave corresponds to the maximum of the other wave. Now multiply the two waves together, and their product is a wave with wavelength l (see above). In a similar manner, starting with four waves of wavelength 4l one can (with suitable phase shifts) multiply them together to end up with a wave of wavelength l. This is really nothing spectacular, just trigonometry, but maybe thought of in the "right" way might give clue as to how multi-particle systems combine to "beat" space in the same way that a single particle "beats" space. Another way to get high frequencies out of low frequencies is to consider drums and drummers. Suppose one person is told to beat drum A with a frequency w, and four other people are told to beat drum B with frequency w /4. Now if the four people beat the drum in phase, some of the resulting sound will have a frequency w /4. But now suppose the four people agree to each beat the drum with the same frequency w /4, but now with a phase difference f between each drummer. If f is chosen to be p/2 then the drummers will produce the same sound as the single drummer who is beating the drum with frequency w. If f is chosen to be smaller so that the drummers beat rapidly in succession, but each drummer still beats with frequency w /4, frequencies much higher then w can be produced. So is it that nature wants drummers to beat one after the other instead of all at once? A less interesting thought. Consider an electron as not one point object but as two point objects which are some how associated, each with some frequency w '. Depending on the relative phase between the components some interesting "sounds" might come out. Also consider two atoms bound to form a molecule. Assume that the coupling between the atoms can be adjusted so that we can reduce the coupling strength to almost zero. Question. How does the wave-length of such a system vary as one reduces the coupling to the point that the system is no longer a single system of two particles but a system of two separate particles? One could imagine that in scattering experiments at energies much smaller then the coupling strength the two atoms would act as one, but as scattering energies approached the energy needed to bust the molecule apart that the wave-length might not be well defined. I'm sure there is stuff written on this, I should check it out.
There is a type of radar, phased-array radar, which consists of an array of electromagnetic radiators whose phase can be adjusted individually. One can easily picture how the ability to control the phase of a large array of radiators can result in a great deal of control over the directionality to the emitted radar beam. When an electron scatters off a crystal, each atom in the crystal scatters electron matter waves in all directions. Where these matter waves interfere constructively is where an electron is likely to be found, and where the matter waves interfere destructively an electron is not likely to be found. Question. Can one think of a way, say by adding some extra variables to the description of spacetime, the crystal, and the electron, so that by a suitable choice of hidden variables we can control where the electron goes?
The ways in which pairs of quanta can be correlated is one aspect of quantum mechanics which seems most hard to model. The "kooky" actions of a pair of correlated photons can be interpreted to suggest that some kind of information is travelling faster then the speed of light. I have a feeling the answer lies not in just a hidden variable of the particle but also some unobserved property of spacetime. More thought needed on this clue to the quantum puzzle.
One important clue comes from the way several Fermi particles can form a system which can have Bose like properties. Consider the two neutrons and two protons which make up a helium nucleus. Not only do they combine to act as one particle which is roughly four times more massive then proton, but their spins, in a similar manner combine to act as a particle with no spin. A big clue!
The hydrogen wave functions are neat to look at, but as interesting to me is the gradient of these functions. With a graph of the hydrogen wave functions as a guide one can easily sketch their gradient. (Mathematica for students used below).
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