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1) In the following video, Steve presents us with a simple case of a free particle at rest, where we have zero momentum and thus know the momentum exactly, but as a consequence we have no idea what the position the particle is in, so the probability density is constant in space. So the wave function for the particle only includes the time component.

Now we take this same particle/wave function and set it into motion relative to a stationary observer or lab frame by introducing a Lorentz transform into the previous time-dependent wave function (see 6:00). The effect of this is to give the particle a wavelength and, thus, a momentum in a left or right direction, and thus a positional dependence added to the wave function. My question here is, "What is the particle we are talking about and where is the particle?" The visualization just looks to me like a continuously oscillating probability density. Does this just mean that the probability of finding a particle oscillates through all space with that sort of distribution? And that that distribution is moving to the right or left? I'm confused.

2) What actually

*is*the particle? Let's forget about about bosons for the moment and just focus on fermions. In terms of a wave packet, what constitutes the particle? Does a particle, say electron, have to take the form of a standing wave? Is this what the Dirac equation is telling us? The mass that characterizes the electron is made up of both a coupled left moving wave function and a right moving wave function. Does the Dirac equation describe a standing wave? If that's the case, then what is Steve talking about in the above video when he adds momentum to the free particle and says it is now representative of a particle moving to the right? I don't see a standing wave there.

Does the standing wave define the embodiment of the particle itself, or does it just define a "pocket" of space-time where an otherwise ambiguous entity called a particle is highly likely to be "found." These distinctions or lack thereof are not very clear to me.

3) Dispersion--When we trap a particle in a potential well, we quantize it's energy, momentum, and position because the "waviness" of the wave packet or wave function must fit integer wavelengths in the well. It seems to me that the "shape" the wave function takes within any given potential energy constraint governs the identity of that particle in the sense that it associates it with a standing wave that has definite energy, say. But what about about a "free" particle? What governs it's identity if it's not under any potential constraint that might shape it? From JMGerton's video here, it would just seem to disintegrate due to "dispersion" as it travels down the road. What does that mean?

In any case, I'm hoping someone can help clear up some of these quandaries for me in a somewhat simplified fashion . Please notice the "B" prefix and don't just list a bunch of equations and axioms, I'm looking for a more intuitive understanding. Thanks