Relativity, confinement and Quantum Mechanics

[DaTario]

Supose for simplicity that c = 1 m/s, and that it is the highest velocity a massive body can achieve.
So if, when t = 0, you are located at x = 0, the limiting velocity c forbids you to reach the position x = 1 (or x = -1) until the clock reaches t = 1 s. Isn't it confinement ?
And, if it is , according to quantum mechanics, confinement implies discretization of the energy spectrum.
I ask then:
Does this relativity postulate imply quantum discretization of energy spectrum of all massive entities?

 

[El Hombre Invisible]

I'd call that a light cone. In the future cone, the edge of the cone at any given time is the furthest away the fastest moving thing (light) can get from the event. It goes without saying that, since a massive object at that event cannot exceed c, this light cone contains that object as well.

 

[DaTario]

But if the light cone structure could be for all practical purposes set equal to a confining structure, then the quantum mechanics would force this relativity fact to have deep quantum implications.

 

[selfAdjoint]

There was somebody (can't remember his name) back decades ago (but after modern quantum mechanics) who had a theory that electrons orbited classically in atoms at close to c, and they couldn't fall into a lower orbit continuously, because they would have to speed up. So they could only change orbits in chunks of energy, ta da! He had a lot more about this - he was a genuine physicsts, not an outsider, and he even had an afternoon presenting his theory to Feynmann. Feynman led him through more and more rigorous derivations and he wound up with what Feynman said was a model of the pion!.

 

[DaTario]

Sorry, but I didn't understand the pion part. The genuine physicist was Feynman ?
The idea of atomic fundamental state connected with limiting speed is beautiful. But you mean it is considered valid?

Best Regards,

DaTario

 

[El Hombre Invisible]

Does this relativity postulate imply quantum discretization of energy spectrum of all massive entities?

I don't really get what you mean by discretization (quantisation?) of energy spectra of massive entities. Massless particles are confined within the same light cone.

 

[DaTario]

When a particle with mass cannot reach regions of space beyond some limit, we say it is confined. If there is a potential which responds for this confinement, the possible values of energy of this particle assumes only discrete values (quantization).

 

[El Hombre Invisible]

When a particle with mass cannot reach regions of space beyond some limit, we say it is confined. If there is a potential which responds for this confinement, the possible values of energy of this particle assumes only discrete values (quantization).

Okay, I'll profess I'm fairly new to QM, but my understanding of it is that if you have a region of space to which a particle is confined, the allowed waves within this well must have an integer number of 'nodes', so the particle can have only specific energies, since this energy determines its wavelength.

Since no massive particle can ever actually attain speed c, it will never reach any edge of the light cone, so the 'wave' has no ends. It can about-face as many times as it likes and never reach an edge. Since there is no end to form a node, the particle can have any energy, and so any wavelength, possible.

Another point is that the energy of the massive body in question must constantly increase, and so its wavelength decrease, as it approaches the edge of its light cone. This, to me, means as the body gets nearer and nearer the edge, the wavelength will get shorter and shorter so that it can always fit in and, as it cannot reach c, always has more distance to travel before it reaches the edge. The actual starting wavelength would be irrelevant. No matter what the starting wavelength is, they will always converge towards 0, and you can get as many of those as you want.

Please correct me if I'm wrong. I've only just started on QM.

 

[DaTario]

Let's try the following approach:

Supose one has made a position measurement in the particle (at time t = 0) and let's suposethat the wave function after the measurement is a very very sharp gaussian centered at x = 0. Supose c = 1 m/s for simplicity. So, as I see the particle at rest, I decide to formulate the question at the time t = 0: What will look like the particles wave function at t = 1 ? Disregarding tunneling effects in this context (which would come about if we agree the light cone is also a barrier...but let's not face it now) we may say that the particle's wave function will have to have width [itex] \Delta x = 2 [\itex].

So, appealling to this somewhat strange "future perspective", we may say that the future perspective of the particle's wave function have reasons to be energy quantized.

 

[El Hombre Invisible]

Disregarding tunneling effects in this context (which would come about if we agree the light cone is also a barrier...but let's not face it now)...

I don't see how this would come about either. The nearest equivilent to potential I can think of in a light cone is inertia which, like potential, increases as the particle nears the edge, but unlike potential increases exponentially right up to that edge. The notion of 'tunnelling' out of the light cone doesn't seem to make quite as much sense. There has to be some lower quantity on the other side of the barrier, but for inertia this is beyond infinity.

... we may say that the particle's wave function will have to have width [itex] \Delta x = 2 [\itex].

? You've lost me. What do you mean by width, and how did you come to this value?

As well as the fact that the particle will never reach the edge of cone, and so the wave has no 'ends', and the fact that the wavelength will diminish towards zero as it approaches this boundary, another thing to take into account is that the path through this light cone towards any point on its edge is dependant on velocity, and so kinetic energy, since the area within the cone at any time increases with time. Even if you could reach the edge of the cone, and even if you could do so with a fixed wavelength, you could find an infinite number of combinations of mass energy and kinetic energy for any given Etot - Epot, all of which take different paths through spacetime to reach the boundary. Essentially, for a particles of different velocities, even if Etot is the same, the light cone has different lengths and so different allowed energies.

 

[DaTario]

The [itex] \Delta x = 2 [\itex] has to do with the assumption of c = 1 m/s and with the assumption we take 1 second later the reality and ask what the wave function should look like.

 

[DaTario]

may be we should appeal to quantum correlation functions, (two times correlation functions) in order to bring this relativity constraint to the workbench of Quantum mechanics, if possible, of course.


Best Regards

DaTario

 

[DaTario]

In short: if my velocity has an upper limit(say vmax = 1), I cannot be two meters from where I am now within one second. This sounds like confinement in some future perspective. Does it imply any kind of eigenvalue discretization ? (as QM says a confined particle must have discrete energy levels).