For a wave function to be valid where it gives a particle a non-zero probability of existence must be separated in time and distance so that it can only go between those points at speed les than light. That includes by quantum tunneling.
The Schrödinger equation describes evolution in the space of states, not spacetime. If the Hamiltonian is non-relativistic, it describes a non-relativistic system, if it’s relativistic, it describes a relativistic system.
Stam Nicolis
I do not understand much of your wording. The time dependent Schrodinger equation is solved for wavefunctions which give the probabilities a particle being found of being at locations in space and time. The time independent Schrodinger equation does not have time in it so it only gives probabilities of being at locations in space.
The statement about the time-dependent Schrödinger equation isn't in contradiction with what I wrote; the statement about the time-independent Schrödinger equation is incorrect, since its solutions are the eigenvalues and eigenfunctions of the Hamiltonian. The time dependence is deduced by expressing the initial condition in the basis of the eigenfunctions.
What matters is that the wavefunction doesn't describe motion in space, but in the space of states of the system. That's why it can't be assigned any speed of propagation-the term doesn't make sense, in association with the solution of the Schrödinger equation.
All this is standard material in any course on quantum mechanics and isn't a research topic since 1926.
Stam Nicolis
I do not know an easy way to type Greek letters onto RG, so let U be a wavefunction. The probability of finding a particle will occupy the space
dxdydzdt is |Uxyzt|2 dxdydzdt for a time varying U.
That not saying you cannot use the wavefunction to find the probability finding some momentum etc.
The dx dy dz dt are obviously differentials distance and time (not my invention). I left out the integral signs in my last post but it is understood.
They refer to the space of states. And there isn’t an integration over time.
But, once more, so what? The speed of light doesn’t have anything to do with the solution 9f the Schrödinger equation, whether the Hamiltonian describes a relativistic system, or a non-relativistic one.
Stam Nicolis
The below is why most physicists believe nothing goes faster than light.
From Einstein's formula observed mass of any particle with a REST mass would need infinite kinetic energy go at the speed of light. Nothing on the Earth has been observed to go faster than the speed of light. Gravitational wave have been clocked at the speed of light. When asked a question (on Researchgate) if something like the above formula was known experimentally before Einstein's mathematical proof because decades before relativity: there where particle accelerators and studies on cosmic rays where also. I got response someone else published such formula. The response did not say if was experimentally derived or not.
From time to time some one says because the mass of particle going faster than light would be a finite imaginary number as it is possible for it created by nature. But the that part of the formula was never experimentally proven. The fact it is an imaginary number also means it is beyond the realm of the formula.
Sure, but this fact doesn’t have anything to do with the properties of the Schrödinger equation.
If the Hamiltonian is that of a non-relativistic system, the speed of light isn’t of any significance; that the non-relativistic description is an approximation isn’t front-page news.
The Schrödinger equation applies to relativistic systems, also, upon using the appropriate Hamiltonian, that can be defined in this case and nothing inconsistent occurs.
The mass of a particle doesn’t depend on its velocity, it’s invariant under Lorentz transformations.
Energy, E and momentum (p_x,p_y,p_z) do depend on the velocity, but what matters is that the combination E^2-|p|^2c^2 does not and it’s equal to m^2c^4.
Stam Nicolis
Additional boundary condition needed:
If the particle has a non-zero probability of being at two places at the same time it means: the gap (sg) is about equal to or less than the particles uncertainty of position and it will be observed to cross the gap at infinite speed. That is even Hamiltonian is suitable for the equation the equation is giving an impossible solution. It appears that that must additional simultaneous conditions to prevent this besides the Schrodinger's equation. In the case of axially symmetric wavefunctions this happens. I am giving a case that is beyond the realm Schrodinger's equation which in this case it additional constraints or call it boundary conditions.
The statement about a quantum particle allegedly being in more than one place at once is, simply, wrong. That the probability of its being somewhere in particular isn’t a delta-function doesn’t mean anything more than that it’s only possible to compute that probability of its being somewhere.
The Schrödinger equation is just a means for computing the probability density for finding the particle somewhere. If the Hamiltonian describes a relativistic system, then Lorentz invariance may not be manifest-but it can be shown to hold; if it describes a non-relativistic system, it’s irrelevant.
The space of states a quantum particle, relativistic or not, can be found in is much bigger ttan the space of states of its classical limit. This is not spacetime, which is the same in all cases.
The probability of finding a quantum particle somewhere isn’t the same as the probabiility of finding a classical particle somewhere, that’s the only difference.
If the particles are relativistic, the description is consistent with Lorentz invariance, if they’re not, the description is consistent with Galilean invariance. The additional states the quantum particle can be found in don’t affect this.
Stam Nicolis
Maybe not in same micro second at two places at once but maybe less than 10 micro seconds. It depends on the case. In case of an axially symmetric case there may be many such particles. They will not have infinite energy as if really moved that fast. But in case of charge particles, the charge will for purposes will be and charged particles moving at relativistic velocities transfers energy to any standing waves they past though which is a reason accelerators are noisy.
I have not did experiments to what laws a subatomic particle follow when it crosses a gap by quantum tunneling.
Whether the particles are charged or not is irrelevant. It’s simply wrong to expect a non-relativistic description to be consistent with special relativity.
However, a relativistic description of quantum particles is consistent-it doesn't lead to violations of Lorentz invariance.
It would be a good idea to actually do the corresponding calculations.
However, once more, the statement about a particle being in two places at once is wrong.
That a particle has a non-zero probability of being somewhere doesn’t mean it’s at more than one place at once. It’s astonishing how this nonsense persists.
``Particle is at more than one place at once'' means that the probability of finding it at more than one place at the same time is equal to 1, which is inconsistent with the properties of any probability distribution. It's a trivial exercise to show that this can't occur with any solution of the Schrödinger equation, whether the system is non-relativistic or relativistic.
A uniform probability distribution on a finite number of states doesn't take the value 1 at all states; it takes the same value, necessarily less than 1, at each state. (And on an infinite number of states is ill-defined; however it can't take the value 1 either, since its integral over all states must be equal to 1.)
The solutions by Schrodinger's equation would 3 a wave functions zero in the gap. The gap surface therefore would be reflectors for the solution on each side of the gap. On each side of the gap it would take the form B(t,x,y,z)-B(t,,x,y,-z) where z is normal to gap wall, and B is a function.
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