Classically a particle cant exist in a region where V>E

[SpectraCat]

Apart from the fact that I'm not sure what the difference is between traveling over the barrier or through it, then sure.. so what's your point?

Well, according to probabilistic QM, there is a finite probability density of the wavefunction describing the particle inside the CFR, but one can never carry out a measurement that "catches" the particle insides the CFR. AFAIK, this is consistent with all experimental evidence. How does dBB deal with this? Is it just that, if a measurement finds the particle in the CFR, then it must have gotten a boost from the quantum potential? This seems like it might be consistent with the uncertainty-based explanation from probabilistic QM, but I don't know enough about dBB to be sure.

I am starting to realize that this is exactly the sort of thing that might make people prefer dBB ...

They do conserve energy, but only if you include the quantum potential energy as well.

OK .. maybe I am starting to get this .. you are saying the the following relation holds for any dBB system:

E_{total} = \frac{p^{2}}{2m} + V + QPE

Where QPE describes a potential, the gradient of which is a field that produces fluctuations in the trajectory of the particle. These fluctuations are fundamentally unpredictable due to hidden variables, but over multiple measurements produce results that are indistinguishable from those of probabilistic QM. Is that about right?

One more thing just for reference, you can't use plane-wave incident, reflected and transmitted waves to analyze the dynamics of the situation, as is normally done in school, since this in no way corresponds to a situation where a particle is incident on a barrier and may or may not tunnel through it. An infinite plane incident wave means the particle can start anywhere in the universe, even on the other side of the barrier. You need to use a proper traveling time-dependent wave packet, if you want to avoid nonsense.

Sure, and I have done this, but the wavepacket results are not fundamentally different from those of the plane-wave simplification. Part of the probability density of the wp gets reflected, and part gets transmitted. The math involved with the time-dependent picture is just more involved, which is why I guess the intro texts stick with the plane-wave description.

 

[zenith8]

..but one can never carry out a measurement that "catches" the particle insides the CFR.

OK - can you explain why?

How does dBB deal with this? Is it just that, if a measurement finds the particle in the CFR, then it must have gotten a boost from the quantum potential? This seems like it might be consistent with the uncertainty-based explanation from probabilistic QM, but I don't know enough about dBB to be sure.

That's about right, yes.

I am starting to realize that this is exactly the sort of thing that might make people prefer dBB ...

Don't get me started or I'll explain the double slit experiment as well.

OK .. maybe I am starting to get this .. you are saying the the following relation holds for any dBB system:

E_{total} = \frac{p^{2}}{2m} + V + QPE

Correct.

Where QPE describes a potential, the gradient of which is a field that produces fluctuations in the trajectory of the particle.

Your QPE is usually written Q, and the formula for it is -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R} where R is the amplitude of the wave function. The negative gradient -\nabla Q is a force (usually called the 'quantum force') which results from the wave field 'pushing' on the particles. The wave field is the objectively existing wave represented mathematically by the wave function. All derivable from the Schroedinger equation under the assumption that particles exist continuously.

These fluctuations are fundamentally unpredictable due to hidden variables, but over multiple measurements produce results that are indistinguishable from those of probabilistic QM. Is that about right?

'Fluctuations' is probably the wrong word to use, since it sounds like you have some kind of random fluctuating field knocking it about like in Brownian motion (which you can add if you want, but it's not necessary). If you know where the particle starts and the initial form of the wave field, then the whole future evolution of the wave-particle system follows from the Schroedinger equation. An individual trajectory will be a nice smooth curve - it's just different from the trajectory which results from the -\nabla V force alone. The uncertainty/probabilistic element just comes from the fact that we don't know where the particle starts its trajectory.

Sure, and I have done this, but the wavepacket results are not fundamentally different from those of the plane-wave simplification. Part of the probability density of the wp gets reflected, and part gets transmitted. The math involved with the time-dependent picture is just more involved, which is why I guess the intro texts stick with the plane-wave description.

The nonsense with plane-waves only appears when you try to analyze the particle dynamics, not the evolution of the wave field.