Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can be shown (Razavy, Quantum theory of tunneling, chapter 2.1; arXiv:quant-ph/0507239 that the position and momentum uncertainties for an electron in such a system are linked by the Heisenberg uncertainty relation. A similar argument can be done for Energy and time observables.

I am aware that here (Reference frame involved in the Schrödinger's equation) a system consisting of an LC circuit can be treated as a quantum harmonic oscillator and in that case, current and voltage are linked by a non-zero commutator, thus implying (as can be seen in arXiv:quant-ph/0507239 that they will also be tied by the uncertainty principle with the right-hand side of the uncertainty being non zero.

What if there is no inductance in the system and we consider a bunch of electrons tunneling through an insulator. Can we say anything about the current and voltage variables at the level of Heisenberg uncertainty? What other variables could be linked in that scenario by an uncertainty relation?

To clarify, I am considering a voltage given by intrinsic properties of the device, rather than an external voltage but I am happy to hear arguments for both cases!

Thanks in advance!

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    $\begingroup$ Please link to arXiv abstract pages instead of pdfs. Otherwise: welcome! $\endgroup$ – Emilio Pisanty May 11 at 16:18
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    $\begingroup$ Can you be more detailed about what exact system you're considering? In particular, what does "voltage" mean in this context? Is it being provided by an external voltage source? Is it an effect of differing electron densities on one side of the barrier vs. the other? Or is it something else? $\endgroup$ – probably_someone May 11 at 16:57
  • $\begingroup$ @probably_someone Thanks for your questions, I amended the main text. $\endgroup$ – JoelleB May 19 at 13:38

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