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Quantum tunnelling is a process that can happen in quantum mechanics but is forbidden in classical mechanics. Roughly speaking, a particle can possibly escape from a potential well or penetrate into a potential barrier. Obviously, the probability for a trapped particle to escape from a potential well should be handled with the time-dependent Schrödinger equation. But in most cases, we simply solve the static Schrödinger equation with proper boundary conditions (and match conditions around the turning points).

I wonder, how can a time-dependent process be handled as a static problem?

Emilio Pisanty
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Wein Eld
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  • You don't always treat tunnelling statically. Optical tunnelling is a long-established field, where a low-frequency field shining on an atom sets up a transient potential-energy barrier as it oscillates, and there full-TDSE methods are crucial. That said, you can still ask what happens with a static potential - the problem is still dynamical, but we do treat it as static. – Emilio Pisanty Apr 12 '17 at 09:51
  • I'm confused what you call "static" here - you're not talking about a time-dependent potential, are you? Then looking at the time-independent Schrödinger equation is, as always, enough to know what happens to every state since you can write every state as the superposition of energy eigenstates. What additional insight would solving the time-dependent equation give you that you cannot gain from superposing the solutions of the time-independent one? – ACuriousMind Apr 12 '17 at 09:57
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    Possible duplicate of Why can we treat quantum scattering problems as time-independent? – sammy gerbil Apr 12 '17 at 10:08
  • Since I suspect this will get closed it may not be worth writing an actual answer. Wien Eld is referring to problems with square barriers and wells. These are static in that the time dependent part of the SE is a phase. The static solution just gives probabilities for the particle on either side of a barrier or in or out of a finite well. The measurement of a particle outside the well is an irreversible process, which is not unitary. This question is related to the measurement problem, which is that the measurement process is not unitary quantum mechanics. – Lawrence B. Crowell Apr 12 '17 at 10:37
  • @ACuriousMind Well, I can accept that quantum tunnelling can be handled as static problem. What I need in this post is some intrigue explanation or proof of the equivalence between the time-dependent solution and time-independent solution to Schrödinger equation. For example, in the time-independent case, we vie $e^{ipx}$ as incident wave, $e^{-ipx}$ as reflecting wave and outside the barrier, we take only the outgoing wave etc... – Wein Eld Apr 12 '17 at 10:38
  • @EmilioPisanty Thank you for your recommendation. I will take a look. The motivation of this post is the above comment. – Wein Eld Apr 12 '17 at 10:43
  • @ACuriousMind Apparently, there are some ``moving of particles"-pictures in the interpretation of the time-independent wave function. And I think that interpretation is rough and hope to get some insights from SE. – Wein Eld Apr 12 '17 at 10:47
  • @WeinEld It seems that you've closed this yourself, so I guess you found your answer. But yeah, it boils down to the fact that if we find a state that obeys $H|\varphi\rangle = \hbar (\omega-i\gamma)|\varphi\rangle$, we can automatically construct a solution of the TDSE of the form $$|\psi(t)\rangle = e^{-\gamma t}e^{-i\omega t}|\varphi\rangle,$$ and bingo, we've got a dynamic solution. For additional reading, see the Fonda et al. paper referenced in this answer. – Emilio Pisanty Apr 12 '17 at 10:58

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