If, say, a particle with energy $E<V_0$, approaches a finite potential barrier with height $V_0$, and happens to tunnel through, where would the particle be during the time period when it is to the left of the potential barrier and to the right of the potential barrier? Surely there must be a finite amount of time for it to travel through to the other side, unless it simply teleports there? If it travels through with energy less than $V_0$, however, doesn't that mean it cannot enter in the region of the potential barrier?
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asked May 14, 2011 at 7:37
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3$\begingroup$ It is precisely where its probability density is telling you it should be; there are no additional special effects here that differ from other QM situations. Why do you think it is a problem for it be located inside of the barrier? That location is only classically unreachable. But this is quantum mechanics... $\endgroup$– MarekMay 14, 2011 at 8:29
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$\begingroup$ Related: physics.stackexchange.com/q/11188/2451 and links therein. $\endgroup$– Qmechanic ♦Apr 7, 2014 at 12:32
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4 Answers
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Isn't the whole point here that one cannot say where the particle IS exactly? One can only calculate the probabilities of it being at one place.
Tunneling means the probability of it being inside the barrier isn't zero (since we want the probability distrubition to be continuous). There is always penetration of the wave function into the barrier.
IMHO tunneling means the penetration goes deep enough to actually reach the other side, so the wave function of the particle is propagated further on that side too, meaning there's a chance the particle went through. During the passing the particle has had a chance of being inside the barrier.
I don't know if it's correct to say that when the particle has passed, it has been inside the barrier, but that just because the notion of the particle actually being somewhere is somewhat wrong.
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As Marek sais, the particle may be found "inside" the barrier, if you like. It means you can really find it there as well as outside. But in QM a particle is a wave and is "created" by the whole volume involved. It is not permanently "localized" or "concentrated".
answered May 14, 2011 at 8:39
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“…where would the particle be during the time period when it is to the left of the potential barrier and to the right of the potential barrier?”
The answer is in the barrier as others above have stated.
I think a key point that would help (IMHO) to explain what is meant by locating a particle in the barrier. Since the particle’s wave function has some probability of being in the barrier then in quantum mechanics one should be able to observe it in the barrier. So to observe this negative kinetic energy particle in the barrier, one has to localize it. To localize implies to shine a short enough photon in order to locate the particle inside the barrier. When this is done, the negative kinetic energy particle now has an increase in energy from the photon; therefore, the negative kinetic energy particle now has acquired a positive kinetic energy and is found outside the barrier, not in the barrier. So the information of this positive energy particle tells us the details of the particle in the barrier.
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The question "how long does it take for the particle to tunnel?" seems to be more interesting than "where is the particle during a tunneling". Quantum mechanics predicts that for an opaque barrier the delay time is independent on the length of the barrier (see Hartman effect) with the obvious problems when the barrier becomes very long.
