If a bunch of electrons, with energy E, traveling in x direction encounter a very wide potential barrier V0 > E, the ensemble wavefunction will exponentially decay within the barrier.
I thought that meant that there was a small probability of detecting an electron within the barrier. But the reflection coefficient of the probability current is unity. So what’s going on here?
Does it mean that there is an instantaneous chance of detecting an electron within the barrier but if TDSE were employed it would show it immediately shooting back out, or what?
I may be way off, but the fact that the reflection coeficiant is one may be due to the fact that the integral of the wavefuntion converges way out in infinity land. But thats just a WAG. My E-mag prof. and my physics prof. both discussed this (from different angles) - I should know how that works, but I don’t. Ah well.
Ok - physics undergrad checking in, so anyone who’s already got their degree can feel free to correct me. When a particle of energy E impacts on a barrier of height greater than E, you’re quite right in saying that the wavefunction decays exponentially inside the barrier. This means that there is a rapidly decaying probability of finding a particle inside the barrier, which in classical physics would have been impenetrable.
Because of the correspondence principle, though, and plain common sense, the particles can’t continue to move on indefinitely through the barrier - thus the reflection coefficient of one. Assuming that the barrier is semi-infinite (starts at some point and continues indefinitely from there) every particle will eventually bounce out of it and head back the way it came. The total flux of particles through the barrier is then zero: every one that goes in comes out again.
The really interesting aspect of this is when it comes to thin barriers, though. Because it’s decaying exponentially, after a finite length of barrier the wavefunction will still be non-zero. Thus, if the barrier ends the wavefunction will change back from a decaying to a propagating state, albeit one with lower intensity. This leads to the weirdness of quantum tunnelling, where particles can penetrate barriers classically supposed to be insurmountable. Hope this helps!
I don’t think that’s right for a semi-infinite barrier. I think the wave function inside the barrier in that case is identically zero. I can’t find the formula online, though. :-/ Anyone?
Sorry, Achernar, I think my phrasing was unclear. You’re correct for a barrier of infinite height; what I meant was a barrier of finite height but semi-infinite in extent. In that case, you get the exponential decay that’s under discussion.
Thanks for the good answers and the good link, but I couldn’t find answers to the following.
Picture a graph with the transmission coefficient (Tc) on the vertical axis and and E/V on the horizontal axis. When E = V the Tc jumps to unity but as E/V increases the Tc dips down below unity and then back up to 1 and then repeats with the dips getting shallower as E increases.
Now picture a second graph the same as the first except with the length of the barrier (L) plotted on the horizontal axis. At L = 0 the Tc, of course = 1, but as L increases the Tc drops way down and then back up in the shape of big U, and it does this at least twice.
I’m struggling here because of a lack of a basic quantum textbook - for those following along at home, French and Taylor’s Introduction to Quantum Mechanics (part of the MIT physics course) is very good, as is Alistair Rae’s ‘Quantum Mechanics’ (Institute of Physics).
To your question: as far as I recall, tranmission resonances such as you’re discussing are solely a property of barriers with finite depth (and finite height, as per Achernar’s point above).
Your question is sort of addressed in the section immediately after the one Achernar linked to above; the short version is that by tuning the height and thickness of the barrier, you can arrange for the waves travelling within that barrier to interfere with each other in such a way as to cancel out reflected waves and amplify transmitted ones. This leads to the variation seen in the transmission coefficient: the TC is clearly 1 when the reflected waves are completely cancelled out, and dips below one as changes in the conditions lead to stronger reflected waves.
Well you can’t directly observe an electron in a classically forbidden region, though you can imply from experimentation that it has passed through a classically forbidden region. If you directly observed an electron in a potential barrier it would defy the conservation of energy (even it’s form corrected by the Heisienburg uncertanity principle). That said the wavefunction is non-zero in the potential barrier.