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What does it mean by an infinite square well being transparent? I have been doing the calculation of the infinite square well and I came up with an answer $T = 1$ where $T$ for Transmission coefficient. But I can't really tell what it actually means in terms of physics. I would imagine a particle to be trapped in a infinite square well to be the inner electron very close to a Big nucleus, So what does it mean by transparent in this context? Does it mean No other wave function can interact with this electron?

If there is an infinite potential, why would wavefunction still be able to pass through it but not getting bounce off the edges? of the well?

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closed as unclear what you're asking by Qmechanic Mar 14 '14 at 14:11

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I put your pictures into the question, since they belong there and not in an answer. They wouldn't have been out of place in a comment, but those are more easily removed. In any case you should never edit an answer to clarify your question. Always do that by editing the actual question. $\endgroup$ – Wouter Oct 14 '13 at 9:51
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    $\begingroup$ There is nothing in this problem about an infinite square well. You have a finite square well here. The wavefunctions for an infinite square well are confined to the interior of the well, so $T$ and $R$ are ill-defined, since $A=B=F=G=0$. The wavenumbers $n\pi/2a$ are special here, not because they happen to be the wavenumbers of an infinite square well, but because they are the ones where an integral number of wavelengths fit in the well. In other words this is a resonance phenomenon. $\endgroup$ – Michael Brown Oct 14 '13 at 9:55
  • $\begingroup$ Echoing @MichaelBrown's comment: Discussing total transparency or reflectionlessness of incoming/outgoing scattering states do not make sense for the infinite square well/particle in a box, as asymptotic scattering states do not exist. I close the question as unclear what you're asking. $\endgroup$ – Qmechanic Mar 14 '14 at 14:11
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If I am understanding your question, the transmission coefficient of an infinite square well should not be zero. As far as what it means, I understand it to relate to the probability of a particle much like the wave function of a particle; however, instead of describing the probability of finding a particle at a point in space it describes the probability of a particle passing a barrier.

You may want to think of it as the transmission of light. A material that allows all light through would essentially have a transmission coefficient of 1; similarly, a barrier that does not block any particles would have a transmission coefficient of 1. Zero in either case would be the case that either no light or no particles pass through the material.

Thinking of this, the transmission coefficient of an infinite square well should definitely not be 1, as no particles should pass the edges of the well. This is also seen in the wave equation for a particle inside of an infinite square well in that the wave function is zero outside of the bounds of the well.

UPDATE: Although Michael may have cleared up some confusion in his comment, I will restate part of that here. The well you are asking about is actually a finite square well, rather than an infinite square well. The wave numbers for which $T = 1$ just happens to be the wave numbers that would be used to satisfy the boundary conditions of an initial square well. As you can see in the picture you posted, one of the dependencies of the value of $T$ is a cosine term. When this term is equal to 1 (when the wave number is $n\pi \over 2a$), the transmission coefficient will also be equal to one.

Note that this is the typical case when $E > U_0$, where $U_0$ is the potential well. Do note that even when $E > U_0$ there is a slight chance that the particle will be reflected by the barrier, although this is not taken into account with the transmission coefficient.

Likewise, if $E < U_0$, there is a slight chance that a particle will pass the barrier; however, the longer the particle is in this barrier, the more chance that the particle will be reflected back into the well. This is a region where the wave function is some multiple of $e^{-\alpha a}$. If the barrier is not an infinite width, the wave function will be non-zero when the barrier ends, and a harmonic wave function will return at this point. This indicates that there is a small probability that a particle will pass through a barrier that it classically should not be able to pass through. This phenomenon is known as quantum tunneling and it may be worth noting that this can happen even if $T = 0$, indicating that no particles travel through the barrier. In an infinite square well, the width of the barrier is infinite, so there is no probability of the particle leaving the barrier, although it may travel some distance inside of the barrier.

That may have been a little more information than you needed, but I find it an interesting exception to the transmission and reflection coefficients of a barrier.

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  • $\begingroup$ Thanks for your comment! I do have the same feeling as well, the infinite sq well should not be transparent. However, my lecture notes showed it is 1 with detailed calculation. I can show you if you are not convinced $\endgroup$ – el psy Congroo Oct 14 '13 at 9:23
  • $\begingroup$ How can quantum tunneling happens if T=0, and why an infinite square well has an infinite width? I supposed infinite square well is a well with finite width and infinite potentials at the boundaries. Sorry, I might need further explanation on this. Thanks $\endgroup$ – el psy Congroo Oct 14 '13 at 15:41

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