Quantum Tunnelling - Transmission Probability

[Silversonic]

Homework Statement

It's not really a given problem, it's more of a section of my lecturing I truly just don't understand.

I'm given that the probability of transmission (T) is the ratio of the intensities of the transmitted wave and the incident wave. However, a bunch of math gives also that;

T = [16E(V-E)/V^2]*e^{-2Ba}

I hope this equation is something many of you guys have met. It's the same equation given here;

http://en.wikipedia.org/wiki/Transmission_coefficient#WKB_approximation

This is for a potential V > E, and the fact that we can approximate e^{-2Ba} << 1. However, my notes tell me that the probability of tunnelling through the barrier can be calculated just by calculating e^{-2Ba}, without the co-efficient in front of it. Maybe I'm being really stupid, but why is this? It doesn't make sense to me. I'm under the impression that "probability of tunnelling" is synonymous to "probability of transmission", but maybe it's not? I know what T is equal to, but my notes eradicate the co-efficient in front of the exponential term, and I don't understand why.

 

[vela]

You're correct that the transmission coefficient is the probability of transmission.

I'm guessing your notes contain an order-of-magnitude calculation. If E and V are comparable, the factor out front is of order 1, so the exponential term is going to dominate. In other words, you don't care if the probability of tunneling is two in a billion as opposed to one in a billion. All you need to know is that it's really small.

 

[Silversonic]

You're correct that the transmission coefficient is the probability of transmission.

I'm guessing your notes contain an order-of-magnitude calculation. If E and V are comparable, the factor out front is of order 1, so the exponential term is going to dominate. In other words, you don't care if the probability of tunneling is two in a billion as opposed to one in a billion. All you need to know is that it's really small.

Thanks for the reply. I guess that last part could be true, but the case is this in my notes; an electron with energy deficit 1eV and barrier of 2 angstroms wide. All it does is calculate e^{-2Ba}, telling me it's roughly equal to T, to come out as 0.13 and tells me this is the probability of the electron making it through the barrier. i.e. if 100 electrons tried to, statistically 13 would make it. That probability is pretty high.

If the factor in front were equal to 1, this would make sense to me. However, I don't see how you could determine the factor in front was anywhere close to 1, with only the value for the energy deficit and no information about the potential or the energy separately. And I'm not sure why my notes tell me that it is.

 

[vela]

If you set E=V-1 and plot the front factor from V=1 up (V has to be more than 1 otherwise the electron's energy would be negative), you'll see it maxes out at about 4, so the front factor is of order 1. At best, you could take 0.13 as a really rough estimate of the probability.

 

[Silversonic]

If you set E=V-1 and plot the front factor from V=1 up (V has to be more than 1 otherwise the electron's energy would be negative), you'll see it maxes out at about 4, so the front factor is of order 1. At best, you could take 0.13 as a really rough estimate of the probability.

Basically, if the front factor did take a value near four, the value for T would come out as about 0.52 instead, which isn't what I'd call 'roughly' equal to 0.13. Considering we're talking about probabilities here and the maximum value we could attain is 1. Unless this is what you meant by "really rough estimate".

This is just all confusing me, because I'm given two questions. One asks me to calculate the approximate fraction of electrons that will succeed in penetrating a barrier, and in this I'm given the separate values of V_b and E.

Another question after that asks me to calculate the approximate probability an electron will jump a certain vacuum gap in an STM experiment, and here I'm given ONLY the energy deficit.

The way I look at it, the fraction of electrons that will succeed in penetrating a barrier is pretty much synonymous to the approximate probability an electron will jump a gap. Basically meaning I'd use the same equation for both. But in one I'm given V_b and E, so I can calculate the factor in front, and in another I'm not - so I can't calculate the factor.

Apologies if this confuses.

 

[vela]

It's not confusing. Your questions are reasonable. It really boils down to how good of an approximation you want. I'd ask your instructor what he or she expects. Without knowing the height of the potential barrier, you can't give an exact answer.