Derivation of quantum tunnelling equation

Some weeks ago, Mr Nick Chan mentioned briefly that his class had some special tip of how to use the tranmission coefficient equation. I was intrigued then, but couldn't really figure what special tip there can be, especially with such gargantuan looking equation.

Well my dear readers, you may or may not be pleased to hear that I have a possible derivation to that particular equation. No more trying to flip through and fro the pages to look for the equation.

We can model the situation of electron tunneling through the barrier as Poisson distribution function. We want to find the probability of finding electrons outside the barrier. Such discrete events can are perfect for the Poisson distribution function.

Let X be the number of electrons that are outside the potential barrier. Then we want:

where λ is the expect number of occurences in a given interval

Now we need λ, so where to find it? Guesses are, this λ is related to the Heisenberg's uncertainty principle. We are about to find out:

Starting from ΔxΔp ≥ ћ/2, we do some manipulation and get 2ΔxΔp/ћ ≥ 1. Abit ad hoc here, but we must press on.

Let λ = 2ΔxΔp/ћ, then the most important coefficient to the Poisson distribution is e^-λ. This coefficient in this case is the transmission coefficient!

Then, T = e^(-2ΔxΔp/ћ) where Δx is the barrier width. We are not finished yet. There's still the Δp to be accounted for. In fact, this Δp is the energy that needs to be 'borrowed' in order for the electron to leap the barrier. We know kinetic energy of the electron is :

Then,

where ΔE is the energy to be 'borrowed'

ΔE = U - E where U is the max. potential of barrier and E is the present energy of electron.

Substituting all back, we get:

which is the exact equation that was printed in our physics notes.

So, recapping, our derivation depended on a few key concepts:

1. Poisson distribution. Rare events distributed in discrete manner in time and space.
2. The Uncertainty Principle.
3. The possibility that the energy can be 'borrowed' for somewhere and returned later. Possibly, if we borrow a small amount, the longer you can borrow this energy for. And thus the more probable the event.

Please remember that this is only my doodling on paper, the day before A levels Physics Paper 2. Anything written here might be possibly a load of utter rubbish, and can only serve at best as a memory guide. A heuristic model as Einstein might say.