Tunneling with broadening of energy levels

[MrPhoenix]

Dear all,
I was wondering about the problem of calculating the probability that an electron will tunnel through a certain barrier (let's assume it is a constant value). Problem is that the initial and final energies of the electron are fixed and possesses a certain broadening (for example gaussian on each of initial and final energies). Does anyone have any suggestion on how to treat such problem?

Any comment is deeply appreciated.

 

[Avodyne]

This is not enough information. By superposing states in a narrow energy band, you can get states with very different physical properties; for example, you can get a wave packet localized (in position) on the left side of the barrier, and moving to the left. Now tunneling is not an issue because the wave packet never hits the barrier!

The interesting problem is to create a wave packet initially on the left, but moving to the right, towards the barrier. It will hit the barrier, and then two packets will emerge, one on the left moving to the left, and one on the right moving to the right. The probability of the transmitted packet will be given (to a good approximation) by the usual formula from the plane-wave analysis. It's possible to show this analytically by making various approximations (I once worked through it, but I'm sorry I can't supply a specific reference where it's done).

 

[HomogenousCow]

Dear all,
I was wondering about the problem of calculating the probability that an electron will tunnel through a certain barrier (let's assume it is a constant value). Problem is that the initial and final energies of the electron are fixed and possesses a certain broadening (for example gaussian on each of initial and final energies). Does anyone have any suggestion on how to treat such problem?

Any comment is deeply appreciated.

What you can do is look at the result for a planewave and then sum them according to your choice of superpositions.