Simple (?) solving of equations re Tunneling

[rwooduk]

Revising for my exams, stuck on how to do this:

http://imageshack.us/photo/my-images/594/mr15.jpg/

please could someone tell me how to get (A/F)?

thanks for any help!

it's not a homework question, just revision so a direct solution would be more than welcome.

 

[vanhees71]

A bit more context would be good!

 

[rwooduk]

A bit more context would be good!

Hi, not sure what you mean, they get A/F from those equations.

If you mean where did they come from, it's an electron approaching a potential barrier and tunneling through it. there are 3 equations for the wavefunction, the incident electron (which has approaching and reflection componants), the tunelling wavefunction, which has a wavefunction with a potential componant (and also a reflective componant), and the exiting electron which has no reflective componant.

approaching the barrier the soln of PSI is:

PSI 1 = A exp ik1x + B exp -k1x

inside the barrier the soln of PSI is:

PSI 2 = C exp -k2x + D exp k2x

where k = sqrt(2m(E-U)) / h-bar

leaving the barrier

PSI 3 = F exp ik1x

at the boundary of the barrier PSI 1 = PSI 2 . thus differential of PSI 1 = differential of PSI 2

at the other boundary of the barrier PSI 2 = PSI 3 . thus differential of PSI 2 = differential of PSI 3

this is where the terms in the photo come from.

know where the terms come from, just don't know how to solve for A / F

 

[rwooduk]

ok this is more complicated than it looks lol

an equally good derivation is for B^2 / A^2 which is the reflection coefficient, if A / F comes up in the exam I am buggered, but i did figure it for B^2 / A^2

here are the steps:

do k2 X [1] + [2] to give [6]
do k2 X [1] - [2] to give [7]

sub [3] into [4] for F exp ik1L to give [8]

put [6] and [7] for D and C into [8] to give [9]

divide through by B, rearrange and take the modulus

QED.

 

[rwooduk]

derivation for A/F in detail here!

http://lynx.uio.no/trine/fys3510/alpha_transmission_eng.pdf

 

[vanhees71]

Note that this is only an approximation for the Coulomb barrier. The Coulomb potential has no plane waves as asymptotic states!