A scanning tunneling microscope (STM) can precisely determine the depths of surface features because the current through its tip is very sensitive to differences in the width of the gap between the tip and the sample surface. Assume that in this direction the electron wave function falls off exponentially with a decay length of 0.108 nm - that is with C=9.26 nm-1. Determine the ratio of the current when the STM tip is 0.489 nm above a surface feature to the current when the tip is 0.512 nm above the surface.
Probability = e^(-2w/n)
The Attempt at a Solution
I assumed neta(n) was constant regardless of how far it was from the surface.
From there, I equated to two probability equations for the first and second distance.
Let P = probability
P1 = e^(-2*0.489/n)
ln P1 = (-2*0.489)/n
n*(ln P1) = (-2*0.489)
P2 = e^(-2*0.512/n)
ln P2 = (-2*0.512)/n
n = (-2*0.489) / (ln P2)
(-2*0.489)(ln P1) / (ln P2) = (-2*0.489)
(0.489)(ln P1) = (0.489)(ln P2)
Now I'm not too sure where to go from here or if my approach is even correct.
Any help would be appreciated, thanks!