Particle Tunneling/ Barrier Potential

1. Nov 8, 2011

tayslayer

1. Show that as the height of the barrier approaches the electron's incident kinetic energy E an approximation to the transmission that includes just a (the barrier width) and the wave number of the incident electron k=((2mE)^1/2)/(h/2pi) can be obtained. The expression has no Vo term. (Hint: use l'Hopitals rule and do a lowest needed order Taylor expansion for T(Vo) about Vo=E)

T=16E/Vo(1-E/Vo)e^(-2αa)
T=[1+sinh^2(αa)/((4E/Vo)(1-E/Vo)]^-1
α=((2m(Vo-E))^1/2)/(h/2pi)
g(z)=g(a)+dg(z)/dz|z=a (z-a)+d^2g(z)/dz^2|z=a (z-a)^2/2!+....

I tried the first equation and derived it. When I substituted it into the Taylor Series equation (last equation) I got zero instead of a function dependent upon a and k.