Question

The potential barrier is defined by V(x) = ((A^2-(x^2)*(b^2))^0.5  where x is less than or equal to A/b.

First we want to sketch V(x)

Then we run a particle into it wth mass m and energy E (which is less than A). Now that we have run the particle into it we are asked to derive an expression for the probability of penetration in terms of D = E/A and sin (y) = bx/A.

Finally, we are to evaluate our expression in numerically for D=0.1 to 1.0 in steps of 0.1 for an electron stricking such a barrier with A=10 eV and b=2eV/angstrom.

Help!! I have no idea how to start this monster. How do I sketch this??

Answer 1

First, the graph of barrier energy is below, at the bottom.

Second the tranmission coefficient for a particle of mass m and energy E through a barrier V(x) is found here:

http://en.wikipedia.org/wiki/Transmission_coefficient#WKB_approximation

You know V(x) = sqrt(A^2-(b*x)^2) and the value of E and need to evaluate the integral

integral (from 0 to 5) sqrt(sqrt(A^2-(b*x)^2) -E) *dx = integral (sqrt(A*sqrt(1- (bx/A)^2)-D))*dx

The integration result is way too long to be written here (you have a double sqrt), but here below is the link to the result:

http://integrals.wolfram.com/index.jsp?expr=sqrt%28sqrt%28A^2-%28bx%29^2%29-E%29&random=false

http://integrals.wolfram.com/index.jsp?expr=sqrt%28A*sqrt%281-%28bx%2FA%29^2%29-E%29&random=false

Observation: There is one trick one can do on this. You can APPROXIMATE the form of graph below with

V(x) =A*sin(y+pi/2) and then compute the integral

integral (from 0 to pi/2) [sqrt((A*sin(y+pi/2) -E)]*dy

However doing this online with mathematica gives an error because of the limited time allotted.

Finally to evaluate the result of the integral above for 10 different value of D you need to write a program. It is imposiible to do it by hand.