How to Calculate Quantum Tunneling Probability for an Electron?

[iAlexN]

Homework Statement

A particle with the energy E < V$_{0}$ (V$_{0}$ > 0) moves in the potential V(x) = 0, x<0 ; V(x)= V$_{0}$, 0<x<d and V(x)= 0, x>d. Measure the probability that the particle will tunnel through the barrier by calculating the absolute value of the ratio squared, |$\Psi$(d)/$\Psi$(0)|$^{2}$ between the values of the wave function at x=d and x = 0

Calculate the probability for an electron, when V$_{0}$- E=1 eV and d = 1 Å.

Homework Equations

$\Psi$(x) = ae$^{\kappa*x}$+be$^{-\kappa*x}$, $\kappa$ = $\sqrt{2m( V_{0}-E)/\hbar^{2}}$ for E<V$_{0}$

The Attempt at a Solution

Firstly I get:

$\kappa$ = $\sqrt{2m(1)/\hbar^{2}}$ for E<V$_{0}$

However, the problem is with this wave function:

$\Psi$(x) = ae$^{\kappa*x}$+be$^{-\kappa*x}$

In order to calculate the ratio, |$\Psi$(d)/$\Psi$(0)|$^{2}$, I think I have to define a and b somehow, but I don't know where to start.

Thanks!

 

[frogjg2003]

You have to solve Shroedinger's equation in all three regions. Then you need to apply the appropriate boundary conditions.