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.