Welcome to lecture 8. In this lecture, we will examine two specific potentials, the finite potential and quantum tunneling. In order to understand how boundary conditions link wave functions so that we can describe quantum effects. This lecture will be broken up into two pieces. The first will focus on the finite potential well, and the second will examine quantum tunneling. First, let's look at the finite potential well. The setup for this problem is very similar to the particle in infinite square potential problem. However in this case, the potential which traps the particle is finite. This is a more realistic model of, say, electrons in bound states given that electrons can escape finite potential wells, like they can escape from atoms and molecules. For our purposes, we will call the magnitude of the potential in regions 1 and 3 U0. For this problem, we will assume that the energy of the particle is less than U0, meaning the height of the walls, so that we have particle trapped in the well. Since the walls of the trap are finite, we now have to explicitly solve psi in regions 1 and 3, since we cannot assume that the wave function is 0 in those regions. The solutions for the three regions are, a growing and decaying exponential in regions 1 and 3 respectively, while inside the well, psi is a combination of sine and cosine functions. Here is a graphical representation of the bound states for this problem. Notice that since the potential is finite, the particle can exist in the walls of the well, being in regions 1 and 3. This of course is an intuitive dust, as we live in a world where objects trapped in rooms do not appear in walls. However, this property of quantum systems is exactly why quantum tunneling occurs. We will examine that in more detail later. For now, let's solve the Schrodinger equation for this problem to get wave functions which describe what we see here. By solving the Schrodinger equation and applying boundary conditions, we will be able to determine among other things the energy levels of the bound states.