Scattering states in one dimension

In the above discussion of the finite depth potential well, we had so far only considered wave functions with energy $E<0$. They gave rise to a discrete spectrum of energies with states that are localized within the potential well, that is outside the well the wave functions decay exponentially. This means that the probability to find the particle outside the well is exponentially small, i.e. the particle is bound to the potential well. The corresponding wave functions are therefore called bound states. Also in the example of the infinitely high potential well, we only had bound states because the particle was restricted within the well.

What about wave functions of the finite depth potential well with positive energies $E$? The discussion of these states leads us to the concept of scattering states with continuous arbitrary energies $E$ (continuous spectrum). Furthermore, we will find the tunnel effect which is an important quantum mechanical phenomenon. We discuss it first again within our general scheme of piecewise constant potentials.