Resonance between two rectangular barriers

Here a particle is caught between two rectangular barriers. If the barriers were infinitely thick, this would just be the rectangular potential well with two bound states. The bound state energies E0 and E1 of the corresponding potential well are indicated on the vertical axis (using the scale defined by the graph of the potential). Due to the finit thickness of the barriers, the particle can leave the well by tunneling through its walls.

The initial state is a superposition of the ground state (energy E0) and the excited state (energy E1) of the potential well. In the well with thin walls, these bound states turn into "resonances". A resonance behaves like a bound state for some time, but finally escapes towards infinity like a scattering state. Notice that the excited state tunnels through the barriers rather quick. So, after a short while only a shape resembling the ground-state wave function of the well remains between the barrier. This resonance has a longer lifetime, that is, it needs much longer to escape.

Due to our method of visualizing the wave function, it appears that the part of the wave packet that tunnels through the barrier is much larger than the wave function that is initially trapped inside. But the flow of probability is determined by the square of the wave function. Squaring the function above increases the contrast between the large and the small parts of the plotted function considerably.