The square well is the simplest particle trap imagineable in one dimension, and as such often serves as the first introduction to the use of Schrödinger's equation in describing the peculiarities of the quantum world. Here we find that a particle bound to the well can have only certain well-defined energies; we say the particle energy is quantized. The energy spectrum consists of one or more discrete levels, and the corresponding [stationary state] wavefunctions are concentrated in the vicinity of the well, as required by an application of the Born probability rule. The finite square well also admits wavefunctions that are not localized in the vicinity of the well. These are the unbound states; they describe particle scattering from the well and have energies that form a continuum [reflecting the physical reality that scattering experiments can be done with particles having any value of kinetic energy].

The finite well employs infinitely wide potential barriers on either side to confine the particle to the well region. If we reduce the width of one (or both) of those barriers, we are left with a leaky well. A particle initially confined to such a well typically will not remain there indefinitely, but instead will tunnel through the barrier(s) to escape the attraction of the binding force. The alpha decay of a radioactive nucleus is a real-world example of this process in action.

The escape probability from a leaky well depends strongly on the transparency of the barrier(s). From a stationary state viewpoint, 'leakage' from a leaky well is accompanied by a continuum of allowed energy levels, with [nearly] all the stationary state wavefunctions delocalized and describing unbound states. But embedded in this continuum are special levels, termed quasi-discrete, for which the wavefunctions remain concentrated in the vicinity of the potential well. These are the quasi-bound states, and systems giving rise to such states are called weakly quantized to suggest a state of affairs intermediate between quantized (having truly discrete energy levels) and unquantized (possessing a continuous, essentially featureless spectrum). These quasi-bound states, being part of a continuum, are unstable; the slightest disturbance will result in transitions [so-called radiationless transitions because they involve little or no exchange of energy] to one or another of the [neighboring] unbound states, leading to decay. The following two applets illustrate the distinction between true and quasi- bound states and how a weakly quantized system differs from an 'ordinary' one.

The first applet below shows the usual finite square well potential V(x) along with the two lowest stationary states of a particle in this well. The well depth U = 30 MeV and width a  = 5.00 fm [1 fm = 10–15 m] are recorded on the Formulas tab of the applet; these values are chosen to model the potential of a particle bound to an atomic nucleus. The mass is that of an alpha particle [a composite object formed from 2 protons and 2 neutrons], m = 4.0026 u = 3728 MeV/c2 (1 u = 931.5 MeV/c2 is the atomic mass unit).

The energy of each state (1.513 MeV and 6.005 MeV, respectively) can be read by right-clicking on its symbol in the list to the right of the graph and noting the value for E in the Function Properties field. These are true bound states of the finite well; even the slightest deviation from either of these energies leads to a waveform that violates the acceptability criteria for quantum wavefunctions (boundedness, continuity). Try changing the energy to see what effect this has on the waveform: select "Change Energy", then use the slider to change the highlighted digits in the text field [the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field].

The next applet shows the leaky well that derives from the square well above when the width of the right-side barrier is reduced from infinity to a finite value, in this case w = b – a = 1.00 fm. H is the barrier height, and is the same as the well depth in this example. All other well properties remain the same as before. The list to the right of the graph has placeholders for two stationary states of the alpha particle in this well. To show a stationary state, right-click on its placeholder, click the visibility icon beside the "Real" label in the Colors | Visibilities field, then choose the "OK" button.

The energies of these states have been "tuned" to produce waveforms that are still localized in the well. The waveforms resemble in every way (symmetry, number of nodes, etc.) their bound state counterparts in the finite well; they are the two lowest quasi-bound states of our leaky well. The energies of these states (now 1.508 MeV and 5.989 MeV, respectively) are among the quasi-discrete values, and slightly lower than the corresponding bound state energies of the finite well. But these quasi-discrete levels are special energies embedded in a continuum of other possibilities. To see what this means, try changing the energy as before, and observe the effect on the waveform. How do the leaky well results differ from the finite-well case? Notice that the leaky well admits acceptable waveforms for every energy. Notice, too, how sharp in energy the transition is between a waveform that is "large inside - small outside" to one that is "small inside - large outside". [This dramatic "inside-outside" behavior could be used to assign a width to the quasi-discrete levels. For instance, the level edges could be taken as those energies above and below the quasi-discrete value that produce waveforms for which the particle has the same probability of being found within the well as outside.]
 

We have seen what weak quantization means for the stationary state description of a quantum system, but what about non-stationary states? In particular, suppose the initial state were one describing a particle that is known with near certainty to be in the well. Does such a particle remain in the well indefinitely? In practice the answer is no, the particle will invariably escape the well. A simple [semi- classical] argument explains why. Within the confines of the potential well, our particle has some kinetic energy and associated [classical] motion that guarrantees it will collide repeatedly with the well edges. But with each such collision at the right-side boundary, there is some [quantum] probability that it will tunnel through the barrier to escape the well. If we think in terms of many particles instead of just one, the 'leakage' rate will be proportional to the number still remaining in the well. This is the classic condition for exponential decay; the decay constant itself depends on the transparency of the barrier to particles of the supposed energy, and to their speed, which determines how often each particle makes a barrier collision and thus has an opportunity to escape. This deceptively simple argument is presented in most texts on beginning quantum mechanics, and is frequently invoked to obtain qualitative and even semi-quantitative results for the decay of radioactive nuclei.

Yet mixing together classical and quantum ideas this way is dangerous, and can be misleading. At the very least, the semi- classical argument just given is less than compelling, and just plain wrong in some cases. For instance, consider what happens if the initial state is a quasi-bound one: this is a stationary state that is localized to the well, and will remain so indefinitely! A particle described by a quasi-bound wavefunction would never escape the well. But this is the exception, and a very unlikely one at that. While the wavefunction for a particle initially localized to the well could be expected to have a large quasi-bound component, at the same time there would be the inevitable contaminants with energies nearly coincident with the quasi-discrete level, and all those states are delocalized to varying degrees. Thus, the initial state more likely is a mixture of localized and delocalized states, and the evolution of this state, though quite complicated, invariably results in decay. The applet that follows illustrates this case.

In the applet below the potential V(x) describes the same leaky well used previously (parameters chosen to model alpha decay from a radioactive nucleus). The initial wave is a mixture of stationary waves that includes the lowest quasi-bound state; in fact, there are 200 such waves in the mixture with amplitudes distributed according to a gaussian profile centered on the quasi-discrete energy value 1.508 MeV, as shown on the (current) Graphics: [E] tab of the applet (see also the expression for ψ(E) on the Formulas tab). The average energy of this state is essentially the quasi-discrete value, 1.508 MeV [right click on the wavefunction entry in the list to the right of the graph and inspect the Function Properties field]. Switch to the Graphics: [x] tab to confirm that the resulting initial waveform is strongly localized in the potential well. Notice the two-color display for the coordinate waveform. The color represents the phase of the wave at each point; since this wavefunction is real, the phase is either 0 or 180° for positive and negative values, respectively. Finally, switch to the Formulas tab; the prob(...) function listed there reports the probability of finding the alpha in the range 0-5 fm, i.e., inside the nuclear well. At the instant t = 0, we see that the alpha particle is found in the well with probability 0.9882.

Return to the Graphics: [x] tab and observe the evolution of this waveform by clicking the Play button on the bottom panel of the applet. Complete control over the developing waveform is afforded by the remaining buttons on the panel: Stop , Reverse , Restart , Step Ahead , and Step Back . [Also, right-clicking on the panel allows for adjusting the frame rate, the number of 'clock ticks' between screen redraws, and the elapsed time between 'ticks'; all affect the speed of animation and can be tweaked (the first two with the 'clock' running) to achieve the most pleasing visual effect.]. The extensive coloring indicates the developing waveform is truly complex-valued. Notice the relentless loss of probability from the well, corresponding to the alpha decay of this 'radioactive nucleus'. Switch to the Graphics: [E] tab and describe the evolution of the spectral components of this wave. Can you account for these observations? Finally, switch to the Formulas tab and monitor prob(...) to estimate how much time passes before the probability for finding the alpha particle in the nuclear well diminishes to half of its starting value (0.9882 ÷ 2 = 0.4941): this is the half-life for decay. While our nucleus unquestionably undergoes decay, careful 'measurements' similar to those described below would confirm that this decay process is in fact not exponential. For comparison, a placeholder for the the pure quasibound state with energy E =  1.508 MeV is also included on the Graphics: [x] tab. Make this state visible and inspect its time evolution. Is it what you expected?

Now contrast the previous observations with those that result from using an initial waveform that does not derive directly from a quasi-bound state.

In the applet below the initial wave is again confined to the potential well, to represent an alpha particle with its parent nucleus. The profile for this initial waveform is recorded as ψ(x) on the Formulas tab of the applet, and is a gaussian. The wave is centered at d = 2.5 fm (the midpoint of the well), and has width w = 1.0 fm, narrow enough to make the initial probability of finding the alpha outside the well negligible. Again the prob(...) function listed on the Formulas tab reports the probability of finding the alpha in the range 0-5 fm, i.e., inside the nuclear well. At the instant t = 0, we see that the alpha particle is found in the well with probability 0.9996, or near certainty. Switch to the Graphics [x] tab and note that the average alpha particle energy this time is 2.622 MeV [right click on the wavefunction entry in the list to the right of the graph and inspect the Function Properties field]. As this is still far below the height of the potential well (30 Mev) [and somewhat above the lowest quasi-discrete level energy 1.508 MeV], the alpha has insufficient energy to surmount the barrier and instead must tunnel through it to escape.

Now examine the evolution of this initial waveform, and observe how it differs from the previous case. Notice the waveform ‘‘pulsations’’ that characterize the early stages of decay. Note, too, that these pulsations gradually fade, eventually transforming the original waveform to one marked by a steady flow of probability from the nuclear well.

As the simulation proceeds, we can monitor the survival probability prob(...), the probability that the alpha particle remains within the confines of the nuclear well. We recorded the survival probability over thousands of time steps and plotted the data at regular intervals on a logarithmic scale; the results are shown here. The graph abscissa is time t in units of attoseconds (1 as = 10–18 s). The data shows clear short-term fluctuations that give way to a steady decline after about t = 0.03 as. The short-term fluctuations accompany the early pulsating waveform, while the straight-line appearance of the long-term data on a logarithmic scale implies the exponential decay that accompanies the steady stream of probability flowing from the nuclear well. The slope of this line, 5.88 × 1018 s−1, is the characteristic decay rate λ, from which we obtain the half-life T1/2 = ln(2/λ) = 1.18 ×10−19 s for this process. How does this half-life compare with that obtained from the previous simulation? Should they be the same? Explain.

We can also compare T1/2 with the standard semi-classical treatment that predicts exponential decay at a rate equal to the collision frequency multiplied by the barrier transparency at this energy. For (kinetic) energy E = 2.622 MeV, the alpha has velocity v/c = [2(2.622)/3728]1/2 = 3.7505 ×10−2, implying a collision frequency f = v/2L = 1.124 × 1021 Hz. And for alpha particles with E = 2.622 MeV, the transmission factor for a square barrier 30 MeV high and 1 fm wide is found to be T = 1.318 × 10−2, giving a predicted decay rate λ = f T(E) = 1.48 × 1019 s−1 and half-life T1/2 = 4.68 × 10−20 s. This order of magnitude agreement is likely all that we can expect from such a simplistic argument.

As these simulations clearly indicate, decay from a leaky well is far more complex than the semi-classical argument would have us believe. We can summarize our observations this way: In the near term, the initial state develops in a way that reflects its precise makeup, and this will differ from case to case. Only after a sufficiently long time has passed will the evolution resemble the expected exponential decay predicted by the semi-classical theory, irrespective of the initial condition. Even the latter assertion is not always true; indeed, our first simulation stands as a counterexample. Moreover, recent model calculations (D. A. Dicus, W. W. Repko, R. F. Schwitters, and T. M. Tinsley, ‘‘Time development of a quasistationary state’’, Phys. Rev. A 65, 032116 (2002)) suggest that the exponential decay law will eventually fail even in those cases where it is observed, ultimately giving way to some form of power law behavior.