Tutorial: Leaky Wells

Background

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 square 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.

Quasi-bound States

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.

No Java support detected The first applet shows the familiar 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] appear on the Math 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 = 3728.4 MeV/c2 (1 u = 931.5 MeV/c2 is the atomic mass unit).

The energies of these states (1.513 MeV and 6.005 MeV, respectively) also can be seen on the Math tab, where they are recorded as E0 and E1. These are true bound states of the finite well. Try changing the energy to see what effect this has on the waveform. From the Math tab, right click anywhere in the value field for either entry and select "Edit Parameter..." from the popup menu to bring up the Parameter Editor. Return to the Graphics: [x] tab and re-position the editor as necessary to allow an unobstructed view of the waveform. Use the slider to change the highlighted digits in the text field while observing the waveform; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. Notice how even the slightest deviation from either of the bound state energies leads to a waveform that violates the acceptability criteria for quantum wavefunctions (boundedness, continuity).

No Java support detected 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? You should find 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 assigned as those energies above and below the quasi-discrete value that produce waveforms for which the relative probability of finding the particle in the well drops to one-half of its quasi-discrete value.

Decaying States

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 suspect, and can be misleading. At the very least, the semi- classical argument just given is less than persuasive, and fails to give a complete description of the decay process. 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, suggesting that a particle described by a quasi-bound wavefunction would never escape the well. But this, too, is misleading inasmuch as a quasi-bound state could never describe true particle localization. [Being part of an energy continuum, such wavefunctions are inherently unnormalizable, so the initial probability of the particle being confined to the well diminishes steadily as more 'outside space' is made available for decay.] But suppose we 'tweak' the quasi-bound state just enough to render it normalizable. While the resulting wavefunction would still retain a large quasi-bound component, there also would be included the inevitable 'contaminants' with energies close to the quasi-discrete value, and all those states are delocalized to varying degrees. Thus, normalization renders the initial state a mixture of localized and delocalized [stationary] waves, and the evolution of this state invariably results in decay. The next applet illustrates this case.

No Java support detected In this applet the potential V(x) describes the same leaky well used previously (parameters chosen to model alpha decay from a radioactive nucleus). The initial wave denoted φ(x) is the lowest quasi-bound state modified by an exponentially decaying envelope that imparts normalizability. ψ(x)) is the result of propagating this initial wave forward in time according to the Schrödinger equation, and coincides with φ(x) at t = 0. The wts(..) function 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 0.9912. Switch to the Graphics: [x] tab where ψ(x) is plotted. For comparison, a placeholder for the the pure quasibound state with energy E = 1.508 MeV is also included. Make the latter state visible and confirm that the initial waveform differs only little from the quasi-bound [stationary] state which it modifies. [Although not shown, the average energy of the initial state remains essentially the quasi-discrete value, 1.508 MeV.] Notice the two-color display for the 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.

Observe the evolution of this modified waveform. Go to the Math tab and right click anywhere in the equation field for ψ(x), then select "Animate..." from the popup menuto activate the Animator Editor. Return to the Graphics: [x] tab and re-position the editor as necessary to afford an unobstructed view of the waveform. Click the Play button at the top right of the editor to begin evolution. Complete control over the developing waveform is afforded by the remaining editor controls: Stop , Reverse , Restart , Step Ahead , and Step Back .

Provision also exists for adjusting the refresh rate and the elapsed time between 'clock ticks'; both can be tweaked (with the 'clock' running) to achieve the desired visual effect.

The extensive coloring indicates that 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'. Return to the Math tab and monitor wts(..) to estimate how much time passes before the probability for finding the alpha particle in the nuclear well drops to half of its starting value (0.9912 ÷ 2 = 0.4956): this is the half-life for decay. Careful 'measurements' similar to those described below confirm that this decay process is in fact exponential (in the long-term -- see below). Note also the evolution of the quasi-bound state ψ0). Is it what you expected (adopt a color-for-phase plotting scheme for ψ0)?

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

No Java support detected In the next applet 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 φ(x) on the Math 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, ψ(x) is the result of propagating the initial wave forward in time according to the Schrödinger equation, and the wts(..) function 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. Although not calculated in the applet, the average alpha particle energy for this case is 2.622 MeV. 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.

From the Graphics: [x] tab, 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 wts(..), 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 time t is reported in 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 the whole story; recent model calculations (D. A. Dicus, W. W. Repko, R. F. Schwitters, and T. M. Tinsley, Time development of a quasistationary state, Phys. Rev. A65, 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.