Ralph Waldo Emerson
The Conduct of Life, 1860, §VII ¶24

     A theoretical and computational analysis in both classical and quantum mechanics demonstrates the existence of nonlinear resonance zones in the 1-dimensional approximation for extreme Stark states of hydrogen perturbed by a microwave field. Comparison of simulations in the “physical” phase space with Koch’s laboratory results provide the first explanation of the widths of his regions of suppressed ionization in terms of the expansion of nonlinear resonance zones. It is also confirmed that the locations of the said regions are predicted with great precision in terms of resonance zones.

     Also included is the first demonstration of the existence of resonance zones in the quantum mechanical Hilbert space, and the overlap of the zones in a manner exactly analogous to the classical behavior.

     Since the mid-1970’s, experimental teams led by James Bayfield and Peter Koch have been studying the behavior of highly excited states of hydrogen perturbed by electromagnetic microwaves.¹ Among their findings has been a series of stable regions in the space of energy levels of the atom:

Figure 1. Plot of the 10% (squares) and 90% (x’s) ionization thresholds versus initial energy level for hydrogen. This is the microwave peak field strength at which the given percentage of the atoms in the beam ionizes. The microwave frequency is 1.509·10-6 (n=9.923 GHz). [877PMK, Figure 5, reproduced with permission of the author.]

These data show the perturbation strength at which the atoms start to ionize, as a function of initial energy level. The overall downward trend reflects the sensible fact that it takes less force to ionize a more highly excited atom. But there are regions where that trend is broken, where the ionization threshold remains roughly constant (or even increases slightly!) over a range of several energy levels. This intriguing feature of the data has been the subject of extended debate for many years. It has been widely agreed that the stable regions correspond to resonance zones [Appendix A] in the phase space of the hydrogen atom, which is a nonlinear oscillator. But this has raised two questions that have gone unanswered until now:

• The excited states of hydrogen are in the quasi-classical regime; they are on the boundary of the correspondence principal and have behavior which may be described by either Hamilton’s or Schrodinger’s equations. The stability can be explained in classical mechanics in terms of resonance zones, but what is the corresponding explanation in quantum mechanics?
• The centers of the stable regions in the space of energy levels match very closely the predicted locations of particular resonance zones. But do the resonance zones also account for the widths of the stable regions?

     The first question is answered here, in Section 4, by showing that the hydrogen atom Hilbert space in quantum mechanics has a resonance structure corresponding to that in the classical phase space.

     The second question is also answered here, in the affirmative, by numerical simulations of the behavior of the atom at various levels of perturbation strength. These simulations, presented in Section 2, show that the resonance zones corresponding to each stable region grow with the strength of the perturbation, reaching maximum widths before they break down and join the chaotic sea. These maximum widths are close to the widths of the stable regions observed in the laboratory.

The harmonically driven Stark states of hydrogen (HSH) model

     The complex behavior that makes hydrogen interesting arises under a perturbation that destroys a constant of its motion, changing it from an integrable system to quasi-integrable[914MBu, §1.2]. The perturbation is a monochromatic, linearly polarized, electromagnetic wave. The energy of interaction is just the scalar product of the electric vector of the wave with the dipole moment of the atom.

     The excited states used in the laboratory are prepared by laser excitation in the presence of a static electric field. These states are therefore extreme Starkor 1-dimensional states.² In these states, the classical orbit or the quantum mechanical orbital of the electron lies almost along a straight line and the electron remains predominantly on one side of the proton. The electrostatic field is used only to prepare the initial states, and is not active in the region of the microwave perturbation. The polarization of the microwave wave is lined up with the Stark axis of the atom. So the energy of the perturbed system can be modeled as:

HSH energy

     The energy of the unperturbed 1-dimensional hydrogen atom is, in atomic units:

,

     When the HSH perturbation is added to this energy in the physical coordinates:

,

Figure 2. Three orbits of the HSH atom, viewed in the phase space of the unperturbed atom. The experimental parameters are F=1.947·10-10 (1.000 V/cm) and w=1.509·10-6 (n=9.923 GHz). Each orbit oscillates within a region near its initial condition. The ionization boundary, (orbit for E=0) is also shown.

     In oscillating within such regions, the orbits necessarily cross themselves. This means that the p-x space cannot be the phase space of this motion. The correct phase space is the 4-dimensional, extended phase space [830AJL, 14], in which the motion for each of the orbits shown above lies on a torus. If the ratio of the frequencies of the perturbation and the atomic motion is irrational, then the motion is aperiodic and fills the torus, as approximated in Figure 3. If the ratio of frequencies is rational, then the motion is periodic, as shown in Figure 4.

Figure 3. The outermost orbit in Figure 2, plotted in its energy subspace of the extended phase space. The orbit lies on a torus with no “donut hole”, and with longitudinal circumferences extending to p=±¥. The sense of motion is: down at the top, spiralling around counterclockwise, and out through the bottom. The asterisks are those points lying in one copy of the original 2-dimensional phase space; they are strobe points, and form a Poincare section of the orbit. This plot may be thought of as made by replotting the outermost orbit in Figure 2 while rotating the 2-dimensional phase space about its p axis, which becomes the central axis of the torus.

I3w=1	I3w=2	I3w=2/3
Figure 4. Trajectories of the HSH atom in its energy subspace of the extended phase space, for three special cases where the ratio of frequencies is a rational number. The asterisks are strobe points, those points appearing when t is an integer multiple of the perturbation period, 2p/w. Although each orbit appears to be a single curve spiralling around from top to bottom, each such curve is actually traversed several times in the calculation, showing that the motion is periodic. The different orientations of the p-axes in the different plots is for visual purposes only and is not physically significant.

     While the proper HSH phase space is 4-dimensional, the most useful physical perspective comes from taking a 2-dimensional slice through that space, known as a Poincare section[914MBu, §1.2]. This is made by strobing the motion at the frequency of the perturbation:

Figure 5. Strobe plot of HSH phase paths for various initial conditions. The perturbing electromagnetic wave has peak field strength F=1.947·10-10 (1.000 V/cm) and angular frequency w=1.509·10-6 (n=9.923 GHz). The fraction labeling each island chain indicates the winding number of the corresponding orbit on its torus.    The outer curve is the ionization boundary (orbit for Eu=0).

     The closed curves in this plot are the cross sections of interstitial families of tori growing in the midst of the original tori illustrated in Figures 3 and 4. These interstitial tori wind around in between the original tori with various winding numbers, which are indicated by the labels in Figure 5.

     In addition to the regular orbits whose Poincare sections are closed curves, there are also orbits whose evolution is chaotic and whose Poincare sections consist of random scatterings of dots. This is caused by the overlap of the families of interstitial tori, as explained in Appendix A. Two such orbits are included in Figure 5.

     In order to determine the effect of the perturbation strength on the HSH behavior, the same series of Poincare sections as in Figure 5 was attempted for a series of higher values of the microwave peak field strength, F. One of the results is shown in Figure 6.

Figure 6. Strobe plot of HSH phase paths for various initial conditions, with F=1.095·10-9 (5.623 V/cm) and w=1.509·10-6 (n=9.923 GHz). This plot is equivalent to Figure 5, except for the increased perturbation strength and an expanded scale on the x axis.

     From the analysis of the series of Poincare sections, including Figures 5 and 6 as well as several others not shown here, the following general statements can be made about the effect of changing the perturbation strength:

• The resonance islands grow wider with increasing perturbation strength.
• The lower order resonances (those for which s and r are smaller) grow faster than the higher order ones.
• As the resonance zones grow, more and more of them decay into the “chaotic sea”.

     The following table compares the data from these computer simulations with the experimental results shown in Figure 1:

     The left hand column gives the center of the regions of suppressed ionization, or enhanced stability, in the laboratory data. The next two columns show how closely these centers match the low order rational winding numbers. Finally, the last three columns compare the widths of these regions with the largest widths reached by the corresponding resonance zones before decaying into the chaotic sea.

     The first three columns confirm what has been claimed by Leopold and Richards [853JGL, 3382 ¶3 and Figure 4], by Jensen [860RVJ, 154 ¶1; 871MMS] and by others for many years: that the locations of the regions of stability are well predicted by the locations of resonance zones in the phase space. However, no one has previously offered an explanation of the widths of these regions, or a correspondence between the widths in the laboratory and numerical data. This is what is found here, in the last three columns of the above table.

     The laboratory and numerical data are in qualitative agreement on the widths of the various resonance zones. The zones may be listed by winding number in order of decreasing width: s/r = 1, 1/2, 1/3, 2/3, …. The significance of this ordering is that the width of the zones decreases as the size of the integers in the numerator and denominator of the winding numbers increases.

     In conclusion, the widths of the regions of suppressed ionization in the Koch data are explained by the expanding widths of the corresponding HSH resonance zones. The zones expand with increasing perturbation strength, and each reaches a maximum width before decaying into the chaotic sea. The widths of the regions of suppressed ionization are on the order of these maximum widths achieved by the resonance zones.

     The plots in Section 3 are made in the physical phase space of the unperturbed 1-dimensional hydrogen atom, and in the extended phase space derived from that space. This section looks at the motion in action-angle coordinates of the unperturbed atom. The action is:

,

     The importance of the action-angle coordinates is the correspondence, for large values, between the action in classical mechanics and the principal quantum number, or energy index, in quantum mechanics.

     In this coordinate system, the HSH energy has the form:

; ,

     From this perspective the HSH perturbation takes the form of an infinite superposition of rotating cosine potentials, indexed by the integers, including a zeroth order, standing cosine potential. The M’th cosine potential in the series has amplitude FI²A_M/2 and rotates (except for M=0) with angular frequency w/M. This frequency is either positive or negative (meaning an either counterclockwise or clockwise sense of rotation), according to the relative sign of M and w. These rotating potentials can resonate with the underlying motion of the atom, which has angular frequency dx/dt=1/I³. The resonance condition is I»(M/w)^1/3 for some M.

     If the HSH perturbation is weak enough, then in each resonance zone the effect of the perturbation is dominated by one particular term. In such a region the effective energy is:

Singly-resonant HSH energy

     Three ways of viewing the motion of a singly-resonant HSH system are shown in Figure 7, for the case M=2. The second primary resonance zone is clearly visible in these plots.

Figure 7. Three views of the second primary HSH resonance zone, made by plotting the behavior of the singly-resonant HSH system with M=2 in three different spaces. On the left are shown phase paths in a frame of reference that is rotating along with the cosine potential, i.e., with angular frequency w/M. In the center is a Poincare section of the 4-dimensional extended phase space, made by strobing the motion at the frequency of the perturbation, w. The right hand view uses the topological equivalence between a torus and a rectangle with opposite sides identified, and plots a single orbit in such a rectangle. The experimental parameters for all three plots are F=2·10-10 (1.0 V/cm) and w=2p/(5·106) (n=8.26 GHz).    In the two polar plots the range of the radial coordinate, the action, is IÎ(78, 162). The rectangular plot covers xÎ(0, 2p] horizontally and tÎ(0, 5·106] on the vertical axis.

     The following table compares the theoretical predictions of the locations and widths of selected primary resonance zones with the corresponding quantities measured in strobe plots like the one in the center of Figure 7:

     The HSH energy with two terms included in the perturbation:

,

Doubly-resonant HSH energy

MÎ{1, 2}, IÎ(95, 115)

Figure 8. Strobe plot of the doubly-resonant HSH system for MÎ{1, 2}. The other parameters are the same as for Figure 7. The region plotted is that between the first and second primary resonances. Several secondary and higher-order resonances are visible.

     The structures seen here are Poincare sections of interstitial tori analogous to those seen in Figure 5 in the physical phase space. The winding numbers of the tori in this plot are reciprocal to those in the physical phase space.

     Simulations were also run in the region below the first primary resonance. The resonances found there are extremely tiny, with widths on the order of 10^-5. In the physical phase space, while it is true that the resonances below the 1/1 resonance are smaller than some of those above, their widths are much wider than in the action-angle space. It is conjectured here that it takes the full series of primary resonances, which are all present implicitly in the energy in the physical phase space, to bring the resonances in the lower region up to their full width.

     This is an important point when making the connection to quantum mechanics. It may be that it is very difficult to reproduce the stable regions in Koch’s experimental data in quantum mechanical calculations without including a very large number of terms in the HSH perturbation. This would be a very time-consuming calculation on even the fastest supercomputer.

     The energy eigenvalues of the unperturbed, bound, 1-dimensional hydrogen atom are, in atomic units:

.

,

     The HSH energy is obtained by adding the perturbation of an electromagnetic wave polarized along the axis of the atom:

.

.

     In classical action-angle coordinates, the energy of this system gives rise to resonance between the rotation of the unperturbed system and the rotation of a set of cosine potentials. In quantum mechanics, resonance is explained in terms of the splitting between energy levels. In the above energy, the kets |n±Mñ represent transitions up and down by M levels, where M is an integer much smaller in magnitude than n. The microwave perturbation is in resonance with the atom if its photon size is a multiple by some M of the level splitting of the atom. So the resonance condition is, for large n, n³w»M, where |M|«n. This is identical to the resonance condition in the classical case, except that the classical theory does not put a limit on the magnitude of M.

     The projection of the HSH energy on the unperturbed energy states can be used in Schrodinger’s equation and integrated on a computer to simulate the quantum mechanical HSH behavior. As in classical action-angle coordinates, in order to study the behavior due to individual resonances and their interactions, the energy used includes a single term or a selected set of terms from the resonance sum. The resulting simulations based on this energy demonstrate a phenomenon very similar to the classical HSH behavior in that there are regions of confinement of probability in the space of energy levels.³ The locations and widths of these regions match those predicted by the pendulum approximation [914MBu, §3.4] for the resonance zones in the classical theory in action-angle coordinates. Furthermore, these regions expand and merge as the strength of the HSH perturbation is increased, as do the resonance zones in the classical theory.

Figure 9. Probability distribution of an approximation to the singly-resonant HSH system (based on the pendulum approximation) with M=1, F=0.95·10-9 (4.9 V/cm) and w=80-3 (n=13 GHz). The initial state is n0=80; the state shown is for t=6.8·106 (1.6·10-10 sec).    The horizontal axis is nÎ[60, 100]. The vertical axis is PÎ[0, 0.08].

     A sample of the data output at a single time-step is shown at the right. This shows the energy configuration of the atom at a particular time after the turn-on of the perturbation. A complete calculation consists of a sequence of many such frames.

     The entire evolution of an atom, from the turn-on of the perturbation through steady-state, may be represented in a single, static picture by using color to represent the magnitude of probability. This frees up one of the two dimensions on the plotting surface to be used to represent time. The result is called here an evolution plot, and is shown in Figure 10. Any vertical slice through this figure contains the same information as the probability plot for one time-frame, although with much lower resolution since there are only nine colors. This technique makes it possible to see many aspects of the motion that are not evident from inspecting series of individual frames. It is also easy to compare numerous time series for different values of a variable parameter, such as initial state, and quickly observe the effect that the variation has on the motion.

Figure 10. Evolution plot for the singly-resonant HSH system in quantum mechanics with M=1 and n0=80. The probability is seen to spread within a specific range of energy levels, and not beyond. This is the first primary resonance zone in quantum mechanics. The experimental parameters are F=0.95·10-9 (4.9 V/cm) and w=80-3=1.9·10-6 (n=13 GHz).

     The probability spreads out evenly both above and below the initial state. After spreading out to fill the region from about n=69 to 95, the probability stays in this region, yet with some oscillation in its structure. The region through which the probability spreads is identified as the first primary resonance zone.

     A further compaction of the data brings out evidence of the resonance zones most clearly. In Figure 10, the essential features of the probability distribution remain fairly stable after steady-state is reached. This suggests that the essential information about a single evolution is preserved if the entire plot is collapsed by averaging the probabilities for each energy eigenvalue over the time after steady-state is reached. This reduces each evolution plot to a vertical slice so that a series can be juxtaposed to form a new type of plot, in just the same way as the color representations of time-frames are juxtaposed to make an evolution plot.

     The result of this technique is called here a distribution plot, with an example being Figure 11. Both the vertical and horizontal axes are measured in the index of the energy eigenvalue. A value on the horizontal axis denotes the initial condition of one particular calculation. A value on the vertical axis denotes a possible state in the energy-index space of each calculation. The color of the plot at the intersection of these two values indicates the extent to which a calculation has predicted the spread of probability from the initial energy state to a state with the energy index indicated on the vertical axis.

Figure 11. Distribution plot for the singly-resonant HSH atom with M=1. On the left is shown the series of initial states from n0=20 to 200. On the right is an expanded view of the vicinity of the resonance zone. This scheme of pairing a large scale plot with a close up view of the resonance zone is used in all the distribution plots. The experimental parameters are the same as in Figure 10.

     The resonance zone shows up as a distinctive square pattern. The probability spreads out to fill the square area in a manner which is almost completely independent of the initial state of the atom.

     This plot is a compact representation of a large amount of data. It comprises vertical slices for time series for each of 181 initial states, with the probabilities in each slice being obtained by averaging over several hundred, and in many cases upwards of a thousand, time-frames.

     The following table compares the classical theoretical estimates (with I®n) of the locations and widths of the resonances with the values measured from the evolution and distribution plots:

The results of the simulations are seen to be in good agreement with the predictions, although the resonance zones seem to be somewhat narrower than predicted.

     While most of the experimental work on excited states of hydrogen has investigated ionization, there are a few results reported [85aJNB, Figure 2] that relate directly to the simulations presented in this section. These are measurements of the final state distribution for a beam initially prepared in a particular excited state, and perturbed in a microwave chamber. Unfortunately, none of this data has been taken with a set of parameters that would test for resonance zones. They do, however, serve as a physical check on the validity of the simulations, with very positive results. For more details, see 914MBu, §5.2.

The doubly-resonant HSH system in quantum mechanics

     The significance of the resonance zones is that their overlap is the route to chaotic behavior in the classical theory of quasi-integrable systems [914MBu, §1.2]. So a very significant question is: “Is there a quantum mechanical process analogous to the overlap of classical resonance zones?” This question is answered here in the affirmative by demonstrating the overlap of neighboring quantum mechanical HSH resonance zones.

     Figure 12 shows a distribution plot for the doubly-resonant HSH system with MÎ{1, 2}. The microwave frequency is the same as is used in Figure 11, but the field strength is reduced by a factor of five:

Figure 12. Distribution plot for the doubly-resonant HSH atom with MÎ{1, 2}, F=1.9·10-10 (0.98 V/cm) and w=80-3=1.9·10-6 (n=13 GHz). The first and second primary resonance zones are visible.

     The calculations for the doubly-resonant system require much more computer time than for the singly-resonant. This is the reason for the gaps in the data seen in this plot, and in the next one.

     This figure shows unmistakable evidence of the first and second primary resonance zones. An atom starting in the region of one of these zones spreads in probability through that zone, and does not enter the other zone.

     When the field strength is increased, overlap occurs:

Figure 13. Distribution plot for the doubly-resonant HSH atom with MÎ{1, 2}, F=0.95·10-9 (4.9 V/cm) and w=1.9·10-6 (n=13 GHz). The first and second primary resonance zones have gone through overlap.

     Despite the gaps due to high computation costs, this plot shows a definite overlap of the resonance regions. The square area extends from about 70 to 120, which energy indices correspond to the lower edge of the first primary resonance zone and the upper edge of the second. For an initial state in either resonance zone, the probability spreads throughout both zones. In most parts, the probability still stays primarily in the original zone, with only very little spreading into the other zone. For n₀=90, however, which is midway between the locations of the two zones, the probability is fairly evenly spread out.

     In both classical and quantum mechanics, harmonically driven Stark states of hydrogen (HSH) exhibit nonlinear resonance between the orbital motion of the atom and the microwave perturbation. In classical mechanics, the resonance occurs when the angular frequency, in action-angle space, of the unperturbed atom is close to the frequency of one or more of an infinite number of rotating cosine potentials representing the perturbation. In quantum mechanics, resonance arises when the energy level spacing of the unperturbed atom is close to an integer multiple of the microwave photon size.

     This resonance is manifested in the evolution of the atom by a structure of resonance zones, regions of stability in the space of states of the atom. The resonance zones grow with the strength of the HSH perturbation. As they grow, neighboring zones overlap, with the result that a system starting out in one zone is free to migrate within the larger total region of the overlapping zones. In classical mechanics this migration is by the evolution of the orbit, in quantum mechanics it is by the spreading of probability.

     These results demonstrate that, despite totally different mathematical foundations, both classical and quantum mechanics do demonstrate the existence and overlap of resonance zones. Moreover, the locations and sizes of these zones are the same in both theories. In classical mechanics these are zones in the atom’s phase space, and may be plotted in either the physical or action-angle coordinates. In quantum mechanics they are zones in the space of the atom’s energy levels, which correspond to subspaces in its Hilbert space.

     In classical mechanics, the growth of the resonance zones have been shown here to explain the widths of the regions of suppressed ionization in the experiments done by Koch. While others have explained the locations of these regions in terms of resonance zones, this is the first evidence that their sizes are explained also by this theory.

     The conclusions of this work may be summed up as follows:

• The HSH atom does demonstrate nonlinear resonance, by the existence and overlap of resonance zones, in both a classical and a quantal analysis.
• Simulations based on the classical equations of motion do reproduce the detailed behavior observed in the laboratory. Specifically, the positions and widths of regions of stability calculated in the physical phase space match those found in the laboratory.
• Simulations based on the quantal equations of motion agree with the spreading of states observed in the laboratory, and with the sizes and overlap of resonance zones calculated in the classical action-angle coordinates.

     Several interesting questions remain to be investigated in this field. These include:

• The quantum mechanical picture of the HSH resonance phenomenon parallels the classical picture in action-angle coordinates. But the calculations in the “physical” phase space provide a more accurate description of the motion. How might one understand the relation between the graded system of primary and higher-order resonances in the action-angle space and the single system of resonances arising in the physical space? An answer to this question might allow for the development of a more accurate quantum mechanical HSH theory. It is possible that some insight into this question may lie in the quantum KAM theory under development by Reichl and Haoming [902LER].
• More runs of the quantum mechanical simulations in Section 4 are needed to fill in the gaps and explore a wider space of parameters.
• More experimental data on final state distributions would be helpful in confirming the physical significance of the simulations in Section 4. It may be possible, although likely quite difficult, to find evidence of individual resonance zones and their overlap by probing a parameter space in which the other resonances are not important.

     It is hoped that the work presented here might contribute some insight into work on these and other ongoing questions.

     In classical mechanics, a linear or harmonic oscillator can be defined in several ways, two of which involve a linearity:

• A harmonic oscillator is a system with a linear restoring force, F(x)=-k_Fx.
• A harmonic oscillator is a periodic system whose energy is linear in its action coordinate: E=E₀+k_EI.

     In a nonlinear oscillator, the energy is a nonlinear function of the action, so the frequency does depend on the action: w=w(I). Here, if the system has action I and is perturbed with a frequency near w(I), then it exchanges energy with the perturbing system. But the resultant change in the value of the action changes also the value of the frequency w. The system moves out of resonance, and so it does not keep exchanging energy indefinitely.

     The behavior in classical linear resonance is a steady increase in the amplitude of oscillation, and a resultant migration of the system through regions of higher and higher energy in its phase space. The more complicated behavior in nonlinear resonance is characterized by the existence of resonance zones in the phase space. Inside each such zone, the system has a pronounced oscillatory response to the perturbation. At moderate perturbation strengths this response is localized within the zone with no migration from zone to zone. Moreover, whereas in a linear system the amplitude of the response is infinite at the exact resonant frequency, the amplitude of resonant oscillation at the center of each resonance zone is zero. Thus, at moderate perturbation strengths, the phenomenon of nonlinear resonance has an inherent stability.

     As the strength of the perturbation is increased, however, the resonance zones grow in size and overlap each other. The boundaries delineating the zones, the KAM surfaces, are destroyed and the phase paths of the system wander from zone to zone. This migration is sensitively dependent on the initial conditions of the system, and is therefore chaotic. At moderate perturbation strengths, some KAM surfaces decay while others remain intact. The migration of the system is then restrained by the surviving KAM surfaces; the behavior is said to be locally chaotic. At higher and higher perturbation strengths, more and more KAM surfaces are destroyed. Eventually, the system becomes free to wander throughout its entire phase space, and the behavior is called globally chaotic.

     There has been some doubt as to whether nonlinear resonance can occur in quantum mechanics. The objection is stated by pointing out that Schrodinger’s equation is linear, so that there can be no nonlinear phenomena in quantum mechanics. This point of view is in error, however, because the linearity of Schrodinger’s equation is linearity in the state vectors, and the significance of that linearity is that the space of quantum mechanical states is a linear (i.e., vector) space. What is relevant to the linearity of a resonance phenomenon is not the linearity of the equation of motion, but the linearity of the energy in the action coordinate.

     There does not exist a pair of operators to correspond, in quantum mechanics, to the action and angle coordinates of classical mechanics.⁴ However, in the classical limit, there is a correspondence between the index of the energy eigenvalues and the classical action. This suggests a way to carry over the concepts of linear and nonlinear resonance to quantum mechanics in the special case of large energy quantum numbers. That is to consider linear resonance to occur in quantum mechanics when the energy eigenvalues of a system with a discrete energy spectrum form a linear function of their index: E_n=E₀+an. The energy level spacing is then independent of the energy index: E_n+1-E_n=a. This means that the dynamics of the system are unchanged by changes in the energy level. If energy is absorbed in the form of photons of energy a, then that absorption can proceed without limit until there is a breakdown which alters the form of the energy. This is indeed the case for the quantum mechanical harmonic oscillator, whose energy eigenvalues are E_n=(n+½)hw/2p.

     With this understanding of quantum mechanical linear resonance, nonlinear resonance is understood to occur when the energy eigenvalues form a nonlinear function of their index, so the energy level spacing depends on the energy eigenvalue of the system: E_n+1-E_n=a(n). If the system has energy E_n, and is perturbed electromagnetically with a frequency 2pa(n)/h, then it exchanges photons with the perturbing system. But the resultant change in the energy level changes also the energy level spacing. The system moves out of resonance, and so it does not keep exchanging photons indefinitely.

     The behavior in quantum mechanical linear resonance is a steady increase in the energy level of the system, similar to the situation in classical mechanics. But what is there in quantum mechanics to compare to the existence of nonlinear resonance zones in the phase spaces of classical mechanics? By studying an appropriate physical system on the border between classical and quantal behavior, extreme Stark states of hydrogen, the present work demonstrates the existence of resonance zones (actually subspaces) in the Hilbert space of the system. In a way that parallels the classical oscillatory behavior, the system undergoes a spreading of its probability within each zone. At moderate perturbation strengths the spreading is localized within the zone with no migration from zone to zone. Thus, at moderate perturbation strengths, the phenomenon of nonlinear resonance has, also in quantum mechanics, an inherent stability. (There is, however, nothing to correspond to the zero amplitude of oscillation at the precise center of the classical resonance zone.)

     As the strength of the perturbation is increased, the quantum mechanical resonance zones grow in size and overlap each other, just as in classical mechanics. Since there are no phase paths, there is no parallel concept for sensitive dependence on initial conditions, and therefore no chaos in the quantum resonance picture. But the net result in the behavior is the same: the probabilistic spread of the system through a larger region of its space of states.

     The state function for the hydrogen atom in paraboloidal coordinates, in the special case n₂=m=0, is [914MBu, §D.6]:

,

Figure 14. Probability distribution for the extreme Stark state |n1, n2, mñ = |39, 0, 0ñ, for which n=40. [872RBl, Figure 1, reproduced with permission of the authors.]    The z-axis in this plot corresponds to what is called the x-axis elsewhere in this work.

where (r₁+r₂)/2 is the radial distance, r, from the proton. This formula shows that the probability drops exponentially with r, except on the positive x-axis, where r₂ is zero and the r₁-dependence of the Laguerre polynomial cancels the effect of the exponential. (On the negative x-axis, r₁ is zero.) A plot of the absolute square of this function for n=40 is given at right (where the x-axis is called “z”). This shows that the probability extends primarily along the positive x-axis, and drops off rapidly everywhere else.

     Shepelyansky [836DLS, §2; 872RBl, >(2.1)] sets out three arguments in favor of the 1-dimensional approximation:

• For small angular momentum Stark states,⁵ the probabilities of n-changing transitions calculated from the 3-dimensional model are within a few percent of those calculated from the 1-dimensional.
• The Coulomb degeneracy causes the sublevels of a single n level to act in unison.
• For small angular momentum Stark states, the time rate of occurrence of transitions in angular momentum is slow.

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   914MBu Nonlinear resonance in the hydrogen atom, Marshall Burns, Dissertation at University of Texas at Austin, May 1991.

     Of the teachers listed in the dedication, the one who stands out most obviously in relation to this dissertation is Prof. Linda E. Reichl, who has motivated and led this research through five years. Her patient guidance and stalwart confidence are as much the cause of any value that may be found in this work as all of my effort and time.

     In addition to the influence of Prof. Reichl, I have also gained valuable insight from conversations with Prof. Luis J. Boya, Steve Cocke, Mark Millonas, Hellen Nelson, Dr. Ben Schumacher and many other colleagues in the Center for Statistical Mechanics and the Physics Department of the University of Texas at Austin.

     When I think about the accomplishment that this thesis represents, I am immediately reminded of the many wise and caring teachers who have advanced and enriched my life to bring me to this stage. I have acknowledged in the dedication those who have had the most pivotal influence on my development. In addition, I am grateful for guidance received from Prof. Robin L. Armstrong, Prof. Anthony P. French, Ms. Shamaan Ochaum and Mr. E. James Rohn.

     The quote in the dedication suits well my intention to thank the teachers in my life. But actually, Emerson was talking about friends when he wrote it. Here is a longer passage from the essay:

     Our chief want in life, is, somebody who shall make us do what we can. This is the service of a friend. With him we are easily great. There is a sublime attraction in him to whatever virtue is in us. How he flings wide the doors of existence!

     Finally, I would like to thank the shareholders of Ennex Corp. (Toronto, Canada) and Ennex Technology Marketing, Inc. (Austin, Texas) for giving me the freedom to pursue this project, and remaining confident that this esoteric track of my business will be turned to profitable advantage.

Marshall Burns

     Marshall Burns was born in Toronto, Canada, on August 24, 1954, to Perry and Evelyn Burns, and attended high school at William Lyon Mackenzie Collegiate Institute in Toronto. In 1971, he entered the Massachusetts Institute of Technology, from which he graduated with a Bachelor of Science in Physics in 1979. During that period, as well as since then, he engaged in various entrepreneurial ventures including advertising, photography and trucking. In 1982, Mr. Burns formed Ennex Technology Marketing, Inc. to market the new IBM Personal Computer to business customers across the United States. This company was quite successful, and provided the funds that he needed to begin graduate school at the University of Texas at Austin later in 1982. In 1990, Mr. Burns was one of 33 U.S. students selected to participate in the third session of the International Space University. Mr. Burns intends to apply the combination of his technical education and business experience to the development of new technologies, particularly in the transportation industry.

     On October 26, 1989, Mr. Burns became a permanent resident of the United States of America.

Ennex

     Marshall Burns is the incorporator and sole shareholder of Ennex Corp. (Toronto, Canada, 1975) and Ennex Technology Marketing, Inc. (Austin, Texas, 1982).

     The name “Ennex” is derived from the two English letters, n and x. In elementary mathematics education, n is often used to represent a quantity which is a given, known number. The letter x is used to represent an unknown. Ennex is intended to represent the infinite potential available from combining what is known with what is not yet known, from combining knowledge with imagination.