Analysis of Orbital Decay Time for the Classical Hydrogen Atom (Figure Captions)

Figure Captions Figure 1: Diagram at t = 0 showing the initial orientation of the velocity vector v, the electric force from the plane wave (−e) E, the Coulombic binding force −e2 r/r3 , the negative of the centripetal acceleration times the mass, mω 2 r, and the radiation reaction force [Eq. (5)], when the classical electron is beginning a circular orbit of radius a. By carefully selecting α, for A > Ac , and by choosing the frequency ω of the plane wave to match the orbital motion, then a perfect balance can be achieved. Figure 2: (a) r vs. t for the classical electron in the scheme starting in the orbital condition of Fig. 1, with A = 300 statvolt and α = −π/4. The pattern shown here is fairly typical, namely, the amplitude of the radial oscillations gradually increase, the period also gradually increases [better seen in Fig. 2(b)], until finally orbital decay sets in. (b) A blown-up view is shown of the r vs. t curve in (a), near the point where orbital decay sets in. td , as indicated, is defined in this article as the point where the radial oscillation only rises to about the halfway point of previous oscillations, before beginning a steady, oscillatory decline. Up until the decay point, the period and amplitude of the oscillation gradually increases. The peaks change shape due to the nonlinear behavior, until finally the transition occurs. After the transition, the oscillations become smaller and smaller. Figure 3: Plot of td vs. α, for several conditions of A = 5.419, 6.0, 10, and 1000 statvolt. The points indicated along the curves were the ones actually calculated, using the method of Fig. 2(b). The dotted lines drawn were curve fits, put in to simply illustrate the trends better here. As analyzed in Sec. II, each of the curves have a near infinite peak for td . The peaks were drawn in here knowing their proper location from the analysis in Sec. II, but they where also verified by using these calculated peak positions in simulation runs to verify that td does appear to be infinite at these locations. Figure 4: Plot of r vs. t with one CP plane wave present, attempting to numerically simulate one of the α− ,ω− situations. The initial radius was 0.5 Å. The conditions imposed on the CP plane wave were A = 1000 statvolt, along with the values of α− and ω− as calculated from Eqs. (11) and (9). The adaptive time-step Burlisch-Stoer algorithm from Ref. [22] was used to compute r vs. t, for the different indicated relative precision conditions. As can be seen, even when a relative precision of exp (−30) was imposed, the algorithm still only predicted that r ≈ 0.5 Å up to about 4 × 10−14 sec, whereas the analytic solution predicts that r = 0.5 Å for all time. Figure 5: (a) A schematic figure is shown here to help better illustrate the effect of α on both td as well as the amplitude of the spiralling motion, for the situation when A À Ac . Each pair of mirror points, such as A and A’, B and B’, have nearly the same value of td for A À Ac . Figures 5(b), 5(c), and 5(d) contain plots of r vs. t, for a = 0.5 Å. Each plot contains two r vs. t curves, one for α = −π/4, and one for the “mirror case” of α = −3π/4. (b) A = 1000 statvolt; (c) A = 300 statvolt; (d) A = 100 statvolt. As can be seen, as A decreases, the behavior between the α = −π/4 and α = −3π/4 curves becomes increasingly different, in agreement with Fig. 3. For A = 100 statvolt, the α = −3π/4 curve isn’t even stable. Also apparent is the decrease in amplitude of the spiraling motion, and the decrease in td as A decreases. Figure 6: Plots of φ = θ − ωt are shown here for the case where A = 1000 statvolt, a = 0.5 Å, and α = −π/4. (a) The main curve to observe here is “Case E”, where φ (t) vs. t is shown for the “exact case”, solving Eqs. (2) and (3). However, cases P1, P2, and O, which are different perturbation approximations discussed in Ref. [15], are also superimposed here to show how well they compare. (b) blown-up view of the transition point D in (a). φ (td ) is seen to agree reasonably well here with prediction. Figure 7: Plots of ∆ (t) = [φ (t) − α] vs. t, for A = 1000 statvolt, for several different values of α. In (a), the “mirror” angles of α = −3π/4, −π/4, as well as α = 0, π, as well as α = 3π/4, π/4, all have nearly identical values of td , which can be identified in 7(a) by the near vertical lines arising at the decay points in these curves [i.e., ∆ (t) rapidly ³ 3 ´ τ am increases when orbital decay begins]. Note that ∆ (td ) = cos−1 ω eA , from Eq. (15), agrees well with prediction. Fig. 7(b) zooms into the early time region of 7(a) to show the differences between the ∆ (t) vs. t curves for α = − 34 π and − 14 π, as well as α = π and 0. Fig. 7(c) zooms in on the last region of the ∆ (t).vs. t curve for α = 14 π. Each of these ∆ (t) curves behave fairly similarly. The ∆ (t) vs. t curve for α = 12 π can also be seen in 7(c), although its amplitude is still very small. As discussed in this article, if we were to repeat this examination for smaller values of A, one would see increasing differences in td for these “mirror” angles, since, as seen in Fig. 3, the td vs. α curves become centered around different values of α+ other than π/2, as A/Ac becomes smaller. Figure 8: (a) Plot of r vs. t for the case where the initial radius is 0.5 Å, and there are two CP plane waves present, both with α = 0. One CP plane wave has A =100 statvolt, with an angular frequency corresponding to an orbital circular motion of radius 0.5 Å, while the other has A =500 statvolt, with an angular frequency corresponding to an orbital circular motion of radius 0.505 Å. The two horizontal lines indicate these radii. As can be seen, “jump” like motion occurs between the radii corresponding to these two angular frequencies. (b) Plot of r vs. t, as in (a), but now six CP plane waves are present, all with α = 0. The circular orbital radii corresponding to their angular frequencies are indicated. These radii and the amplitudes of the plane waves are, respectively: 0.485 Å and 500 statvolt; 0.490 Å and 500 statvolt; 0.495 Å and 500 statvolt; 0.500 Å and 100 statvolt; 0.505 Å and 500 statvolt; 0.510 Å and 500 statvolt. Figure 9: (a) Plot of r vs. t for two situations, both where the electron starts at r = 0.525 Å, but one where no CP plane wave is present (A = 0), while the other case has a CP plane wave present with A = 100 statvolt, α = 0, and an angular frequency corresponding to an electron in a circular orbit of radius 0.5 Å. (b) The maximum radius minus the minimum radius for each fluctuation in the r vs. t curve in Fig. 9(a), versus r, are plotted here for three situations. In each case the electron starts out in a circular orbit, with a CP plane wave acting, with A = 100 statvolt and α = 0; the three curves are due to the presence of a single CP plane waves of angular frequency corresponding to a circular orbit of radius 0.48 Å, 0.49 Å, and 0.50 Å, respectively. As can be seen, these “response” curves are sharply peaked. Changing the amplitude of the applied CP plane wave changes the magnitude of this response, although the shape stays fairly similar for A not too large. Figure 10: (a) Plot of r vs. t for a classical electron starting in a circular orbit of radius 0.53 Å, with ≈ 2.2×106 plane wave acting during the simulation. The amplitudes and phases were chosen at the beginning of the simulation and then held fixed throughout the remainder of the simulation. The values of these amplitudes and phases were chosen to represent one stochastic realization of a section of the classical electromagnetic ZP radiation spectrum. Ref. [21] discusses the specifics in more detail. (b) Nine classical electrons were tracked in time, each starting at 0.53 Å in a circular orbit, but then subsequently not constrained. A different realization of the classical electromagnetic ZP radiation spectrum was assumed for each simulation, although all had 771, 692 plane waves acting, ranging in angular frequency from 5.0 × 1017 s−1 to 4.6 × 1011 s−1 . The top three curves show the times up to which point the percentage of time was spent at each radius by the ensemble of nine electrons, as each evolved in trajectories like those in Fig. 10(a). The bottom curve was calculated from the ground state of hydrogen via Schrödinger’s equation: P (r) = 4πr2 |Ψ (x)|2 = 4r2 a3B ³ ´ exp − a2rB , where aB = ~2 /me2 . Carrying these lengthy simulations out farther in time, we expect the ³ ´ 2 2r − , as histogram curves computed by simulation to converge closely to P (r) = 4r exp aB a3 B they did in Ref. [21].