The microcomputer simulations are available from Global View, Inc., Rt. 1, Box ... The arrays. (R;, V;) can be specified with the graphics cursor or one of ... on cursor input to plot wavefunctions, scattering phases, total cross ... range of energies.
Computers in Physics Scattering in a spherical potential: Motion of complex‐plane poles and zeros Richard A. Arndt and L. David Roper Citation: Computers in Physics 3, 65 (1989); doi: 10.1063/1.168318 View online: http://dx.doi.org/10.1063/1.168318 View Table of Contents: http://scitation.aip.org/content/aip/journal/cip/3/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Implementing digital holograms to create and measure complex-plane optical fields Am. J. Phys. 84, 106 (2016); 10.1119/1.4935354 Electrode effects in positive temperature coefficient and negative temperature coefficient devices measured by complex‐plane impedance analysis J. Appl. Phys. 80, 1628 (1996); 10.1063/1.362961 The acoustic scattering by a submerged, spherical shell. III: Pole trajectories in the complex‐k a plane J. Acoust. Soc. Am. 90, 2705 (1991); 10.1121/1.401866 The acoustic scattering by a submerged, elastic spherical shell—Pole trajectories in the complex k plane J. Acoust. Soc. Am. 84, S185 (1988); 10.1121/1.2026029 Zeros of Hankel Functions and Poles of Scattering Amplitudes J. Math. Phys. 4, 829 (1963); 10.1063/1.1724325
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Scattering in a spherical potential: Motion of complex-plane poles and zeros Richard A. Arndt and L. David Roper Virginia Polytechnic Institute and State University, Blacksburg, Vtrgtma 24061
D~p~rt_ment ofPhysics,
(Received 15 July 1987; accepted 12 September 1988)
Scatte~ing.of spi~ess nucleons in a spherical potential is examined with the use of a computer graphics sim~latiOn VSCA'~. The poten~ial is defined stepwise and the Schrodinger equation is solved to obtam wavefunctions, scattermg phases, partial-wave total cross sections, and differential cross sections, which are then displayed graphically. For the particular case of a square well, partial-wave amplitudes are displayed over the complex momentum plane in a three-dimensional plot. The well depth is then varied to follow the motion of poles in the complex momentum plane as they become resonances and then are bound states. Also displayed are the partial-wave zeros, which are required to satisfy Levinson's theorem for multiple states. The requirement on well depth is developed to produce a specified number of bound states and enumerate the energies which, at a given well depth, create equal scattering phases in adjoining partial waves {j 1 _ 1 = {j1 = {j1 + 1 • This symmetry of scattering phases exists for both repulsive and attractive square potentials. A square repulsive core is also studied ' which has the same triple-point symmetry as the square well. '
INTRODUCTION Over the last decade we have been developing physics simulations using computer graphics.• At first they were on VAX and IBM mainframes and were viewed by monochrome Tektronix or compatible terminals. In recent years we have used eight-color terminals and more recently have converted many of our simulations to run on color medium-resolution microcomputers (NEC-APC, NEC-APCIII, and IBM-PC/AT-EGA). The microcomputer simulations are available from Global View, Inc., Rt. 1, Box 282, Blacksburg, VA 24060. The VAX versions of the simulations are available from the authors. This paper describes a simulation of quantum scattering off a central potential. Many of the details of such scattering are depicted in computer-generated pictures in the excellent book The Picture Book ofQuantum Mechanics.2 We do not present the theory of quantum scattering here; there are many excellent books on the subject.3 Complete details about how to use this simulation program are not given here; the simulation itself contains much information about how to use it. Some of the operations required to run the simulation are given here to provide some flavor of what running the simulation involves. Section I contains a description of a computer graphics simulation VSCAT, which uses a stepwise spherical potential to calculate, through the Schrodinger equation, a spinless particle-particle interaction; displayed quantities include wavefunctions, scattering phases, differential cross sections, and partial-wave total cross sections. In Sec. II we elaborate on the solutions for a spherical square well of radius R and depth V0 • We indicate the conditions on well depth necessary to produce threshold bound states, and we point out an interesting symmetry of the scattering partial waves: the periodic (in energy) equality for states of adjoining D1 _ 1 = D1 = D1 + 1 • A square repulsive core is examined in Sec. III, where the same triple-
point symmetry is seen in the scattering phases as was seen in the case of a spherical well. In Sec. IV we display the motion, in complex momentum k, of the scattering amplitude poles and zeros when the spherical well depth is varied. We show that in each partial wave there are an infinite number of poles that move as the well is deepened to become resonances and then through the origin to become bound states. The complex-plane zeros move along paths that ensure satisfaction of Levinson's theorem as progressive numbers of bound states are formed. In Sec. V we show the complexplane structure and pole/zero migration for a square repulsive core as its height is increased. The figures for this paper are generally taken directly from the NEC-APCIII (a partial IBM-PC clone, but with higher resolution than EGA) computer screen to faithfully represent what one sees in the simulation. There are seven colors (red, yellow, green, cyan, blue, magenta, and white) on a black background. The colors are used in "rainbow order" (rygcbmw) to represent items (e.g., curves) for increasing values of some parameter (e.g., energy). (Most computer graphics hardware and software do not represent the color spectrum as numbers in a rational order relative to frequencies.) Only two of the figures are shown in color, but all figures are in color on the computer screen. The figure captions speak of colors as seen on the computer screen. Of course, time dependence cannot be reproduced in static figures.
I. VSCAT: A SIMUlATION OF SCATfERING IN A SPHERICAL POTENTIAL VSCAT is one of about thirty computer graphics FORTRAN programs developed at Virginia Polytechnic Institute and State University for the purpose of illustrating various aspects of interesting physics problems. (These programs
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were originally developed on a VAX computer, but recently· many of them have been converted to run on color microcomputers.) The simulations use Tektronix-type graphics output, enhanced by color for some terminals and microcomputers, to illustrate the consequences of varying physical parameters over a wide range of values. In VSCAT a spherical potential is specified stepwise (potential V; on the ith step which ends at R; ). The arrays (R;, V;) can be specified with the graphics cursor or one of several "standard" setups can be used. The algorithm for solving for wavefunctions and scattering phases is quite simple: on each step the radial wavefunction must satisfy the radial Schrodinger equation R/'
+ ZR/!r +
[k2 -I(/+ 1)/rZ]R1 = 0,
where k 2 = 2m(E - V; )/fzZ, m = particle mass, and E = kinetic energy of the particle. This equation has as solutions the spherical Bessel functionsj1(kr) and n 1(kr), so R 1 can be written as R 1(r) = Aj1(kr) - Bn 1(kr).
Starting on the innermost step with A = 1 and B = 0 (to insure regularity of the wavefunction at the origin), we match R 1 and R/ at the endpoint of each radial segment, thereby obtaining A and B at the next step out until the exterior region, where V = 0, is reached. The partial-wave scattering amplitude ft is then obtained from the last values for A and B as
ft(E) = B/(A - iB). It is usually expressed in terms of the scattering phase 81 as
ft
= exp(i81 )sin 81,
where tan 81 = B/A. The full scattering amplitudej(E,O) is expanded in partial waves as j(E,O) =
k1 L (21 + 1)f;(E)Pt(COS 8).
The differential cross section is du!dO = 1/(E,O)f. The total cross section is u, = ~u1 , where the partial-wave cross section is U 1(E)
= 411i21
+ 1)sin
2
81 /k 2 •
Once a potential has been specified, it is displayed along with a menu showing which keys to strike on cursor input to plot wavefunctions, scattering phases, total cross sections, or differential cross sections. All of the figures shown here are essentially what VSCAT shows on a computer screen when one runs it interactively, except that in most of them we suppress the menu of commands and color is only shown in two figures. Figure 1 illustrates a potential that is similar to the two-nucleon interaction containing a shallow well and a repulsive core. This potential produces a shallowS-wave (/ = 0) bound state and a large, nearly resonant, P-wave (I= 1). In addition to the potential, Fig. 1(a) contains the I= 0 (S-wave) wavefunction and, on the right, a vertical plot of the I = 0 scattering phase versus energy for positive energies. For this potential the S wave is 180• at threshold, as required by Levinson's theorem for a system with one bound state. Wavefunctions are obtained by positioning the graphics cross-hair cursor vertically to specify an energy and then stroking a "W." In Fig. 1(a) are plotted the wavefunctions for energies slightly below and above the
bound-state energy, which is indicated by the lower circle on the left vertical axis. The wavefunction divergence at large distances, where the spherical Bessel functions can be combined for negative energies to produce exponentially increasing and decreasing functions, changes direction as the energy crosses the bound-state energy. One can then relate the bound-state energy, as is customary, to normalizability of the wavefunction. The condition for a bound state to exist is that A+ B = 0. In Fig. 1(b) a larger energy ( > 0) has been chosen for the wavefunction, which may be compared to the "unscattered" wave (dashed curve) obtained by stroking a "Z" on input. At large radii the wavefunctions oscillate with the same period but are phase shifted from each other by the scattering phase 81 • The phase shift is positive because the potential is attractive. Figure 1(c) shows the scattering phases for this potential versus energy. This interaction is distinguished by an S-wave phase shift beginning at 1so• at threshold, a P-wave phase that peaks at about 70• near E = 20 MeV and a D wave that rises to nearly so• at about 100 MeV. A plot of partial-wave total cross sections in Fig. 1(d) shows an S-wave cross section of nearly 2 b at threshold, which falls rapidly as the energy increases to about 20 MeV. The P wave shows a very prominent resonance structure near 15 MeV, and there is aD-wave ''bump'' near 50 MeV. In Fig. 1(e) are plotted differential cross sections for a range of energies. The angular distribution changes from a nearly isotropic S-wave form at low energies to a strongly peaked forward shape as the P-wave resonance is crossed. II. SPHERICAL SQUARE WELL
For the case of a square well with radius R and depth V0 , the scattering amplitude can be expressed in terms of a dimensionless momentum parameter p and a dimensionless well depth parameter p0 , which are defined as
p=
.,
R~2ME
, Po=
.,
R~ 2MVo
.-----::-
, and Pb =,}p2 +p~.
The width of the well R is seen to be a momentum scaling factor and the strength of the interaction is then specified in terms of a single dimensionless number p 0 • Upon matching boundary conditions, the scattering amplitude can be expressed as
ft =A/(A- iB), where
A= iit(p)jt(pb)- nt(p)jt(pb ), and
-
-
B = jM)jt(pb)- jl(p)jl(pb),
where nl(p)
pnl-1 (p)
and jl(p) pjl- 1 (p).
For the S wave (/ = 0) this becomes A =
sinp sinpb
cosp cospb + _....:__.:__
Pb
Pb
and B
= _c_o_s.:...p_c_o_s...:.p_b_
sinp cospb
p p Poles in the scattering amplitude occur whenever the function D =A - iB is zero. A bound state is a pole on the
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Imp axis; threshold bound states are produced when D = 0 atthreshold, where p b = p 0 • The condition on p 0 for production of a threshold bound state in the lth partial wave is ) 1 _ 1 = 0. For I= 0 this occurs when p0 = 1T/2, 37T/2, ... , (m + 1/2)1T. For I= 1 this occurs whenp0 = 1T, 21T, 37T, ... , m1T. In Fig. 2 we have repeated the plots of Fig. 1 using a 3 fm, 60 MeV square well as the interaction. We see in Fig. 2(a) that the Pwave has a bound state near threshold with a scattering phase starting at 180• at zero energy. The scattering phases plotted in Fig. 2(b) reveal
FIG. 1. Scattering in a "nuclear" potential represented as a step-wise potential. (a) Wavefunctions (/ = 0) for energies above and below the S-wave bound-state energy. The energies for the curves are listed at the appropriate places on the left vertical axis. The scattering phase is plotted vertically at the right. The circles on the vertical axes show energies for bound states or oo• phase shift. (The commands menu is shown in this figure but not in the figures that immediately follow it, although it always appears on the screen.) (b) Wavefunction (/ = O) at 75 MeV (solid curve) and unscattered wave (dashed curve). The energy is listed at the appropriate place on the left vertical axis. The scattering phase is plotted vertically at the right. The circles on the vertical axes show energies for bound states or go• phase shift. (c) Scattering phases from I= 0 (solid curve) to I= 3 (least solid curve) in rainbow color order (see text). The bottom scale is incident momentum k in inverse fin and the top scale is incident kinetic energy in MeV. (d) Partial-wave total cross sections; lowest curve isS wave, next lowest isS+ Pwaves, etc., in rainbow color order. The area between the curves represents the effect of the added partial wave. (e) Differential cross sections from 3 MeV (solid curve) to 18 MeV (least solid curve) in 3 MeV intervals with colors in rainbow order.
S-wave and P-wave bound states and aD-wave resonance
around 20 MeV. They also show an interesting triple-point symmetry, a succession of energies at which states of adjacent I become equal to each other (t51 _ 1 = t51 = t51 + 1 ). By setting K 1 = K 1 + 1 , where K B/A, and by using the recurrence relations for the functions), n,J, and ii, it can be shown that this will occur whenever } 1+ 1 (pb) = 0. This specifies the energies which, for a given well depth, will produce the symmetry. The partial-wave total cross sections in Fig. Z(c) reveal the low-energy D-wave resonance. The angular distribu-
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l•
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-
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z
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X
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come bound states. Also plotted is the trajectory of the zero that lies closest to threshold for p 0 = 0. It moves to the real axis about one unit from threshold where it meets with a zero moving down from the upper half-plane; then one zero moves toward threshold while the other moves outward along the real axis. The outward moving zero stays on the real axis and is necessary to satisfy Levinson's theorem as the system changes from a one-bound-state well to a twobound-state well. In Fig. 6 are plotted the trajectories of all complexplane poles and zeros for the S- [Figs. 6(a) and 6(b)], P- [Figs. 6(c) and 6(d)], and D-wave states [Figs. 6(e) and 6(f)] as the well depth is increased from a minimal value to a value sufficient to produce a third bound-state pole in each of the partial waves. This is accomplished with VSCAT by increasing the well depth with the cursor and then stroking an "F" (for "follow"). In each of the displayed states Fig. 6 reveals an elaborate pattern of poles moving from deep in the complex plane to trajectories that run nearly
FIG. 5. Complex-momentum-plane view of S-wave scattering from a 1 fin 40 MeV square well whose potential is indicated by a small inset at top right. The values of Po which produce threshold bound states (1.6, 4.7, and 7.9 MeV) are indicated in the box on the right of the drawing. The on-shell scattering amplitude is indicated by the dashed curve (Re F) and the + curve (lm F). Poles ofF are indicated by X and zeros of F are indicated by Z. (a) For a slightly unbound system (e.g., the 'Snucleon-nucleon state). The + on the right of the vertical box indicates the value of p0 • (b) For a slightly bound system (e.g., the deuteron 3S nucleon-nucleon state). Note that the pole position (XJ for the bound state .in (a) has moved from the negative lm p axis to the positive axis. The + on the right of the vertical box indicates the value of p0 • (c) Pole and zero motion as well depth is changed. The pole followed is the one which, at p 0 = 4. 7, produces the second threshold bound state. The arrows show the direction of motion of the pole and zero. (The colors in rainbow order show the direction of motion of the poles and zeros and correlate with the energy colors in the vertical box; on the computer screen the motion is shown by successively drawing the arrows.)
parallel to the physical axis; during this phase the poles would be manifested as physical resonances. The poles then meet at the origin with poles from the left half-plane to become bound states, which then proceed up the imaginary axis. The zeros move along trajectories nearly perpendicular to the physical axis until they meet zeros moving down from the upper half-plane at the physical axis; they then split and move along the physical axis in a manner required to satisfy Levinson's theorem as progressively more bound states are formed. It is clear that in all partial waves there are an infinity of poles which would, with sufficient coupling strength, move close enough to the physical axis to produce the measureable effects that we associate with resonances. Of course, the zeros also produce prominent effects on the real physical axis. At some energies a pole and zero are close by each other and produce a combined effect in physical observables. (At some well depths a zero is behind a pole and at other well depths it is in front of a pole relative to the real axis.)
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