provocative opinion An Elementary Pulsating-Sphere Hydrogen Atom Model Robert L. Wadlinger 7 Lynewood Building, Middletown PA 17057 J a m e s H. Lawler 4068 lronton Drive, West Richland, WA 99352 Charles R. Brent University of Southern Mississippi, Haniesburg, MS 39406 Rememher the "good old days"in atomic structure study, prior to the Heisenberg Uncertainty Principle's impact on the field, when one could enjoy imagining possible details of the make-up of atoms? Let us return to those nostalgic times briefly, to glance at a channel of theory of the hydrogen atom which might have been taken by Erwin Schrodinger-had he stayed on his original course. Niels Bohr's H Atom Rememher the Bohr H atom? Its simplicity? The creative hlendinn of coulombic and rotatorv-dvnamics. coupled with . quantu& mechanical relations for stability? o n e ofthe most fantastic accom~lishmentsin the historv of science took place when Bohr delived the Rydherg constant-the empirical constant known to the greatest numher of significant figures-in terms of so many more elementary universal constants to so many different powers, as The reader who has not placed each of the magnitudes of m, the mass of the electron, e, the electron charge, h, Planck's
constant and c, the velocity of light into eqn. (1)to calculate RH has missed the startling fact of the greatness of Bohr's accomplishment. Bohr's basic model had its drawbacks-as does even the currently accepted statistical model-so scientists had fun expanding on the circular model to include elliptical motions which themselves were quantized, using subsidiary quantum numbers 1, m l , and m, to account mainly for the spectral nuances which were observed as better optical resolution showed finer lines in the spectrum in conjunction with electric
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Figure 1. Four de Broglie electron waves imposed on Bohr circular orbit.
and magnetic field studies. These, with the Bohr principal quantum numher, n , provided Pauli the key to show that any one electron in an atom has a unique set of such quantum numbers to characterize it. Louis de Broglie's Modification A rather mysterious relation written by Bohr, was shown by Louis de Broglie some years later to ,he physically based in terms of a wave nature to the electron in its circular orbits. T o prevent wave interference, de Broglie allowed an integral numher of wavelengths to fit into a given circumferential path by
wherein u is the freauencv of vibration with time (number of vibrations per second). ~e placed eqn. (8) into eqn. (3,then operated on this D usina ean. (6). to vield the corresnondine
Schrodlnger Wave Functions for Hydrogen Atom ( 12)
Schrodinger next took a rather hizarre theoretical path (5). whwrby relation (9). which holds for macroscopic bodies, was
+2
~ + 1 d ~ - ~ ~ ~ -
4 = radius of sohr n atom tor n = 1, of magnitde 0.529 X 1 0 P O m . z= 1 = number 01 pmitive charges in nucleus.
then applied his (now well-known) matter-wave relation
X = hlmu
(4)
to eqn. (31, deriving eqn. (2). This result caused researchers to consider eqn. (2) as expressing the concept that, in a circular orbit, an integral number of h unite of action are involved. (Previously, eqn. (2) was written the way Bohr first presented i t as mvr = nhl2s
(5)
which expressed that the orbital angular momentum, mur, equalled an integral number of units h12a.) See Figure 1for n = 4. Erwin Schrodinger's Genlus Both Bohr and de Broglie were imagining models, and making headway in elucidating the possible detailed nature of things at the atomic level for the simplest atom. De Broglie's work insoired Erwin Schrodineer to imagine the ~ossihilitv that the H atom might be a tit&, pulsatrng sphere, with its natural surface waves made up of electron waves. He had a full command of the study of vibrating bodies (wires, membranes, snheres) from his sunervisors a t the University of Vienna; hence, he endeavored to convert the classical differential euuation for macroscodc vibrating bodies to one which wt,uld saksfy the quantized'atomic sy&ems. Basic to his creative thinking was the realization that integers, such as those which represent n, 1, and ml, are inherent in the mathematics of the solutions to the D'Alembert Equation
which describes the pulsating sphere. In eqn. (6). D is the disolacement of the surface wave: x . v. and z are the Cartesian coordinates; t is the time; and v is'ihe wave propagation velocity. Classically, the study was called spherical harmonics, a good presentation of which is made by Kauzmann ( I ) . Relation (6) is frightening a t first glance to the reader illversed in mathematics being first introduced in the standard advanced calculus course. Indeed, the theory of Schrodinger was received historically by manyscientistsas a bewilderhg network of ideas. Even Max Planck expressed, "I am reading i t as a child reads a puzzle," (2) when first examining the Schrodinger . papers .~ (3).Perhaps this very brief excursion into the Schrodinger domain will cause some readers to peruse some elementary vibration concepts such as presented by D'Ahro ( 4 ) , to step gently into that rewarding region of math-science. Schrodinger recognized that the displacement D could be expressed as the product of aposition amplitude $ and a time amplitude X , as ~~~
~
.~~~~ ~
D(x$,r,t) = \I(x,Y,z)X x ( t )
which is the famous (time-independent) Schrodinger Equation for submicroscopic systems. In reference (51, Boorse and Motz comment on the loose. free-thinkine manner of deducing eqn. (10) that Schrodinger presented. T K ~reader who wish& asimple argument for eqn. (10) from eqn. (9) needs only refer to Castellan's creative treatment (6). Both derivations are dissatisfviug insofar as they do not allow anv -nhvsical. . . real interpretation of things atomic in terms of surface wavks. Indeed, the elementary perusal of the first several Schrodinger wave functions, listed in the table, will allow for as physical a comprehension as possible of the $ functions. (Recognize that the nature of $, and the restrictions on its properties, have been continuously studied since its inception . . (7). So, while the reader has beenintroduced to the prohahilistic meaning of I$ ',the meaning of $ is ytill less than certain.'l'he best minds in historv have never auite . eiven d more than a mathematical meaning.) Thus, as one proceeds into Schrodineer's H-atom theorv. one moves awav from nhvsicdv real elecGon waves as envisioned by de ~rogiie-vk'though Schrodinger orieinallv olanned to follow an avoroach similar to that orde ~riglie.'i'etus quote ~chrodin& (8): ~~
~
It is, of course,strongly suggested that we should try to connect the function +with some uibration process in the atom, whichwould more nearly approach reality than the electronic orbits, the real existence of which is being very much questioned today. I originally intended to found the new quantum conditions in this more intuitive manner, but finally gave them the above neutral mathematical form, because it brings more clearlv tolight what is really essential. The essential things&mstome to be, that the patulation of "whole numbers" no loneer enters the auantum rules mvsteriouslv. hut t h a t WP have trawd the marrcr a step iurrhrr hack,and found t h ~ '.inwardness" ru have its urigin in the finiteness and singlr-valurdnesq of 3 c~rminspace funclion 1 do not wish w discuss further the possible representations of the vibration process, before more complicated eases have been calculated successfullyfrom the new standpoint. I t was a t this point, then,that Schrodinger took a theoretical channel which led to his concept of an "electron cloud," the elementarv electron charge and mass of Millikan disintegrated into extremely fine particles and charges (9). This model had its disadvantages, which are brought out clearly by D'Abro (10).Most of the disadvantages were taken care of when Max Born proposed that the "cloud" was in reality the probability of finding the electron in the space surrounding the nucleus, the prevalent concept today. We shall return to the point of Schrodinger's departure
(7)
He first considered the time amplitude to he periodic in 2a, hrnre let x t t ) he represented by such an Figure 2. Great circle of spherical surface of~onstant+. Volume 62 Number 4
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295
from de Broelie's electron matter waves and insnect a wssihle vein of stud; Schrodinger could have taken, whereby a pulsatine-snhere H-atom model could be develoned-a small sten hackwaid, hopeful of an ultimate step forward. Pulsating-Sphere H Atom De Broelie claimed that the electron "followed the rav of its phase wave" (11). He could not attribute reality to &ch phase waves, hence considered them fictitious until further research could improve on the theory. Since electron diffraction (Davisson and Germer experiments) showed the realness of the wave nature of electrons traveling freely, then the bound electron of the H atom should have a wave basis in reality as well. Let us now show that Schrodinger's theory lays a basis for that electron wave hehavior in the H atom, using only the siml~lestwave funrtionu ( 1 = 0).The reader can elaborate on more comdex. anele-denendent wave functions. Consider the table more criticky, now. Notice that there is no antular dependence of the functions qlm, qzw, and q 3 ~$.depends only on radial distance r from the nucleus. So, whatever vhvsical . . .oropertv . "the amnlitude & is measurine.-. its mamitude is the same a t a given distance r from the nucleus no matter what direction of travel one takes from the nucleus. Choosing a given, arbitrary value of r, Figure 2 then depicts aspherical surface of identical $ values. Should an electron he traveling along the great circle of such a surface of constant JI, no undulatory properties would he exhihited-a Bohr circular orbit wbdd b e represented. However, the surface depicted in Figure 2 is that existing constant in time! Since the total displacement D of the p u c sating sphere takes place with time passage (it cannot pulsate without time passing), the D surface must he used to best represent a dynamic atom. Inspecting eqn. (7) for the total disnlacement. and insertine"ean. . (8) . . therein. we see that the sphere "inflates" out to a maximum radial amplitude, then "deflates" hack to a minimum amnlitude in a cvclic fashion. as represented by Figure 3. The solid line represents t he "zero'; disrllacement state uf the s~hericalsurface. each dashed line representing the maximum or minimum, respectively; thus, three instants are devicted. I t is the total displacement D that dictates the hehavior of a traveling electron, not the position 6.Let us allow an electron particle to begin an orbit along'the great circle of a $ surface that is oscillating radially with time. That electron would he obliged to travel a wave path! This is best shown in Figure 4, for each n = 2 and n = 3. Electron path amplitude variation follows D variation. The electron "follows the ray of its phase surface," analogous to de Broglie's "phase waves." That surface is not fictitious if Scbrodinger's theory has any basis in physical, dynamic reality. Historically, the amplitude i(/ has been stressed, especially in the form J\I.I2,the probability density for finding the electron in space around the nucleus. Insufficient effort has been made at inspecting the detailed mechanism with time-the kmetics of the dynamic atom-which involves per se as an indicator of what is oscillating with time.
Figure 3. Spherical D surface oscillating-3
-
instants. D = $4
2mt.
-
Pulsating Electric Field Speculation can he made as to what physical property oscillates with time. Is it the positive electric field of the nucleus? That would involve a pulsating nucleus, of course. This is the only apparent real field that could he represented by the wave function 1C.that would affect electron orbital behavior directly. Perhaps another reader might speculate on another property. Is this detailed outlook comnatihle with the currentlv accepted prohahilistic outlook? h i s author believes so, subject to further studv. Precise atomic radii values using.the Bohr circular orbits become imprecise within the range of dis.
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Journal of Chemical Education
Figure 4. A: n = 2: B: n = 3. Elemon follows the path of its oscillating Dsurface.
placement D maxima and minima of the electron wave orbits, thus the tendency is toward the statistical already within the D framework. This should be compatible with IJ.12 statistics. Further investigation would involve ensuring that the sphere oscillation frequency is compatible with pertinent integral numbers of electron wavelengths in the respective stationary states. Also, application of these concepts to angle-dependent wave functions must he made. Conclusion This glimpse a t a possihle physical explanation of the de Broglie electron waves, using the Schrodinger mathematics for the pulsating H-atom sphere, is presented in hopes that it may prove to he a small spark to light up the imaginations of those physical chemists who would enjoy considering the possibility of more detailed knowledge of atomic structure. Literature Cited (1) Kammann, W., "Quantum Chemistry,"Academic Press. New York, 1951,pp. 8% ,nn A " " .
12) Jammer. M. "Conaptual Ikvclopment of Q-tum Mshanic4," Mffirar-Hill, New York ,1966,~. 271. 13) Schralinger. E."CoUsted Psplson Wave Mechanics," Chelsea PUM.Co., New Ymk, 1 ...1..111. I41 D'Abro, A, "The Rise oftbe New Physics,"Vol2, Dover, New York, 1951.~~. 67&
680.
~
~
15) Bao~e, H.,and Mot.,L..,(Editom),"The W w w l d c York, I966.p~.10M)-1076. 16) Csstellan,G.."PhysiealChemistri,"2ndod.,Addiaon-Wealey,Rcading,MA.I97I,p.
465.
17) Jammer, M., "Conceptual Ikvelopment of-tum Mechanics," Mffiraw-Hill, N o r York, 1966, p. 270. (6) Schrdnger, E . " C o U d Pawn oo Wave Mechanim."Chelsea Publ. Ca., New York, 1978, p 9. (9) Sehradinger, E."Cousted papen an wave Mechanics,"C h h h h l Co, New York,
".
,07R,y.." 79"
110) D'Abro.
A., "TheRiae oftbe New Physica,!'Vol. 2, Do-.
NOWYork. I95lIpp. 127-
779
111) Jammer, M, "Conceptual Dwelopment of Qvantum Mechanics," Mffirsw-HiU, New York. 1966,~. 245. 112) Eisbrg. R., and Reanick, R. "Quanwm Physic4 ofAtoms, Molecules.. ,"JohnWiey, New York, 1914, p. 263.
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