Richard E. Powell 1 Relativistic Quantum University of California Berkeley, 94720 I The electrons and the M a n y students in their first exposure to quantum mechanics are puzzled by the nodes in the position probability density, i.e., of surfaces in space at which the probability an electron be there is exactly zero. They ask, "How do the electrons get across the nodes?" A particularly satisfying answer has been at hand since 1928,when Paul A. M. Dirac published a two-part paper ( I ) which has recently been described (2,p . 187) as "undoubtedly one of the great papers in physics of this century." Namely, there aren't any nodes. When an electron problem is solved by Dirac's relativistic treatment, as it properly must be, the solution differs in a number of major respects from that of Schrodinger's nonrelativistic treatment. One of the minor, but interesting, consequences is that the nodal surfaces which resulted from the Schrodingerapproximation are replaced by regions in which the position probability density, although small, is alwaysfinite. This simple result has a surprisingly scanty pubha^tion history. In 1931,Harvey White published (3),as a sequel to his famous electron-cloud pictures of a Schrodinger atom, the electron-cloud pictures of the corresponding Dirac atom. He there commented on the nonexistence of radial nodes in the Dirac electron cloud, and, too, on the nonexistence of angular nodes, although, as we shall see later, the reason for the nonexistence of angular nodes is somewhat different. A great many textbook authors, including myself, have copied White's Schrodingerpictures instead of his more correct Dirac pictures, and we all seem to have ignored his observations about the nodes. In a 1965paper (4) devoted to the computation of relativistic self-consistent-field eigenfunctionsfor all the atoms of the periodic table, Waber and Cromer made passing comment on, and gave the correct explanation for, the nonexistence of radial nodes. Seel, in his little book "Atomic Structure and Chemical Bonding," (6, p 14),raised the question of how the electrons pass through the nodes, and said that the answer was to be found in Dirac's treatment of the atom. However, he did not carry the explanation far enough for a reader to see why this is so, and with the comment, "Unfortunately, the elegance of this more general theory is bought at the expense of considerably more mathematical effort," he dropped the cnhiwt...-','... tn fact, Diroc's mathcma1i~:iIlediniquf?~<\I. I t is usually denoted by a subscript following the azimuthal quantum number. Fourth, m, the magnetic quantum number, which can take all half-odd-integer values from - j to +j, The energy level diagram for a Dirac hydrogenlike atom (Fig. 1) differs slightly from that for the SchrodThe reader may satisfy himself that these operators do giveunity when squared and that any pair of them anti- commute. This matrix notation for the Dirac Hamiltonian is really a shorthand way of saying that not one but four differentialequations have to be satisfied,and, when one actually has to carry out the solution, he multiplies out the vectors by the matrices in full and solves the four simultaneous equations that result. The explicit solution of the Dirac Hamiltonian for a hydrogenlike atom, viz., an atom with a central scalar potential V = - Ze/r, was first published by Darwin in 1928 (2). An excellent modern presentation of it is in Bethe and Salpeter's monograph (8). The procedure is much like that used to solve the correspondingSchrodinger problem. One first expressesthe eigenfunction as a product of a radial factor and an angular factor, and finds that this leads to the separation of the Hamiltonian into an angular equation and a radial equation. The angular equation can be solved in terms of spherical harmonics and certain angular quantum numbers, the radial equation solved in the form ecC' multiplied by a terminating power series in r, with certain radial quantum numbers. Four quantum numbers are required to specify the state of a Dirac hydrogenlike atom, but these are not exactly the same as the four familiar quantum numbers of the Schrodinger atom, because orbital angular momentum and spin angular momentum, taken separately, are not constants of motion and therefore cannot be represented by good quantum numbers. The Dirac quantum numbers are: Figure 1. Dime pigeonhole diagram for a hydrogenlike atom. Not drawn to scale: in particular, the energy separation between fine-structure subgroups is greatly exoggereated in this diagram. Each energy level is more negative than in the Sch;6dinger treatment by a fraction of its d u e equal to IZ/137l2 multiplied by the following: for lsi/i, '/" for 2pVa 7 0 4 ; for 2S/i and 2pi/;, Vei; for 3d6I2, /iw for 3pVs and 3dVt, Vm; for 3sVs and 3 p / ~ ,"AM. The general expression for the fractional lowering is inger atom, in that the three p levels, the five d levels, etc., are no longer degenerate. There is a small splitting in energy between those levelsof higher and those of lower j value. This "fine-structure" splitting had long been known experimentally, and it was a great triumph of Dirac's treatment to predict this splitting and even to give its correct numerical value. The phenomenon is sometimes, even today, ascribed to "spin-orbit" interaction, perhaps an unfortunate name inasmuch as spin angular momentum and orbital angular momentum are not, separately, defined in Dirac's theory. Moreover (and this had been known earlier) the shift of energy is greatest for s states, which were not supposed to have any orbital angular momentum. Fortunately, there is a way to resolve this seeming discrepancy (cf. sectionon the Zitterbewegung Approach). A display of the Dirac eigenfunctions for the Is, 28, and 2p states is presented in Table 1. A reader who wishes to try his mettle by computing, say, the 3d functions, is encouraged to turn to the Appendix where he will find the general expression. The four components of a Dirac eigenfunction are familiarly named, reading from top to bottom, as follows: large spin-up component, large spin-down component, small spin-down component, small spin-up comVolume 45, Number 9, September 1968 / 559 ponent. The two components called "small" are smaller than those called "large" by a factor of the order (speed of the electron)/(speed of light), namely Z/137, usually written aZ. The Radial Nodes The same large radial factor, called g, applies to both the large components, and the same small radial factor, f,to both the small components. The only instances in which f is merely a multiple of g are those in which both are nodeless anyway, namely the Is./,, 2py,, 3d.,,, etc., states. For these states the power series in r terminates with its first term, so that both g and f are proportional to rn-I e-"". The proof of this result is assigned as a homework problem by Messiah in his textbook ((O), v. 2, p. 958) with the hint that it can be done by a systematic search; in fact, it can be seen by inspection of the general expression for the radial factors (Appendix). When g has two or more terms in its power series, so has f. But the coefficients in the power series are always different, so that the nodes in g never fall in the sameplace as the nodes inf. Consider,for example,the 2si,, orbital, which on the Schrodinger treatment has a radial node at r = 2. In the Dirac treatment the large component has a radial node at r = 2, but the small component has its radial node at r = 4. Consequently, the position probability density at r = 2has a finite contribution fromthe small component (Fig. 2) This result is general. In summary, at the Schrodinger radial "nodes" the relativistic treatment leads to a residual probability density of the order of (Z/137)2 r2"-2 e-2"n. The Angular Nodes The question of angular nodes must be examined in a little more detail, because we must distinguish between the angular dependence of position probability density for an isolated atom ("spectroscopists' atom") and that foran atom within a molecule ("chemists' atom"). Consider first the isolated atom, perhaps in a weak field. The problem can be simplified by Hartree's distance Figure 2. Relativistic position probability density for a 2s state. Probability density expressed in units of its maximum value, radial distance in units ifo/z. 560 / Journal of Chemical Education observation (Iff),on the basis of general properties of the sphericalharmonics, that the angular dependenceof position probability density of the small components is identical with that of the large components, so that we need only consider the two large components. This simplificationis equivalent to reducing the Dirac fourcomponent eigenfunctions to the Pauli two-component eigenfunctioni. A few explicit angular functions are given in Table 1. Others can readily be calculated (Appendix). White, in his 1931 naner (3 calculated and drew arranhs to illustrate ail the angular distributions up through the g states ((3),pp. 516-517). The qualitative features can he summarizedasfollows: (a) Orbitals with the same value of ,)' and m have the same angular distribution. Thus the pila distribution is spherically symmetrical, like the S,,,. The pair of d,,, orbitals has the same angular distribution as the pair of pi,,. The triplet of fsh has the same as the triplet of And soon. (h) The si/,distribution is spherically symmetrical. For other states with this lowest possible value of m, namely, m = 1/2, the angular distribution is stretched out along the z axis but has no nodes. The pv,, m = state, for example,has an angular dependenceof cos2 6 + so it more resembles a dog-bone than two touching ellipsoids. The angular distribution and the corresponding contour map of position probahility density for the 2p.,,, m = 'I2state are shown in Figure 3. Figure 3. Left, angular factor of the position probability density for the 2 p ~ / ~m = 'A state of on isolated atom. Right, probability contours for that state, counting downward at 10% increments from its maximum d u e . When the probability density is 25% or less of its mommum value, there is no discontinuity between the regions of space. (c) The state with the highest possible m value for its value of 1,such as the pi,,, m = the m = 5/. thefvIÃm = 51%states, always has a distribution in the shape of a toroid. A doughnut. No nodes. (d) The states with m fallingbetween its lowest and highest possible value have distributions resembling two-lobed, three-lobed, etc., toroids, hut which never approach zero. No nodes. In short, the angular dependence of position probability density for an isolated atom may have mild minima, hut never has anything even resembling an angular node. This nonexistenceof angular nodes does not arise from any consideration so subtle as that of the Dirac small components, but is merely a natural consequence of the correct handling of the angular dependence for an electron with spin. The correct result would be obtained if the calculations were earned through with the Pauli two-component theory. The familiar Schrodinger angular distributions are, then, those for the nonexistent spinlesselectron. Let us turn next to the question of the angular distribution for an electron on an atom within a molecule. Here the distinguishing physical circumstance is that the atom is subjected to the intense electrostatic field resulting from the neighboring nuclei, or, if one of the atoms of the environment be paramagnetic, to an intense magnetic field. Chemists have long known that, in most molecules, the orbital angular momentum is "quenched" by the electrostatic or magnetic field, so that they have not used eigenfunctionsappropriate to the isolated atom. but linear combinations of eiarenfunc-u tions so chosen as to give zero net angular momentum. We need, then, to look at the Dirac eigenfunctionsin intense electrostatic or magnetic fields, i.e., to the limiting cases of Stark effect and Zeemau effect. In such a field,the electron will be in a state approximating a pure spin state-one of the large components will be fully excited, while the other becomes zero to the order of the ratio (fine structure splitting)/(Zeeman or Stark splitting) ((8)p. 205, et seq ). We seek, then, to form linear combinationsof eigenfunctionsof a given n and I, suchthat oneof the two large componentsbecomes zero. For the 2p orbitals, the large spin-down component is zero for the combinations: V ~ P Z / , ,m = 'A -V'/~P,A,m = -I/, +V'APv,, m = - $ / a and the large spin-up is (l/n)"* r e-'I2 mutliplied by, respectively, cos 6, sin 6 cos 0,and i sin 6 sin 0. But these are just the familiar pz, -p,, and pv. So we show that it is possible to construct Schrodinger-typeorbitals, provided the requirement be dropped that j and m be good quantum numbers. Now, the choice of linear combination which has made one of the large components vanish does not make either of the small components vanish. Even in the nodal plane of the large isolatedpoints). In summary, the situation with regard to angular nodes of an atom within a molecule is much the same as that with regard to the radial nodes: at the angular Schrodinger "nodes" the relativistic treatment again leads to a residual probability density of the order of (Z/137)2r2"-' e-2''n Circulating Current and the Electron Spin It is sometimes forgotten that every quantum-mechanical problem, relativistic or not, has two solutions: a position probability density and a probability current which need not vanish, even in the stationary state, hut must then satisfy the condition that its divergence he zero, i.e., that probahility be locally conserved. In a Dirac atom, the probahility current does not vanVolume 45, Number 9, September 1968 / 561 ish. Its three space components are given, in units of c, by **as4 **av* **a** With the aid of the Dirac velocity operators (videsupra) it is straightforward to evaluate the prohability current. Hartree (10) was the first to do so explicitly, showing that the x-component is proportional to sin 6, the y-component proportional to -cos 6 with the same coefficientof proportionality, and the z-component is zero. This clearly represents a nonvanishing probability current circulating about the axis of z. Table 2 gives the tangential magnitude of this circulating current for the atomic stateslisted in Table 1. Two features of the circulating current are worth comment: Its radial dependence is always that of the productfg, sothat the circulating current reverses direction wherever either f or g passes through zero. Its angular dependence is always that of the probability density itself multiplied by sin 8, so that there are no angular reversals of the current. Hartree went a stepfarther (10). He worked out the general expression for the magnetic moment which results from such a circulating probability current. The procedure is as follows: multiply the probability current by -e to convert it to electriccurrent; multiply by its axial distance (r sin 8) and divide by 2, which according to classical electrodynamics gives the magnetic moment; divide by c to convert to electromagneticunits; and sum over all space. Hartree carried out the calculationto obtain the followingexpression 3 + '/!magnetic moment = (eaaI2)m2m The first factor in this expression is just the Bohr magneton. The second is the Land6 splitting factor. The third is the magnetic quantum number. Taken together, they give correctly all the magnetic levels of the electron in an atom. In other words, the circulating current is exactly enoughto account for all the magnetic properties of an electronin an atom. This remarkable result deserves greater fame than it has. A number of textbook authors have asserted, without bothering to givetheir evidence,that the "spin" of an electron is to be thought of as a rotation of the electron about an axis through it, rather like the daily rotation of the earth around the axis between its North Pole and South Pole. As we see, the evidence contradicts that statement. Inasmuch as the magnetic properties of the electron are already exactly accounted for by its circulational motion, any additional effectarising from its rotating around an axis through it would destroy the agreement with experiment. If a physical visualization of "spin" be needed, it is perhaps better to think of it as the lower limit of circulational motion below which the electron will not drop. The electron appears to be a particle which doesn't like to move in straight lines-it insists on moving in helical spirals wherever it goes. How can an electron with a fixed total angular momentum 1 give rise to severaldifferent possible values of magnetic moment? The old answer, the "vector model," was that there was a total magnetic moment corresponding to the total angular momentum, but that only quantized values of its projection along the z axiswere observable. Hartree (10)already in 1929 had complained about such an interpretation. The prohability current gives both a qualitative and a quantitative explanation. As has already been mentioned, the circulating current passes through zero and reverses its sign wherever g or f is zero. Thus the circulating current does not reverse fora l s state. It reverses at r = 2 and again reverses at r = 4 for a 2s state. It reverses onceat r = 6 in a 2pil,state. And so on. These various reversals serve to cancel one another's magnetic effects, and so lead to various possible intermediate magnetic moments. The Zi'Werbewegung Approach There is another way of looking at the relativistic problem which, although mathematically equivalent to Dirac's, puts it in quite a differentperspective. It is possible to solve for the eigenvaluesof the Dirac velocity operators, which turn out to he  1, i.e., the instantaneous value of any one component of the electron's velocity is either +c or -c. The frequency of the motion with this velocityis,however, at least 2mc2/fi and thus very high ((S),p. 205, et seq.),sothat the average amplitude of the motion is less than l/m of a Bohr radius. The electron is thus executinga kind of motion which Schrodinger called "jitter-movement," Zitterbewegung ( I I ) , a very rapid motion of very small amplitude, periodic for a free electron and almost periodic. for a bound electron, superimposed on whatever slow motion it may be executing. In 1950, Foldy and Wouthuysen set out to find a mathematical representation of the electron's eigenfunction that did not have Dirac's small components. And they succeeded ((IS),p. 51 et seq.; (IS), p. 152 et sea.; (2),p. 206). But their function turned out not to represent the probability of an electron's being at a position, hut instead the probability of the center of gravity of its Zitterbewegung being at a position. In the FoIdy-Wonthnysen representation, there can indeed he nodes. However, this only means that the center of gravity of the Zitterbewegung cannot sit right on a node; it can sit nearby, and the Zitterbewegung carry the electron across. One advantage of the Foldy-Wouthuysen approach is its generality. It showsus, without our having to solve every separate problem in the Dirac way, that there cannot ever be any true nodes in the positionprohability density. Another advantage is that it serves as a bridge between the rigorous Dirac approach and the spin-orbit approach sowidely used in the theory of atomic spectra. As the reader will recall, spin angular momentum and 562 / Journal of Chemical Education orbital angular momentum, taken separately, had to be discarded from the Dirac theory because they are not constants of motion. In the Foldy-Wouthuysen representation, they now reappear in the form of "mean spin angular momentum" and "mean orbital angular momentum." The Foldy-Wouthuysen Hamiltonian can be expanded in powers of aZ, and when this is done as far as the second order, two small energy terms appear: One of them has the form of a mean-spin-mean-orbit interaction, and corresponds correctly to Dirac's finestructure shift. The other is a contact term which vanishes except for s states, and corresponds correctly to Dirac's relativistic shift of s states. The Electron in a Box The Dirac treatment of an electron in a one-dimensional "box" (i.e., potential well with infinitely high walls) is given in detail in Sherwin's textbook ((7), p. 301 et seq.; p. 373, et sea.). The problem is somewhat simplified by the circumstance that the electron can be in a pure spin state, i.e., we need only consider one large and one small component. The reader will recall that the energy levels of a nonrelativistic electronin a box are given by Em= nSh'/SmL' and the eigenfunctionsby *, = (2/L)'/- sin B M / ~ , Let us write p for the absolute value of the momentum of the electron, namely nh/2L. Then p/m is its speed, and p / m its speed compared to the speed of light. The energy levels of the relativistic electron are then slightly more negative than the nonrelativistic, each by a fraction of its own value equal to (p/m)%. Compare the correspondingresult for the energy levels of an atom, given in the caption to Figure 1. The Dirac eigenfunctions,with normalization correct to first order in p/mc, are so that wherever the large component has a node, the small component remains nonzero. Thus there are no nodes in the position probability density of an electron in a box, either. Other Particles The other two constituents of ordinary matter, the proton and the neutron, are, like the electron, spin particles that obey the Dirac equation. Consequently, there are no nodes in the position probability density for any ordinary matter. It may be mentioned that the circulating probability current does not account for the magnetic moment of the proton entirely, nor of the neutron at all. Discussionof the other elementary particles of physics would carry us too deeply into relativistic quantum field theory. A general principle can, however, be stated. Any particle of finite rest mass and spin ,I' requires a theory of 4j+2 componentsto describe it. If the particle has zero rest mass, then any process by which it is liberated leaves half the components unexcited, so that only 2.i + 1 components are required. The photon, a massless particle of spin 1, requires a 3componenttheory. The neutrino, a masslessparticle of spin can be described by a 2-component theory, a slightly simplified version of the Dirac treatment. Appendix The general expression for the Dirac eigenfunctions of a hydrogenlike atom is the following where N is a normalization factor, the symbol -I- denotes the sign of j - I, a andf are radial factors defined below, and Y is the indicated spherical harmonic ((a),p. 5). The large radial factor g and the small radial factor f are, to first order in aZ,given by g = ç-'1 X {r"-'1% + '1,  1  '/,I where r is measured in units of a/Z and each series terminates with the zeroth power of r. If these radial series are evaluated to the order of a2Z2 or higher, various of the factors of the coefficients become slightly different from integers; but the difference is so small that it can he ignored for any ordinary purpose (cf. (81, P. 69). The normalization factor N can be worked out in general m terms ((a), p. 69), but it is handier merely to require that N JU (g2+W d r be equal to unity. Literature Cited DIRAC,P. A. M., Proc. Roy. SOC.(London), A M , 610 and A118, 351 (1928). DARWIN,C. G., Proc. Ray. Sac. (London), A118,654 (1928). HARTREE.D. R.. Proc. Camb. Phil. Sac.. 25.225 il9291. SCHB~DINGER,E.,Silzungsber.~ e r l i n~kad.;418 (1930). WHITE.H. E.,Phys. Rev., 38, 513 (1931). FOLDY,L. I., AND WOUTHUYSEN,S. A., Phya. Rm., 78, 29 (1950). BETHE,H. A,, AND S-VLPETER,E. E.,"Quantum Mechanics of One- and Two-Electron Atoms," Academic Press, New York, 1957. SHERWIN,C. W., "Quantum Mechanics," (Holt, Rinehart and Winston, New York, 1959. MESSIAH,A,, "Quantum Mechanics," (2 vol.), NorthHolland, Amsterdam, 1961, 1963. CORINALDESI,E.,AND STROCCHI,F., "Relativistic Wave Mechanics." North-Holland. Amsterdam. 1963. SEEL, F., "Atomic Structure and Chemical Bonding," Methum. NPWYnrk. 10113~ ~ , - - - ~ ~ -, BETHE,H., "Intermediate Quantum Mechanics," (W. A. Benjamin, Inc., New York, 1964). WABER,J. T., AND CBOMER,D. T., J. Chem. Phvs., 42,4116 (1965). Volume 45, Number 9, September 1968 / 563