Substantia. An International Journal of the History of Chemistry 6(2): 7-13, 2022 Firenze University Press www.fupress.com/substantia ISSN 2532-3997 (online) | DOI: 10.36253/Substantia-1630 Citation: Lekner J. (2022) The spinning electron. Substantia 6(2): 7-13. doi: 10.36253/Substantia-1630 Received: Apr 16, 2022 Revised: May 23, 2022 Just Accepted Online: May 24, 2022 Published: September 1, 2022 Copyright: © 2022 Lekner J. This is an open access, peer-reviewed article published by Firenze University Press (http://www.fupress.com/substantia) and distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medi- um, provided the original author and source are credited. Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The Author(s) declare(s) no conflict of interest. The Spinning Electron John Lekner School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand E-mail: john.lekner@vuw.ac.nz Abstract. The notion introduced by Ohanian that spin is a wave property is implement- ed, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor compo- nents, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equa- tions is illustrated. Keywords: electron, spin, spinor. 1. INTRODUCTION In his article “What is spin” [1], Ohanian argues that ‘spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron’, and that ‘the spin of the electron has a close classical analog: It is an angular momentum of exactly the same kind as car- ried by the wave field of a circularly polarized electromagnetic wave.’ Oha- nian credits Belifante [2] for establishing that ‘this picture of spin is valid not only for electrons but also for photons, vector mesons, and gravitons.’ Dirac [3,4] regarded his four-by four matrices as ‘new dynamical vari- ables…describing some internal motion of the electron, which for most purposes may be taken to be the spin of the electron postulated in previous theories’ [4]. This is how the concept of spin is presented in most texts, as intrinsically relativistic, a mysterious internal angular momentum for which there is no classical analogue. For example, in his “Introduction to quan- tum mechanics” [5] Griffiths states ‘…the electron also carries another form of angular momentum, which has nothing to do with motion in space (and which is not, therefore, described by any function of the position variables r,θ,ϕ) but which is somewhat analogous to classical spin…’. We shall construct, for a general relativistic or non-relativistic wavepack- et, an eigenstate of the component of total angular momentum in the net http://www.fupress.com/substantia http://www.fupress.com/substantia mailto:john.lekner@vuw.ac.nz 8 John Lekner direction of propagation, with eigenvalue ℏ/2. Such eigenstates are four-component spinors, of which two components have eiϕ azimuthal dependence. In these formulations the phenomenon of spin is incorporated into ordinary space-time: the twist is in the azimuthal dependence of two of the wavefunctions. To the ques- tion: what does a spinning electron look like? we answer, in brief, that spin in the spinor formulation, relativistic or nonrelativistic, resides in the azimuthal dependence of two of the spinor components. This contrasts with the usual spin-space formulation, and the decoupling of spin from space-time. In Sections 2 we construct genera l relativistic wavepackets with spin half; these are four-component spinors. An important aspect of spin is that it is not purely a relativistic effect: Levy-Léblond [6] has proved that the Galileo group has irreducible representations with non-zero spin. A Reviewer has pointed out that Galindo and del Rio [7] show that Galilean fermions are possible, with a four-component spinor lineariza- tion of the non-relativistic wave equation and a cor- rect (to lowest order) g-factor. The Galindo and del Rio paper anticipates some of the work of Lévy-Leblond [6] and Gould [14]. Lev y-Léblond ’s four-component nonrelativ istic spinors are implemented in Section 3, to construct gen- eral angular momentum eigenstates with spin half. An explicit example of a non-relativistic spinning wavepack- et is illustrated in Section 4. 2. DIRAC SPINORS The wavefunction Ψ(r,t) of an electron wavepacket in free space is to satisfy the Dirac equation HΨ(r,t)=iℏ∂tΨ(r,t), H=cα∙p+βmc2, p=-iℏ∇ (2.1) The 4×4 matrices α,β are written in terms of the Pauli spin matrices σx,σy,σz and the unit 2×2 matrix I as (2.2) The wave equation (2.1) thus consists of four coupled first-order partial differential equations. We consider wavepacket motion, predominantly along the z direction. In cylindrical polar coordinates is the distance from the z-axis, ϕ is the azimuthal angle, and (2.3) The four time-dependent free-space equations for the spinor Ψ read, with mc/ℏ=K, (∂ct+iK)ψ1+e-iϕ(∂ρ-iρ-1∂ϕ)ψ4+∂zψ3=0 (2.4a) (∂ct+iK)ψ2+eiϕ(∂ρ+iρ-1∂ϕ)ψ3-∂zψ4=0 (2.4b) (∂ct-iK)ψ3+e-iϕ(∂ρ-iρ-1∂ϕ)ψ2+∂zψ1=0 (2.4c) (∂ct-iK)ψ4+eiϕ(∂ρ+iρ-1∂ϕ)ψ1-∂zψ2=0 (2.4d) When the spinor components ψj are independent of ϕ, solutions exist only for the ψj also independent of ρ. These are the well-known plane wave solutions ψj=ajei(qz- ωt), where the wavenumber q and the energy ℏω are con- strained by (ω/c)2=K2+q2. To attain localized wavepacket solutions, we need to consider azimuthal dependence. The angular momentum operator L=r×p does not commute with the Hamiltonian, but the combination J=L+ Σ does, where Σ= . The z component of the total angular momentum operator is (2.5) Let the spinor components ψj have azimuthal dependence eiνjϕ; the Jz eigenstate equations for ψ1,ψ2 read (2.6) This will be an eigenstate of Jz if ν1+1/2=ν2-1/2, ν2-ν1=1, with eigenvalue (ν1+1/2)ℏ. Similarly for ψ3,ψ4 we shall have an eigenstate of Jz if ν3+1/2=ν4-1/2, ν4-ν3=1, with eigenvalue (ν3+1/2)ℏ. Hence the choice ν1,3=0, ν2,4=1 makes Ψ an eigenstate of Jz with eigenvalue ℏ/2. (The choice ν1,3=-1, ν2,4=0 makes Ψ an eigenstate of Jz with eigenvalue -ℏ/2.) It is necessary to have integer νj, since the spinor components are in real space-time (not in some abstract spin space) so we must have ψj(ϕ+2π)=ψj(ϕ). The eigenvalues of Jz are thus ±ℏ/2,±3ℏ/2 etc. With spinor components ψ1,3=f1,3(ρ,z,t),ψ2,4=eiϕf2,4 (ρ,z,t), the azimuthal dependence cancels out, and the equations (2.4) read (∂ct+iK)f1+(∂ρ+ρ-1)f4+∂zf3=0 (2.7a) 9The Spinning Electron (∂ct+iK)f2+ ∂ρf3-∂zf4=0 (2.7b) (∂ct-iK)f3+(∂ρ+ρ-1)f2+∂zf1=0 (2.7c) (∂ct-iK)f4+ ∂ρf1-∂zf2=0 (2.7d) The combination (∂ct-iK)(2.7a)-(∂ρ+ρ-1)(2.7d)-∂z(2.7c) gives (∂2ct+K2-∂2ρ-ρ-1∂ρ-∂2z)f1(ρ,z,t)=0 (2.8) Likewise (∂ct-iK)(2.7b)-∂ρ(2.7c)+∂z(2.7d) gives us (∂2ct+K2-∂2ρ-ρ-1∂ρ+ρ-2-∂2z)f2(ρ,z,t)=0 (2.9) The equations (2.8) and (2.9) are solved respectively by ei(qz-ωt)J0(kρ), ei(qz-ωt)J1(kρ), k2+q2+K2=(ω/c)2 (2.10) The function f3 satisfies the same equation as f1, and f4 satisfies the same equation as f2. The transverse and longitudinal wavenumbers k and q are real, and ω≥cK, or ℏω≥mc2. The wavenumbers k≥0 and q≥0 are related to K=mc/ℏ and ω by k2+q2+K2=(ω/c)2; the maximum value of both k and q is Q= . Hence the general form of the spinor eigenstate of Jz with eigenvalue ℏ/2 is 0 ψ1,3(ρ,z,t)= dω dk A1,3(ω,k)ei(qz-ωt)J0(kρ) (2.11) ψ2,4(ρ,ϕ,z,t)=eiϕ dω dk A2,4(ω,k)ei(qz-ωt)J1(kρ) (2.12) These are analogues of the acoustic and electromag- netic wavepackets, for which simple closed forms exist ([8], Section 2.6). The author has not found amplitudes Aj(ω,k) which lead to closed forms for the relativistic spinor components. Bessel beam wavefunctions (not localized enough transversely to have finite energy per unit length) have been studied by Bliokh et al. [9]. 3. NON-RELATIVISTIC SPINORS Lévy-Leblond [6] has shown that four coupled non- relativistic wave equations, equivalent to the Schröding- er equation, are spinors representing spin 1/2 particles, with g-factor 2 (see also Greiner [10]). We shall again construct a general eigenstate of Jz with eigenvalue ℏ/2: it is a four-component spinor. It is based on localized wavepacket solutions of the time-dependent Schrödinger equation, with no restriction on the wavepacket param- eters. In Section 4 we shall explore some properties of exact Gaussian solutions of the equations satisfied by the spinor components. Let Ψ(r,t) be the four-component spinor, Ψ= , with ψ,χ each having two components. The Lévy-Leb- lond non-relativistic coupled spinor equations are, with E=iℏ∂t, p=-iℏ∇, Eψ+σ∙pχ=0, σ∙pψ+2mχ=0 (3.1) σ are, as before, the Pauli spin matrices defined in (2.2). Note that the ψ,χ in (3.1) have dimension differing by a speed; we could make them the same by inserting factors e2/ℏ or c in front of χ, but choose not to do so, in order keep the Lévy-Leblond formulation. Note also that the lower spinor component χ can be eliminated, giv- ing the Pauli-Schrödinger equation Eψ= (σ∙p)2ψ, with Hamiltonian H= (σ∙p)2. For comparison, the Dirac equations (2.1), with ψu= , ψv= , may be written in the form (E-mc2)ψu=cσ∙pψv, cσ∙pψu=(E+mc2)ψv (3.2) The non-relativistic limit is obtained from (3.2) by setting ψj(r,t)=e-imc 2t/ℏ)Fj(r,t). Then Eψj=iℏ∂t ψj=e (mc2+iℏ∂t)Fj, and the equations (3.2) have the dominant terms EFu=cσ∙pFv, cσ∙pFu=2mc2Fv (3.3) These are the same as (3.1) if we identify Fu with ψ, and cFu with -χ. Returning to solutions of the Lévy-Leblond equa- tions (3.10), we write ψ= , χ= , and consider wavepacket motion, predominantly along the direc- tion, but of course converging onto or diverging from the focal region, which we shall centre at the space-time origin. Again in cylindrical polar coordinates ρ,ϕ, and with use of (2.3), the four time-dependent free-space equations (3.1) for the spinor Ψ read -∂tψ1+e-iϕ(∂ρ-iρ-1∂ϕ)ψ4+∂zψ3=0 (3.4a) -∂tψ2+eiϕ(∂ρ-iρ-1∂ϕ)ψ3+∂zψ4=0 (3.4b) ψ3+e-iϕ(∂ρ-iρ-1∂ϕ)ψ2+∂zψ1=0 (3.4c) ψ4+eiϕ(∂ρ-iρ-1∂ϕ)ψ1+∂zψ2=0 (3.4d) When the spinor components ψj are independent of ϕ, solutions exist only for the ψj also independent of ρ. These are the plane wave solutions ψj=ajei(qz-ωt), where the wavenumber k and the energy ℏω are constrained by 10 John Lekner ℏω=ℏ2q2/2m. To attain localized wavepacket solutions, we need to consider azimuthal dependence. The angular momentum operator L=r×p does not commute with the free-particle Hamiltonian H= (σ∙p)2, but the combination J=L+ Σ, Σ= does, as may be verified from the commutators σ×σ=2iσ, [L,σ∙p]=iℏσ×p, [σ,σ∙p]=-2iσ×p. J satisfies the angular momentum com- mutation relations J×J=iℏJ. The z component of the total angular momentum operator is again Jz=Lz+ =-iℏ diag(1,1,1,1)∂ϕ+ diag(1,-1,1,-1) (3.5) We shall now construct the non-relativistic spinor eigenstates of Jz. Let the spinor components ψj have azimuthal dependence eiνjϕ; the Jz eigenstate equations for ψ1,ψ2 are the same as in (2.6): (3.6) The equations (3.5) and (3.6) have the same form as in the relativistic case, equations (2.5) and (2.6). Hence as before the choice ν1,3=0, ν2,4=1 makes Ψ an eigenstate of Jz with eigenvalue ℏ/2 and the choice ν1,3=-1, ν2,4=0 makes Ψ an eigenstate of Jz with eigenvalue -ℏ/2. With spinor components ψ1,3=f1,3(ρ,z,t), ψ2,4=eiϕf2,4(ρ,z,t), the equations (3.4) read -∂tf1+(∂ρ+ρ-1)f4+∂zf3=0 (3.7a) -∂tf2+∂ρf3-∂zf4=0 (3.7b) f3+(∂ρ+ρ-1)f2+∂zf1=0 (3.7c) f4+∂ρf1-∂zf2=0 (3.7d) The last two equations give f3,4 in terms of deriv- atives of f1,2, which in turn satisf y the free-space Schrödinger equation for azimuthal orbital quantum number 0 and 1: (iℏ∂t+ [∂2ρ+ρ-1∂ρ+∂2z])f1(ρ,z,t)=0 (3.8) (iℏ∂t+ [∂2ρ+ρ-1∂ρ-ρ-2+∂2z])f2(ρ,z,t)=0 (3.9) Equations(3.8) and (3.9) are satisfied by Jn(κρ)einϕeiqz e-iℏk2t/2m, with n=0,1 respectively, and κ2+q2=k2; Jn are the regular Bessel functions of order n. Hence spinor com- ponents of forward-propagating wavepackets have the form einϕ dk e-iℏk2t/2m dq Fn(k,q)eiqzJn(κρ) (κ2+q2=k2) (3.10) The amplitudes Fn(k,q) are complex functions, sub- ject only to the existence of the norm and of the expec- tation values of energy and momentum of the wave packet. A similar expression gives the wavefunctions of scalar and of electromagnetic pulses [8]. To sum up this Section: a general non-relativistic eigenstate of Jz with eigenvalue ℏ/2 has been found: it is a four-component spinor, of which two components have ‘twist’, with eiϕ azimuthal dependence. In this formula- tion the spin resides in the azimuthal dependence of two of the wavefunctions, in real space-time. Any spinor based on localized wavepacket solutions of the time-dependent Schrödinger equation, construct- ed as above, will be an eigenstate of Jz with eigenvalue ℏ/2. The next Section gives an explicit example. Station- ary states (energy eigenstates) of the hydrogen atom are briefly discussed in Appendix A. 4. SPINNING GAUSSIAN WAVEPACKETS A free-particle wavepacket solution of Schrödinger’s time-dependent equation dates back to the early days of quantum mechanics (Kennard [11], Darwin [12]). This is the Gaussian wavepacket. It is a compact exact solu- tion, but with a physical flaw, to be discussed below. For propagation along the axis, and with cylindrical sym- metry, it has the form g(ρ,z,t)=b3/2[b+ivt]-3/2exp{iQ(z- )- } (4.1) The Gaussian wavepacket (4.1) is normalized so that g*g=1 at the space-time origin. In (4.1) the spatial origin ρ=0, z=0 is the position of maximal |g| a time t=0, Q is the dominant z component wavenumber, m is the mass of the particle, u=ℏQ/m is the group speed, and v=ℏ ⁄2mb is the spreading speed. The length b gives the spread of the wavepacket at t=0. Earlier and later the longitudinal and lateral spread of the packet is greater, proportional to [b2+(vt)2]1/2. Thus ρ=0, z=0 can be thought of as the centre of the focal region of the wavepacket, occupied at t=0. As t increases towards zero the wavepacket con- verges to its most compact form, reaches it at t=0, and then expands as it continues to propagate in the positive z direction. The packet used by Ohanian [1] is equivalent to (4.1) evaluated at Q=0 (zero momentum expectation value) and t=0. For the Gaussian wavepacket g the momentum oper- ator has the expectation values (see for example [13]) 11The Spinning Electron ‹pz›=-iℏ‹∂z›=ℏQ, ‹px›=0=‹py›, ‹p2›= =‹-ℏ2∇2›=ℏ2 (4.2) The wavepacket g is neither an energy nor a momen- tum eigenstate, but it is an eigenstate of the orbital angular momentum operator Lz=xpy-ypx=-iℏ(x∂y-y∂x)= -iℏ∂ϕ. The orbital angular momentum eigenvalue is zero, because g is independent of the azimuthal angle ϕ. Eigenstates of the z component of orbital angular momentum, with eigenvalues which are integer multi- ples of ℏ, may be generated from any such g by differen- tiation, as shown in [13]. The probability density of the scalar wavepacket is g*g: the probability that the particle described by g(r,t) is within the volume element d3r is d3r g*g. The norm N=∫d3r g*g (integration over all of space) is independent of time. The probability density flux, or the probability current density vector S, satisfies the conservation law ∇.S+∂t(g*g)=0, S(r,t)= Im(g*∇g) (4.3) What are the corresponding relations for spinors? The conservation law is now (Lévy-Leblond [6], Section IIIe, and Appendix B) ∇.S+∂t(ψ+ψ)=0 (4.4) S(r,t)=-ψ+σχ-χ+σψ= Im(ψ+ ∇ψ)+ ∇×(ψ+σψ) (4.5) The first term in the second expression for S corre- sponds to the Schrödinger current in (4.3), the second is a spin current. Ohanian [1] derived the relativistic ana- logue of last term in (4.5). He showed that it leads, in the nonrelativistic limit, to an azimuthal current. In his words, “such a circulating flow of energy will give rise to an angular momentum. This angular momentum is the spin of the electron.” We shall calculate the radial, azimuthal, and longi- tudinal components of the probability current density, Sρ,Sϕ,Sz in the simplest case, in which the spinor com- ponents are ψ1=f1(ρ,z,t), ψ2=0, ψ3~∂zψ1, ψ4~eiϕ∂ρψ1. From Appendix B, the components of the probability current density are given by Sρ=Im{ f*1∂ρf1}, Sϕ=- ∂ρ|f1|2, Sz=Im{ f*1∂zf1} (4.6) With f1(ρ,z,t)=g(ρ,z,t) the probability density and current components are given by g*g=b3[b2+(vt)2] exp (4.7) Sρ= g*g, Sϕ= g*g, Sz= g*g (4.8) The components Sρ,Sz are the same for the scalar wavepacket, the azimuthal component Sϕ is zero in the scalar case based on g. The conservation law (4.4) is sat- isfied. A problem with the Gaussian solution is appar- ent in Sz: for positive z and negative t (or vice versa) the longitudinal component is negative if the magnitude of vtz exceeds that of 2Qb3. The probability current then flows backward. Far from the focal region (here centred on the space-time origin) there should be no backward flow for free-space propagation. Note that the Gaussian wavepacket cannot be put in the purely forward-propa- gating form (3.10). Nevertheless, the Gaussian packets demonstrate the azimuthal current component which arises in the spinor formulation. Figures 1 and 2 show the current components in the focal plane, and at a transverse plane cutting through the wavepacket center at a later time. The azimuthal part gives the electron wavepacket its spin. Figure 1. Focal plane section through a Gaussian spinor wavepack- et, at t=0. The contours give the probability density, the arrows the transverse current density (the longitudinal current is not shown). The direction of motion is out of the page. The transverse current density is purely azimuthal at this instant. 12 John Lekner 5. SUMMARY The spinning electron may be described by a four- component spinor, depending on space and time coor- dinates, in both relativistic and non-relativistic quan- tum theory. The non-relativistic quantum theory and its azimuthal dependence is similar to the relativistic Dirac spinor formulation of Section 2. In both cases the spin is contained in the azimuthal dependence of wavefunc- tions in ordinary space-time. Gould [14] used the Ham- iltonian H= (σ.p)2 to show that the magnetic moment follows (correct to lowest order), just as in the Lévy- Leblond spinor formulation. There is thus an alternative formulation to the usual ‘spin degree of freedom’, and the total wavefunction being a product of space and spin parts, as is done in nonrelativistic quantum theory. Nev- ertheless, the non-relativistic decoupling of space and spin is usually simpler, as is illustrated by the spinor ver- sion of the Hydrogen atom, Appendix A. ACKNOWLEDGMENT Constructive comments of t he referees have improved the paper, and are much appreciated. REFERENCES [1] Ohanian H C “What is spin?”, Am. J. Phys. 54, 500- 505 (1986). [2] Belinfante F J “On the spin angular momentum of mesons”, Physica 6, 887-898 (1939). [3] Dirac P A M “The quantum theory of the electron”, Proc. Roy. Soc. A 117, 610-624 (1928). [4] Dirac P A M “The quantum theory of the electron. Part II”, Proc. Roy. Soc. A 118, 351-361 (1928). [5] Griffiths D J, Introduction to quantum mechanics, 2ed (Prentice Hall, New Jersey, 2005), Section 4.4. [6] Lévy-Leblond J-M “Nonrelativistic particles and wave equations”, Commun. Math. Phys. 6, 286-311 (1967). [7] Galindo A and del Rio C S “Intrinsic magnetic moment as a nonrelativistic phenomenon”, Am. J. Phys. 29, 582-584 (1961). [8] Lekner J, Theory of electromagnetic pulses (Institute of Physics Publishing 2018), Section 2.6. [9] Bliokh K Y, Dennis M R and Nori F “Relativis- tic electron vortex beams: angular momentum and spin-orbit interaction”, Phys. Rev. Lett. 107, 174802 (2011). [10] Greiner W, Quantum mechanics, 3ed (Springer, Ber- lin, 1994), Chapter 13. [11] Kennard E H “Zur Quantenmechanik einfacher Bewegungstypen”, Z. Physik 44, 326-352 (1927). [12] Darwin C G “Free motion in wave mechanics”, Proc. Roy. Soc. Lond. A117, 258-293 (1927). [13] Lekner J “Rotating wavepackets”, Eur. J. Phys. 29, 1121-1125 (2008). [14] Gould R J “The intrinsic magnetic moment of ele- mentary particles”, Am. J. Phys. 64, 597-601 (1996). [15] Lévy-Leblond J-M “The total probability current and the quantum period”, Am. J. Phys. 55, 146-149 (1987). [16] Mita K “Virtual probability current associated with the spin”, Am. J. Phys. 68, 259-264 (2000). [17] Landau L D and Lifshitz E M, Quantum mechanics, 2ed. (Pergamon, Oxford, 1985). Figure 2. Gaussian spinor wavepacket, at z=ut=2b. The trans- verse current density now has radial and azimuthal components. The group speed is u, so the section is through the centre of the wavepacket. The longitudinal current density is not shown. 13The Spinning Electron APPENDIX A. THE HYDROGEN ATOM IN SPINOR FORM The equations (3.1) become, with E now an energy eigenvalue, no longer a time derivative, (E+ )ψ+σ∙pχ=0, σ∙pψ+2mχ=0 (A.1) ψ=Eψ or [ ∇2-e ]ψ=Eψ (A.2) Considering the non-degenerate ground state, with Jz eigenvalue , ψ1 and ψ2 must satisfy the same equation. This is not possible if we choose ψ2 to have azimuthal dependence eiϕ, as in Section 3, unless we also take ψ2 to be zero. The ground state spinor now consists of ψ1, the hydrogenic ground state 1S, and ψ2=0, ψ3~∂zψ1, ψ4~eiϕ∂ρψ1. Because the Lévy-Leblond probability den- sity is defined in terms of the first two spinor compo- nents ψ1,ψ2, and the probability density current can be expressed in terms of ψ1,ψ2, the hydrogenic ground state is, at least in the probability density and the prob- ability density current, equivalent to the scalar ground state. The azimuthal dependence is hidden in the fourth spinor component. For the first excited states we have a choice of 2S and 2P. The former is set up as above, the latter with ψ1=0, and ψ2 with e±iϕ dependence. Lévy-Leblond [15] and Mita [16] discuss the electron probability current of the ‘stationary’ states. APPENDIX B. PROBABILITY DENSITY AND FLUX In the Dirac case (Section 2), Ψ+Ψ is the probabil- ity density, and S=cΨ+αΨ, with α is defined in (2.2). In the nonrelativistic formulation of Lév y-Leblond we have a time derivative of ψ but not of χ: iℏ∂tψ+σ∙pχ=0, σ∙pψ+2mχ=0, or ∂tψ-σ∙∇χ=0,-iℏσ∙∇ψ+2mχ=0. To keep the norm time-independent Lévy-Leblond defines the prob- ability density in terms of ψ only, as ψ+ψ. The conserva- tion law is now (Lévy-Leblond [6], Section IIIe) ∇.S+∂t(ψ+ψ)=0 (B.1) ∂t(ψ+ψ)=ψ+(σ∙∇χ)+(∇χ+∙σ)ψ=∇∙(ψ+σχ+χ+σψ) (B.2) Hence S(r,t)=-(ψ+σχ+χ+σψ). We may express this cur- rent purely in terms of the top two spinor components ψ, since χ= σ∙∇ψ. This gives S(r,t)= {ψ+σ(σ∙∇ψ)-(σ∙∇ψ)+σψ} (B.3) On using the commutation relations of the Pau- li matrices, σ×σ=2iσ, the probability density current becomes S(r,t)= [ψ+∇ψ-(∇ψ+)ψ]+ ∇×(ψ+σψ) (B.4) The first term in this expression for S corresponds to the Schrödinger current in (3.3), the second is a spin current, which gives the correct g factor at leading order [6]. The spin term is the curl of a vector, and so does not contribute to the conservation law (B.1). See also Landau and Lifshitz [17] Section 114, and Mita [16] for the spin current term. We shall calculate the radial, azimuthal, and longi- tudinal components of the probability current density, Sρ,Sϕ,Sz. The corresponding spin matrix components are σρ=σ. = , σϕ=σ. = , σz= (B.5) Let f1,f2 be solutions of (3.8) and (3.9), respectively, and ψ1=f1, ψ2=eiϕf2. We can set f2=a∂ρf1 [12]; a is a length parameter. We shall first calculate ψ+σψ; this has the cylindrical components (2aRe{ f *1∂ρf1}, 2aIm{(∂ρf *1)f1}, | f1|2-a2|∂ρf1|2). Note that there is no ϕ dependence. The curl of this vector is ∇×(ψ+σψ)=(-2a∂zIm{(∂ρf*1)f1}, 2a∂zRe{ f*1∂ρf1}- -∂ρ[|f1|2-a2|∂ρf1|2], 2a∂ρIm{(∂ρf*1)f1}+2aρ-1Im{(∂ρf*1)f1}) (B.6) When the length a is zero, just the azimuthal com- ponent remains, ∇×(ψ+σψ)a=0=(0,-∂ρ| f1|2,0). In that spe- cial case the Schrödinger current is proportional to Im{ f *1∇f1}=Im{ f *1(∂ρf1,0,∂zf1)}, and the components of the probability current density are given by Sρ=Im{ f*1∂ρf1}, Sϕ=- ∂ρ|f1|2, Sz=Im{ f*1∂zf1} (B.7) As in the hydrogen ground state, the a=0 spinor now consists of ψ1, and ψ2=0, ψ3~∂zψ1, ψ4~eiϕ∂ρψ1. The fourth component contributes to the azimuthal current, and to the angular momentum.