arXiv:1908.03276v2 [quant-ph] 14 Oct 2019 hogottepeetwr) is work), present the throughout non-relativistic Uh- the and [ Goudsmit modifying lenbeck’s for in account to matrices Schr¨odinger equation spin mous relevant the of some of review work. selective earlier a with begin We [ articles nal urn) hc a eevdcnieal teto [ attention considerable spin received (the has which term current), include spin-dependent to additional out unexpected turns an elec- density current non-relativistic probability its with trons, associated results perimental neeti unu ehnc etok [ textbooks mechanics quantum in interest aefnto.Tesadr omo al’ qainas Ref. equation example, Pauli’s for of in, form given standard spinor The two-component function. a wave function by momentum Schr¨odinger wave equation angular the scalar of complex spin of the this Introduction replacing describe necessitated magnetic to momentum. matrices a angular to spin proportional of spin moment intrinsic energy magnetic its electron interaction electromag- an an the with in field viz., moving usual field, electron the netic to an term for phenomenological a Hamiltonian add to a momentum was ansatz implied angular numbers spin quantum integer half-integral by able ∗ ri,Sur,F 49,USA 34994, FL Stuart, Trail, [email protected] sbcgon,i 97Pui[ Pauli 1927 in background, As al’ qainfrteeeto satpco general of topic a is electron the for equation Pauli’s 2 1 m ( p 11 ˆ h al n ´v-eln qain,adteSi Current Spin the and L´evy-Leblond Equations, and Pauli The − 4 h pntr ntecretdensity. current the mig in that term work spin L´evy-Leb experimental the the for non-relativistic of possibilities derivation was on simplified current commenting deri somewhat spin density a current the present the itself, We of spin was feature the current prominent the with a in was show term term already L´evy-Leblond spin this had of com the work We implying earlier it. aware the for However, are account vario we to although which forward put h of equation, were follow, Pauli explanations not the hoc prob does ad from term associated density a the current Such of the term. non decomposition spin-dependent app a is Gordon rela term contain electron a the interaction an of spin limit, derivation Pauli’s as this Dirac’s that such following demonstration particle stated quent spin-1/2 often a was of establi what spin was to the It Pauli Schr¨odingerthat of equation. the version of phenomenological linearization original the from sion , , 12 q erve h ieaueo h al qainadiscurren its and equation Pauli the on literature the review We 5 A .INTRODUCTION I. .Atog tepan ayo h ex- the of many explains it Although ]. yohssta pcrldt explain- data spectral that hypothesis ] ) 2 1 + emnn drs:22 WTilia NW 2127 address: permanent ; qφ 1 2 − cnetdt Iuis used units, SI to (converted µ s · o h lcrn His electron. the for 10 B nrdcdhsfa- his introduced ] ψ E ˆ ( 1 r t , ψ 1 ( = ) – r 3 t , n jour- and ] Dtd coe 5 2019) 15, October (Dated: ) , ae .Wilkes M. James 5 half- – (1) 9 ]. E where hr eepoietecpin oti tne(Ref. stance this to exceptions although prominent phenomenon, were relativistic there a as spin electron gard 17 L´evy-Leblond [ by extensive compiled the quotations example, of for list (see, textbooks physics the many of triumph a was assumptions, theory. hoc Dirac prediction, ad This in- no equation. electron field, requiring Pauli’s the in magnetic One hand” involving a “by term freedom.” troduced with the of interacting course, degree moment new of magnetic its was, to these energy due of potential electron additional the the as of regarded be terms electro- “can two contained include that use it to that making showed replacement by interactions coupling magnetic and electron, minimal free the a of for equation wave tic .11.Aogteeecpin,eg,Refs. e.g., exceptions, these Among 141). p. by mag- external defined with fields interaction electric the and describe to netic 205) p. al pnvco ihPuisi matrices spin Pauli with vector spin Pauli understood), be to and but omitted, (often matrix identity p steeeto antcmmn prtr where operator, moment magnetic electron the is n nryoeaos h perneo (real-valued) of potentials, vector appearance and The scalar electromagnetic operators. energy and scomponents. as iia coupling minimal A ˆ ( = h bevto a aeta h pn12prop- 1/2 spin the that made was observation the , fe ia’ icvr tbcm omnlr in lore common became it discovery Dirac’s After h olwn erDrc[ Dirac year following The = r t , q − hdcnlsvl yL´evy-Leblond’s work by conclusively shed − list h otaynotwithstanding. contrary the to claims , (= ,i osqec fuigtegueinvariant gauge the using of consequence a is ), i ∗ ψ ∇ ~ tidct esrbecneune of consequences measurable indicate ht ,wt oadtoa supin,that assumptions, additional no with n, spyial oiae u otherwise but motivated physically us a L´evy-Leblond from by derivation its ∇ iitcwv qain n i subse- his and equation, wave tivistic ae ntennrltvsi ii.In limit. non-relativistic the in eared = e rmhseuto.Hne utas just Hence, equation. his from ved odeuto n t urn density, current its and equation lond φ − blt urn est a on to found was density current ability wvr rmteuuldrvto of derivation usual the from owever, − eto h nyecpint these to exception only the on ment e and rltvsi nntr,cnrr to contrary nature, in -relativistic r h asadcag fteeeto,and electron, the of charge and mass the are ) est,dsusn h progres- the discussing density, t ψ ψ nfc o-eaiitci nature. in non-relativistic fact in ∂t ∂ 2 1 E A ˆ Finally, . sa2cmoetspinor, 2-component a is = supin(e,freape Ref. example, for (see, assumption i µ ~ s ∂t ∂ = r h sa iermomentum linear usual the are 2 q 13 m ~ rsne i relativis- his presented ] σ Density B = σ 1 ∇ i , ste2 the is φ 14 i × ( 1 = r )t re- to ]) σ t , A 4 sthe is and ) , , 2 and 15 × 14 (2) , m 2 3, – 2 , , 2 erty could be incorporated into the spin-0 non-relativistic may be left with the feeling that if not for the purely Schr¨odinger equation by simply replacing the kinetic mo- mathematical and fortuitous existence of the spin iden- mentum operator πˆ = pˆ − qA appearing there with the tity, the Pauli equation would seem a slightly strange product σ · πˆ. To see how this is possible, we follow concoction forced upon the Schr¨odinger equation. This Sakurai [16] by beginning with the spin-0 non-relativistic situation is somewhat alleviated by the fact that the min- free-electron Schr¨odinger equation of wave mechanics: imally coupled Schr¨odinger-Pauli equation, from which Pauli’s equation follows, can be shown [3, 16, 20] to pˆ2 Ψ(r,t)= Eˆ Ψ(r,t) , (3) be the non-relativistic limit of Dirac’s equation. Such 2m a derivation is convincing, yet fails to satisfy the de- where Ψ is Schr¨odinger’s complex scalar wave function. sire for a purely non-relativistic account. The work of Recall the important spin-vector identity L´evy-Leblond, to be discussed later, seeks to fill this gap.
(σ · aˆ)(σ · bˆ) = (aˆ · bˆ)1 + i σ · (aˆ × bˆ) , (4) II. THE MISSING TERM IN THE PAULI where aˆ and bˆ are any two vector operators that com- CURRENT DENSITY mute with σ. For aˆ = bˆ = pˆ one finds from Eq. (4) the operator identity (σ · pˆ)2 = pˆ21. Sakurai refers As with the Schr¨odinger equation, one would hope to to the theory that results from making this substitu- determine from the Pauli equation a probability density tion in Schr¨odinger’s equation (3), and replacing the ρ and current J ′ satisfying the probability conservation wave function Ψ by a 2-component spinor ψ, as the (or continuity) equation Schr¨odinger-Pauli theory (Ref. 16, footnote, p. 79), i.e., ∂ρ + ∇ · J ′ =0 . (8) (σ · pˆ)2 ∂t ψ(r,t)= Eˆ ψ(r,t), (5) 2m Such derivations are offered, for example, in Refs. 1, 2, which we refer to as the Schr¨odinger-Pauli equation. It and 4–7. A canonical derivation of the current density has the plane wave solutions ei(k·r−ωt) χ, where χ is from Pauli’s equation is that of Ref. 2, p. 340. Defining † † ∗ ∗ a constant but otherwise arbitrary spinor, giving rise the probability density by ρ = ψ ψ, where ψ = ψ1 ψ2 to the dispersion relation ω = ~k2/2m characteristic denotes the Hermitian conjugate or adjoint of ψ, the cur- of non-relativistic particles. The effects of spin manifest rent density is found there to be given (in our notation by allowing the electron to interact with an electromag- and SI units) by netic field. This is accomplished by introduction of the i~ q gauge invariant minimal coupling replacements of pˆ with J ′ = − ψ† (∇ψ) − (∇ψ)† ψ − Aψ†ψ . (9) πˆ = pˆ − qA, and Eˆ withε ˆ = Eˆ − qφ, in which case the 2m m h i Schr¨odinger-Pauli equation (5) takes the form Since the Pauli equation is the non-relativistic limit of Dirac’s equation, one might expect that Eq. (9) would (σ · πˆ)2 ψ(r,t)=ˆε1 ψ(r,t). (6) be the corresponding non-relativistic limit of the current 2m density of Dirac’s equation, but such is not the case. Applying the identity (4) with aˆ = bˆ = πˆ = pˆ − qA, we Relativistic quantum mechanics shows that in this limit have in the numerator of the last line: there appears an additional term, referred to as the spin current density. The details are given, for example, in (σ · πˆ)2ψ = πˆ 21ψ + i σ · [(pˆ − qA) × (pˆ − qA)] ψ , Sakurai’s discussion, Ref. 16, p. 107, of the Gordon de- composition of the Dirac current and its non-relativistic = πˆ 21ψ − q~ σ · [∇ × (Aψ)+ A × (∇ψ)] , limit. It is found (see also Ref. 6) that this limit yields = πˆ 21ψ − q~ (σ · B)ψ , (7) instead the following expression:
′ where the cross products pˆ × pˆ and A × A both give J = J + Jspin , (10) vanishing contributions, we have replaced pˆ = −i~∇, and used the well-known vector identity ∇ × (Aψ) = where ∇ψ×A+(∇×A)ψ, where ∇×A = B. Substituting Eq. ~ † (7) together withε ˆ = Eˆ − qφ in Eq. (6) shows that the Jspin ≡ ∇ × ψ σψ (11) 2m minimally coupled Schr¨odinger-Pauli equation (6) repro- duces precisely the Pauli equation (1). The result follows is the additional spin current density . This term must be without regard for the Dirac equation, obtaining Pauli’s a feature of the current density for any spin-1/2 particle spin interaction term directly from the minimally cou- described by the Dirac equation, even when the particle pled Schr¨odinger equation and the replacement of πˆ by carries no charge, e.g., a neutron described by the Dirac σ · πˆ, as suggested by Feynman [15]. equation with the addition of a non-minimal coupling The conclusion, then, is that the Pauli spin interaction term (as in Ref. 20, p. 241). It is important to note, with term is an intrinsically non-relativistic effect. Still, one Nowakowski [6], that this spin current is also manifest in 3 the reduction of the Dirac equation for a free particle, i.e, the correct expression for the spin current. As admitted in the absence of any interactions. in Ref. 9, referring to the current density, “This argument Various writers have suggested methods for supplying is not rigorous, but it does hint that another term should the spin current missing from Eq. (9). For example, in be added ... .” the venerable and oft-cited textbook of Landau and Lif- As a final example, we consider the work of Shikakhwa, shitz [21], the problem of a charged particle moving in et al. [8]. They discuss Nowakowski’s [6] spin current a magnetic field is considered. A variational principle term, denoting it by JM and referring to it as a mag- is applied, varying the expectation value of the Pauli netization current. These authors are clearly aware that Hamiltonian of Eq. (1) (ignoring the scalar potential) the spin current cannot be derived from the Pauli equa- under a variation in the vector potential A. From this tion (1) in the usual way (as, for example, in Ref. 2, p. they derive the charge current density qJ, where J is 340), so they instead begin with the minimally coupled given precisely by Eq. (10), including the spin current. Schr¨odinger-Pauli equation (6), ignoring the scalar po- Without the minimal coupling to a magnetic field via A tential term: and the charge q, however, such a derivation apparently (σ · πˆ)2 ∂ψ would fail to provide the desired spin current term, which ψ = i~ . (15) we know from Eq. (10) should appear even in the absence 2m ∂t of such an interaction. The key to the success of their approach was to use Eq. Of other works known to the present author, each relies (15) in deriving a probability conservation equation be- in some way upon the following line of reasoning. Since fore expanding (σ · πˆ)2 (cf. Eq. (7)), since otherwise the continuity equation (8) involves only the divergence the Pauli equation (1) follows. Note that although their of the current density, it is always possible to add a diver- goal is to obtain Nowakowski’s expression for the non- genceless term to the current density without changing relativistic limit of the current density for a free spin- the equation. The spin current proportional to a curl is, 1/2 particle, given in their Eq. (2), they actually work of course, just such a term. But since the Pauli equation with Eq. (15), above, for such a particle interacting with does not provide this, one must rely on reasonable phys- a magnetic field. However, they emphasize more than ical arguments to supply it, which should hold even in once their use of πˆ as merely a convenient bookkeeping the absence of interaction potentials. device in arriving at the desired result; the vector poten- As an example, Mita [7] takes the novel approach of tial is ignored in the final step of their derivation. Their beginning with the expectation value of the spin angular ~ motivation for using πˆ rather than pˆ, which naturally momentum S = σ, and showing that it can be written 2 appears in the free particle Hamiltonian, is the concern as the volume integral of an angular momentum, viz., that using pˆ in their derivation would be problematic due ~ to the commutation of its components. We show in the hSi = m r × j d3r , j = ∇ × ψ†σψ . (12) s s 4m next section that their concern can be addressed by a ZV0 different derivation that avoids this problem. Mita considers this js to be “the virtual probability cur- In concluding this (in no way comprehensive) review, rent density that gives rise to the spin angular momen- we are compelled to attempt clarification of certain asser- tum of the electron.” He also notes, however, that “the tions made in the Nowakowski [6] and Shikakhwa, et al. proper gyromagnetic ratio of the particle cannot be ob- [8] papers. In Ref. 8 one reads “Nowakowski correctly tained in the context of our analysis based on nonrela- states that the additional term JM cannot be derived tivistic quantum mechanics.” In fact, his current density from the non-relativistic Pauli equation, because the co- gives a gyromagnetic ratio that is half the correct value. variance argument can be applied only for the fully rela- Greiner [2] and Parker [5], as well as Hodge, et al [9] tivistic Dirac equation.” The first clause of this sentence (following Greiner), call upon a result from electromag- appears to be based on Nowakowski’s [6] Eq. (4) for the netic theory, viz., that a magnetization field M induces current density derived from Pauli’s equation, in which an additional (electric) current density qJM = ∇ × M the spin current is notably absent. But this is already (see, for example, Refs. 18 and 19). The expectation well known, and no surprise (e.g., as cited earlier in Ref. value of the electron magnetic moment is 2, p. 340). The second clause refers to the Lorentz co- variance of the 4-vector probability current density of the q~ Dirac equation, whose form can be shown [22, 23] to be hµ i = ψ†µ ψ d3r= ψ†σψ d3r , (13) s s 2m unique. Z Z What the authors of Ref. 8 have shown, however, is the integrand of Eq. (13) defining the magnetic moment that if one starts from the non-relativistic minimally cou- density, which these authors identify with the magneti- pled Schr¨odinger-Pauli equation of Eq. (15), computing zation M of electromagnetic theory. One thus finds for the current density before expanding the term (σ · π)2, the associated probability current density: precisely the same form of the free particle current den- sity found in the non-relativistic limit of the Dirac current ~ 1 † JM = ∇ × M = ∇ × ψ σψ , (14) density is obtained, including the correct spin current (af- q 2m ter setting A to zero). Contrary to what Nowakowski [6] 4 suggests, their work shows (although they seem reluctant be written as to make this claim) that there is no ambiguity in the 1 ∂ρ i i † definition of the current density derived from the non- = ψ† (σ · pˆ) χ − (σ · pˆ) χ ψ , relativistic Schr¨odinger-Pauli equation. True enough, the c ∂t ~ ~ uniqueness of the non-relativistic current density was not = ψ† (σ · ∇χ) + (∇χ ·σ)† ψ . discussed there, but in fact its uniqueness was found in earlier work of Holland [22, 23] and Holland and Philip- The divergence identity pidis [24]tobe inherited as the limit of the unique current density of the relativistic case. The situation is reminis- ∇· ψ†σχ + χ†σψ = ψ† (σ · ∇χ) + (∇χ · σ)† ψ cent of the claim by early proponents of the Dirac equa- ∇χ † χ χ† ∇ tion that relativity was necessary to explain the electron’s + ( · σ) + (σ · ψ) , spin. The claim that relativity is required to explain the spin current is similarly unfounded. We believe that allows us to replace the right-hand side of the last equa- Shikakhwa, et al. [8] have shown it to be derivable from tion, yielding the non-relativistic Schr¨odinger-Pauli equation (15), al- 1 ∂ρ though just as with the spin, relativity proves to be in- = ∇· ψ†σχ + χ†σψ − (∇χ · σ)† χ − χ†(σ · ∇ψ) . valuable in confirming its existence and uniqueness. c ∂t From equation (17) and its adjoint, however, we have
III. DERIVATION OF THE SPIN CURRENT ~ ~ i † i † DENSITY FROM THE SCHRODINGER-PAULI¨ χ = σ · ∇ψ , χ = − (σ · ∇ψ) , 2mc 2mc EQUATION FOR A FREE SPIN-1/2 PARTICLE which, when substituted in the previous equation elimi- The authors of Ref. 8 derived the non-relativistic cur- nates the last two terms, bringing it to simply rent density for a free spin-1/2 particle beginning with 1 ∂ρ the minimally coupled Schr¨odinger-Pauli equation, and = ∇ · (ψ†σχ + χ†σψ) . (18) later set A = 0 in the result. We show here that the cur- c ∂t rent density can be derived directly from the free-particle † Schr¨odinger-Pauli equation (5). From this equation we Substituting for χ and χ in (18) (thereby eliminating have the dimensional factor of c) yields
∂ψ i 2 ∂ρ i~ = − (σ · pˆ) ψ , = ∇ · ψ†σ(σ · ∇ψ) − (∇ψ · σ)†σψ , ∂t 2m~ ∂t 2m = −∇· J , (19) and its Hermitian conjugate equation: allowing identification of the components of the proba- ∂ψ† i † = (σ · pˆ)2 ψ . bility current density J: ∂t 2m~ h i ~ † i † † Taking ρ = ψ ψ as the probability density, we find for Ji = (∇j ψ) σj σiψ − ψ σiσj (∇j ψ) , 2m its time derivative summation over repeated indices being understood. Us- ∂ρ ∂ψ ∂ψ† = ψ† + ψ , ing the Pauli matrix identity σiσj = δij + iǫijkσk, the ∂t ∂t ∂t current density components can be written as † i † 2 2 = − ψ (σ · pˆ) ψ − (σ · pˆ) ψ ψ . (16) ~ 2m~ i † † Ji = (∇j ψ) (δij − iǫijkσk) ψ − ψ (δij + iǫijkσk) ∇j ψ , h i h i 2m At this point, our derivation departs from the one given i~ = (∇ ψ)†)ψ − ψ†(∇ ψ) − iǫ ∇ (ψ†σ ψ) , in Ref. 8, where our momentum operator pˆ in Eq. (16) 2m i i ijk j k was replaced by the kinetic momentum πˆ in their corre- sponding Eq. (8b). We instead retain pˆ and introduce an or in vector form: auxiliary spinor χ defined by i~ J = − ψ†(∇ψ) − (∇ψ)†ψ 1 2m χ = − (σ · pˆ) ψ , (17) ~ 2mc + ∇ × (ψ†σψ) , (20) 2m the speed of light c included only to ensure dimensional consistency (in no way to be construed as introducing a the correct current density for an external interaction- relativistic concept into the theory). Eq. (16) can then free spin-1/2 particle. 5
IV. THE LEVY-LEBLOND´ EQUATION Following L´evy-Leblond [25], we seek a wave equation linear in both the energy and momenta operators: Nearly 40 years after Pauli and Dirac’s seminal work, Aˆ in a paper [25] “devoted to a detailed study of non- θˆΨ(r,t) ≡ Eˆ + Bˆipˆj δij + mc Cˆ Ψ(r,t)=0 , (24) relativistic particles and their properties, as described by c ! Galilei invariant wave equations,” J.-M. L´evy-Leblond, inspired by Dirac’s heuristic derivation [13] of his rel- where Aˆ, Cˆ, and Bˆi (i =1, 2, 3) are dimensionless linear ativistic wave equation for the electron, obtained from operators to be determined. They are assumed to de- the free-particle Schr¨odinger equation a non-relativistic pend on neither the spatial coordinates nor time, hence analog now referred to as the L´evy-Leblond equation. commute with Eˆ and eachp ˆi, but not necessarily with From it the Pauli equation follows as an immediate con- each other. The factors 1/c of Aˆ, and mc of Bˆ, where c sequence after calling upon the minimal coupling replace- is an arbitrary constant speed, are introduced to insure ment. In addition, the probability current computed the correct units of linear momentum in each term. It from the L´evy-Leblond equation includes the correct spin will later be identified with the speed of light for com- current contribution. patibility with the non-relativistic reduction of the Dirac A derivation of the L´evy-Leblond equation eventu- equation. From Eq. (24) we obtain, by operating on both ally involves the introduction of a 4-component spinor sides with θˆ, (bispinor), and 4 × 4 matrices satisfying the Dirac alge- 2 bra, which may be considered at too high a level to be ˆ 2 A introduced in an early course on quantum mechanics. As θˆ Ψ(r,t) ≡ Eˆ + Bˆipˆj δij + mc Cˆ Ψ(r,t)=0 . c we show, however, these matrices are easily constructed ! from the Pauli matrices, so if the Pauli theory has been (25) introduced, there is really no great hurdle to be overcome in using them. Even so, if the decision is made to avoid In doing so we have departed from L´evy-Leblond’s [25] their use, we should mention that having made plausible and other derivations (e.g., Ref. 2, Ch. 13, and Refs. [26, ˆ the Pauli equation by, e.g., the Sakurai argument lead- 27]) by omitting primed counterparts of the operators A, ing to Eq. (6), a coupled set of equations linear in the Bˆi, and Cˆ. Conditions are now set on these operators momentum operator, i.e., first order in the spatial deriva- by requiring that Eqs. (23) and (25) agree. Squaring the tives, can be obtained by simply introducing, similar to linear operator, we obtain what was done in the last section, an auxiliary spinor χ Aˆ2 1 defined by Eˆ2 + (AˆBˆ + Bˆ Aˆ)Eˆpˆ + m(AˆCˆ + CˆAˆ)Eˆ c2 c i i i 1 1 χ = − [σ · (pˆ − qA)] ψ , (21) + (BˆiBˆj + Bˆj Bˆi)ˆpipˆj + mc(BˆiCˆ + CˆBˆi)ˆpi 2mc 2 2 2 2 + m c Cˆ =p ˆipˆj δij − 2mE,ˆ It then follows from Eq. (6) that where the fourth term on the left-hand side was obtained −c [σ · (pˆ − qA)] χ + qφ ψ = Eˆ ψ. (22) by interchanging summation indices to rewrite it as a symmetrized sum. This polynomial equation in Eˆ and ˆ ˆ ˆ This pair of equations is precisely the system derived by thep ˆi is satisfied for all A, C, and Bi if and only if each L´evy-Leblond in Ref. 25, that is, the L´evy-Leblond equa- of its coefficients is zero, yielding follow tions from the Pauli equation. L´evy-Leblond’s dis- ˆ2 ˆ ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ tinguished contribution, on the other hand, was to A = 0 , C = 0 , AC + CA = −2 I, provide a derivation of these equations from the non- AˆBˆi + BˆiAˆ = 0ˆ , CˆBˆi + BˆiCˆ = 0ˆ , (26) relativistic Schr¨odinger equation, from which Pauli’s Bˆ Bˆ + Bˆ Bˆ =2δ I,ˆ (i, j =1, 2, 3) . equation then follows. A derivation based on his insights i j j i ij is offered in what follows. where Iˆ is the identity operator. Along with the operators Bˆi, whose anticommutation relations are satisfied by the three Pauli spin matrix rep- A. Linearizing the Schr¨odinger Equation resentations Bi = σi (the “hat” will be omitted in denot- ing matrix representations of operators), two additional ˆ ˆ We begin with the Schr¨odinger equation (3) for a free operators A and C have been introduced. We see in particle of mass m, multiplying both sides by 2m to write Eqs. (26) that the matrix representations of both opera- it in the more convenient form tors must anticommute with each of the Bi matrix rep- resentations, but unlike the Pauli matrices, they neither pˆ pˆ δ − 2mEˆ Ψ(r,t)=0. (23) anticommute with each other nor does each square to i j ij the identity matrix (both are in fact singular). Noticing 6 that anticommutation with the Bi is maintained when so we have the representations A and C are each replaced by any 2 linear combination of the two, we seek such linear com- B5 = (B1B2B3B4)(B1B2B3B4), binations, B and B , having the same anticommutation 2 4 5 = −B2B3B4B1 B2B3B4, relations as the Pauli matrices in order to take advantage = −B B B B B B , of their well documented properties. It is not difficult to 2 3 4 2 3 4 2 demonstrate that they must have the general forms, up = −B2B3B2B3B4 , to a choice of sign on i, = −B2B3B2B3, 2 2 = B2 B3 = I. 1 A C i A C B4 = + , B5 = − − , (27) 2 a b 2 a b By similar manipulations, one finds that B5Bp = −BpB5 for each p taken separately. provided the nonzero dimensionless constants a and b We emphasize that our matrices Bµ, µ = 1,..., 5 satisfy are not the same as those introduced by either L´evy-Leblond in Ref. 25, Greiner in Ref. 2 (Ch. 13), Hladik [26], or Du Toit [27]. In all four references their 2ab = −1 . (28) matrices are intimately tied to primed counterparts that we have not introduced (and, incidentally, are different in That is, when A, C, and the Bi satisfy Eqs. (26), then each reference), and do not satisfy the anticommutation Eqs. (27) and (28) insure that the five matrices Bµ, µ = relations (29) of the Dirac algebra. 1,..., 5, satisfy the anticommutation relations
B B + B B =2δ I, µ,ν =1,..., 5 . (29) B. Constructing Representations of the Dirac µ ν ν µ µν Algebra
Our situation is reminiscent of Dirac’s derivation of The first four of our five matrices could be chosen to his relativistic wave equation, where he found that the be Dirac’s original matrices, i.e., Bp = αp, p = 1,..., 4. three 2×2 Pauli matrices were not sufficient to satisfy his They can be constructed as Kronecker products of Pauli anticommutation relations. It was necessary to generalize matrices [28–30], where the Kronecker product of two the three Pauli matrices to four 4 × 4 matrices αp, p = 2 × 2 matrices M and N is a 4 × 4 matrix defined by 1,..., 4, for which m m m N m N M ⊗ N = 11 12 ⊗ N = 11 12 , m21 m22 m21N m22N αpαq + αqαp =2δpq I, p,q =1,... 4, (30) m11n11 m11n12 m12n11 m12n12 where I is the 4 × 4 identity matrix (a demonstration of m n m n m n m n = 11 21 11 22 12 21 12 22 . (32) the necessity for 4 × 4 matrices is given in, for example, m21n11 m21n12 m22n11 m22n12 Ref. 20, p. 8). We see now, in light of Eq. (29), that m21n21 m21n22 m22n21 m22n22 our proposed linearization of the Schr¨odinger equation can be completed by finding five complex 4 × 4 matrices Recalling the Pauli matrices in their standard represen- satisfying Dirac’s algebra. The defining equations (27) tation, are then easily inverted to get expressions for the original 0 1 0 −i 1 0 matrices A and C of Eq. (24), viz., σ = , σ = , σ = , (33) 1 1 0 2 i 0 3 0 −1 A = a(B4 + iB5) , C = b(B4 − iB5) . (31) Dirac’s original matrices are the following Kronecker products (i =1, 2, 3): Fortunately, if four initial matrices satisfying Eq. (30) 0 1 0 σi are known, then the matrix B5 = B1B2B3B4 is the de- αi = σ1 ⊗ σi = ⊗ σi = , (34a) 1 0 σi 0 sired fifth matrix, i.e., a calculation using the properties 2 1 0 of the Bp matrices, p =1,..., 4, shows that B5 = I, and 1 0 β = α4 = σ3 ⊗ 1 = ⊗ 1 = , (34b) B5Bp = −BpB5 for each p taken separately. For example, 0 −1 0 −1 the demonstration that B2 = I can be accomplished in 5 0 1 three steps: (1) focusing on the first factor of B5 in the where is the 2 × 2 zero matrix, and is the 2 × 2 square, use the anticommutation relations to move B1 identity matrix introduced earlier. A fifth matrix of the 2 three places to the right, and replace B1 = I, (2) move Dirac algebra is then given by 2 B4 two places to the right, and replace B4 = I, and (3) 2 2 0 −i1 commute B3 with B2, replacing B2 = B3 = I in the re- B = α α α α = , (35) 5 1 2 3 4 i1 0 sult. Each commutation changes the sign on the product, 7 using the Pauli spin matrix identity σ1σ2σ3 = i1. One (Ref. 2, Eqs. (13.28)), and Hladik’s (Ref. 26), versions easily verifies that any pair of the five matrices Bi = αi, of the L´evy-Leblond equations correspond to the choice i =1, 2, 3, B4 = α4, and B5 satisfies Eqs. (29). a = i/2, hence b = i, while Du Toit’s equations (Ref. A more convenient set of matrices, however, is obtained 27) correspond to a = 1/2, hence b = −1. In all that from the following Kronecker products (i =1, 2, 3): follows, we adhere to L´evy-Leblond’s original equations (providing the missing factors of c), hence 1 0 σi 0 Bi = σ3 ⊗ σi = ⊗ σi = , (36a) 1χ 0 0 −1 0 −σi σ · pˆψ +2mc = , (41a) 0 1 0 1 1Eˆ ψ + cσ · pˆχ = 0 , (41b) B = σ ⊗ 1 = ⊗ 1 = , (36b) 4 1 1 0 1 0 in agreement with the non-relativistic limit of the Dirac in which case the fifth matrix is equations if c is identified with the speed of light. If the particle has a charge q and is moving in an exter- 0 i1 nal electromagnetic field derived from a vector potential B = B B B B = . (36c) 5 1 2 3 4 −i1 0 A and scalar potential φ, then Eqs. (41) are modified to include the interaction using the minimal coupling pre- These matrices also satisfy Eqs. (29), and with their scription, as was done in Eq. (6): choice the matrices A and C of Eqs. (31) take the simple forms σ · (pˆ − qA) ψ +2mc1 χ = 0 , (42a) 0 0 0 1 cσ · (pˆ − qA) χ + (Eˆ − qφ)1 ψ = 0 . (42b) A =2a , C =2b . (37) 1 0 0 0 χ Eliminating between the two equations by multiplying Eq. (42b) by 2m, then using Eq. (42a) to replace 2mcχ in the result, we obtain a wave equation for the spinor ψ: C. The Levy-Leblond and Pauli Equations 2 σ · (pˆ − qA) ψ − 2m(Eˆ − qφ)1 ψ = 0 , (43) In the representations of Eqs. (36) and (37) the lin- earized Schr¨odinger equation (24) takes the matrix form the factors of c obligingly falling out (showing once again that special relativity is not involved). Divid- A ing both sides by 2m, we see that this is precisely the Eˆ + B pˆ δ + mc C Ψ c i j ij minimally coupled Schr¨odinger-Pauli equation, Eq. (6), from which we have shown that Pauli’s equation (1) fol- 2a 0 0 σi 0 0 1 = Eˆ + pˆi +2bmc Ψ , lows directly. Thus, as emphasized by L´evy-Leblond in c 1 0 0 −σi 0 0 Ref. 25, the linearized non-relativistic (Galilean covari- 1 c σ · pˆ 2bmc21 ant) Schr¨odinger equation together with minimal cou- = Ψ= 0 , (38) pling predicts the correct value for the intrinsic magnetic c 2a1Eˆ − c σ · pˆ moment of a spin-1/2 particle: spin is not an intrinsically where we have omitted the position and time argu- relativistic phenomena. ments of Ψ for convenience, and note that Ψ must now be a bispinor, i.e., a 4-element column matrix of com- plex functions (0 is the zero bispinor). Eq. (38) is the V. DERIVATION OF THE SPIN CURRENT DENSITY FROM THE LEVY-LEBLOND´ L´evy-Leblond equation for a free particle of mass m. Var- EQUATION ious forms of it can be found in the literature, depending on the choices made for a and b subject to condition (28). We write the bispinor Ψ as It seems not well known, although L´evy-Leblond pub- lished it over 40 years ago, that his equation predicts ψ unambiguously the form of the spin current showing Ψ= χ , (39) that, like the intrinsic spin itself, it is an inherently non- relativistic phenomena. where ψ and χ are two-component spinors (sometimes re- To demonstrate this we replace the energy operator in ferred to as semispinors), to obtain the component equa- the components, Eqs. (42), of the L´evy-Leblond equa- tions tion, yielding 1 ∂ψ ic i σ · pˆψ +2bmc χ = 0 = σ · (pˆ − qA) χ − qφ1ψ , (44a) ∂t ~ ~ 1 ˆ 0 −2a E ψ + cσ · pˆχ = . 1 χ = − σ · (pˆ − qA) ψ , (44b) When we choose a = −1/2, hence b = 1, these reduce 2mc to L´evy-Leblond’s Eqs. (25) of Ref. 25, up to missing obvious generalizations of the free particle equations in- factors of c required for correct SI units. Both Greiner’s troduced as an ansatz in section III. In this form we see 8 that χ plays an auxiliary role, as it is given in terms Substituting this result in Eq. (47) then gives of the component ψ through the constraint (44b). On 1 ∂ρ the other hand, Eq. (44a) is a true dynamical equation = ∇· ψ†σχ + χ†σψ − (σ · ∇ψ)†χ describing the time evolution of the state Ψ via its inde- c ∂t q pendent component ψ. Defining the probability density + χ† (σ · ∇ψ) + ψ† (σ · A)(σ · ∇ψ) again by ρ = ψ†ψ, we have 2mc n + (σ · A)(σ · ∇ψ)† ψ . (48) ∂ρ ∂ψ ∂ψ† = ψ† + ψ . (45) ∂t ∂t ∂t o We next focus attention on the second term, in brackets, The adjoint of (44a) is of Eq. (48). Substituting for χ and χ† in this term yields
† ∂ψ ic † i σ ∇ †χ χ† σ ∇ = − σ · (pˆ − qA) χ + qφ ψ†1 , (46) ( · ψ) + ( · ψ) ∂t ~ ~ 1 = − (σ · ∇ψ)† [σ · (pˆ − qA) ψ] hence from Eqs. (44a), (46), and (45): 2mc n † 1 ∂ρ i † χ χ † + [σ · (pˆ − qA) ψ] (σ · ∇ψ) = ~ ψ σ · (pˆ − qA) − σ · (pˆ − qA) ψ , c ∂t o n iq o i~ † = ψ† (σ · ∇χ) + (σ · ∇χ)†ψ − ψ† (σ · A) χ = (σ · ∇ψ)† (σ · ∇ ψ) − (σ · ∇ ψ) (σ · ∇ψ) ~ 2mc n † nq o − (σ · A) χ ψ , + (σ · ∇ψ)† (σ · A ψ) 2mc o n † the terms involving qφ cancelling out. The first two terms + (σ · A ψ) (σ · ∇ψ) can be rewritten in terms of a divergence, to obtain o The first two terms on the right cancel, so we can sub- 1 ∂ρ = ∇· ψ†σχ + χ†σψ − (σ · ∇ψ)†χ − χ†(σ · ∇ψ) stitute the last two in Eq. (48) to obtain an expression c ∂t with two additional terms in braces: iq † χ χ † − ~ ψ (σ · A) − (σ · A) ψ . (47) 1 ∂ρ q = ∇· ψ†σχ + χ†σψ + ψ† (σ · A)(σ · ∇ψ) n o c ∂t 2mc From Eq. (44b) and its adjoint, however, we have + (σ · A)(σ · ∇ ψ)† ψ −n (σ· ∇ψ)†(σ · Aψ) 1 χ = − σ · (pˆ − qA) ψ , † ∇ 2mc + (σ · Aψ) (σ · ψ) , † 1 † o χ = − σ · (pˆ − qA) ψ , 2mc and again, since σ · A is Hermitian, the terms involving A vanish, leaving simply which, when substituted in the last two terms of equation (47) involving A yield 1 ∂ρ = ∇· ψ†σχ + χ†σψ , c ∂t iq † − ψ† (σ · A) χ − (σ · A) χ ψ ~ hence the current density can be identified as n o iq † † = ψ† (σ · A) σ · (pˆ − qA) ψ J = −c (ψ σχ + χ σψ) . (49) 2mc~ n † † − (σ · A) σ · (pˆ − qA) ψ ψ Substituting once more for χ and χ , we obtain from Eq. (49): o q † ∇ ∇ † = ψ (σ · A)(σ · ψ) + (σ · A)(σ · ψ) ψ 1 † 2mc J = ψ†σ σ · (pˆ − qA)ψ + σ · (pˆ − qA)ψ σ ψ n2 o 2m iq † + ψ (σ · A)(σ · Aψ) i~n o 2mc~ = − ψ†σ(σ · ∇ψ) − (σ · ∇ψ)†σψ n 2m − (σ · A)(σ · A ψ)† ψ . q − ψ†σ(σ · Aψ) + (σ · Aψ)†σψ. 2m o Here and in what follows we will make frequent use of the fact that σ · A is Hermitian, in which case the last From the derivation leading to Eqs (19) and (20), the two terms vanish, leaving first two terms in braces can be replaced to yield
iq † i~ ~ − ψ† (σ · A) χ − (σ · A) χ ψ J = − ψ†∇ψ − (∇ψ)†ψ + ∇ × ψ†σψ ~ 2m 2m n q o q † † = ψ† (σ · A)(σ · ∇ψ) + (σ · A)(σ · ∇ψ)† ψ − ψ σ(σ · Aψ)+ ψ (σ · A)σψ . 2mc 2m n o 9
The ith component of the terms involving A can be writ- In recent work [34, 35] directed toward the prediction of ten, making use of the Pauli matrix identity σiσj +σj σi = arrival times in time-of-flight experiments, the possibility 2δij , very simply as of observing rather remarkable effects of the spin term is discussed. In these papers, time-of-flight experiments are † † † ψ σi(σj Aj ψ)+ ψ (σj Aj )σiψ = ψ (σiσj + σj σi)Aj ψ proposed and carefully analyzed for electrons prepared in =2ψ†δ A ψ =2A ψ†ψ, various initial states, moving in a cylindrical waveguide. ij j i Analytical and numerical methods are used to obtain pre- which, when substituted in the previous equation, yields dictions of their arrival time distributions at a detector. the correct non-relativistic current density for a spin- For a particular initial state, with spin vector oriented 1/2 particle interacting with an external electromagnetic perpendicular to the waveguide axis, the predicted dis- field: tributions show a striking result, a cutoff, or maximum arrival time, after which no further electrons would be de- i~ ~ J = − ψ†∇ψ − (∇ψ)†ψ + ∇ × ψ†σψ tected. Such an experiment, or one similar to it, seems to 2m 2m be well within the scope of present day technology. The q − A ψ†ψ, (50) results would be a welcome test of the ability of Bohmian m mechanics to predict arrival times of spin-1/2 particles, given that a standard quantum mechanical prediction of verifying Eqs. (10) and (11). the same is still ambiguous [36]. Finally, as further examples of the possible impact of VI. DOES THE SPIN CURRENT HAVE the spin term, we mention recent publications [37, 38] MEASURABLE CONSEQUENCES? applying the Dirac theory to experiments with relativistic electron beams. In these, a velocity field of probability flow is defined by precisely the same equation used in Although the spin current has no effect in the prob- Bohmian mechanics to describe particle trajectories, i.e., ability conservation equation, it has been convincingly proportional to the current density, and used to describe demonstrated to be an inherent feature of the probabil- results of such experiments (or suggested ones). As in ity current density of a spin-1/2 particle described by the non-relativistic case, the spin term makes a definite the Schr¨odinger-Pauli equation. If the particle carries a contribution that cannot be ignored. charge q, then it should contribute to the charge current density, hence be measurable in experiments sensitive to the effects of such currents. After all, paraphrasing Nowakowski in Ref. 6, electric current is a physical ob- VII. CONCLUDING REMARKS servable. In Ref. 6 an interesting experiment is proposed and an- After reviewing previous work, our goals here were alyzed to demonstrate the relevance of the spin current. two-fold. First, to emphasize that although the Pauli It is determined that in such an experiment there would equation can be constructed from the Schr¨odinger equa- be a contribution to the electric current density “which tion by judicious use of spin operator identities, thus is of purely quantum mechanical origin and which can be demonstrating the non-relativistic nature of the elec- traced back to the ‘spin term’ ... .” The author further tron spin, L´evy-Leblond pointed the way to a di- speculates that “this term might play a role in problems rect derivation of the Pauli equation via a lineariza- concerning conductivity in solids.” tion of the Schr¨odinger equation, from which the non- Du Toit, in his Honors Thesis [27], discusses a sim- relativistic nature of the electron spin naturally follows. pler experiment, an electron in a homogeneous magnetic Second, to direct attention to the work of Shikakhwa, field (the Landau problem). His analysis is rather in- et al. [8], who showed conclusively and unambiguously conclusive, the main effect being described as a “swirl” that the spin term of the probability current density of charge current creating a current loop/dipole moment is a non-relativistic phenomenon derivable from the that could possibly interact with the field to produce a Schr¨odinger-Pauli equation, contrary to what is some- torque on the dipole, presumably observable as some sort times claimed in the literature [6]. Furthermore, we em- of precessional motion. phasized the importance of the L´evy-Leblond theory in A field of physics where the current density plays a this regard, showing that the spin term is directly deriv- critical role is Bohmian mechanics (or de Broglie-Bohm able from his equation, which is properly viewed as a theory; see Refs. 31–33 for excellent general accounts). precursor of the Pauli equation. Well-defined concepts of point particles and trajectories We conclude by cautioning readers about the casual are fundamental features of this theory, where particle use of the ubiquitous minimal coupling prescription in motion takes place along trajectories that are integral this and other work, invoking electromagnetic poten- curves of a velocity field proportional to the current den- tials that are inherently relativistic (since they originate sity (Ref. 33, Ch. 10). Thus, a particle’s equations of in the Lorentz covariant Maxwell equations), and us- motion are directly determined by the current density it- ing them in a non-relativistic quantum theory involv- self, hence the spin term can directly influence its motion. ing the Schr¨odinger, Pauli, or L´evy-Leblond equations. 10
Constraints on the allowable electromagnetic fields as a ACKNOWLEDGMENTS result of requiring Galilean covariance of both the quan- tum theory and electromagnetism were first discussed by My thanks to Siddhant Das for his encouragement in L´evy-Leblond in Ref. 25, p. 305. His results have been pursuing a better understanding of the Pauli equation sharpened and developed considerably in later work [39– and its spin current density, and for critically reviewing 43], but a discussion of these important articles is beyond the manuscript. I am also grateful to Detlef D¨urr for his the scope of the present paper. A review of this work is interest in this work, and for his collegial grace in mak- highly recommended to anyone applying non-relativistic ing me feel welcome as a long-distance collaborator on quantum theories to the analysis of experimental results research efforts at the Mathematisches Institut, Ludwig- involving electromagnetic interactions. Maximilians-Universitat M¨unchen.
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