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Nonexistence of ”Spin Transverse Force” for a Relativistic Electron ∗ Wlodek Zawadzki Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland UsingthecompleteDiracHamiltonianforarelativisticelectronitisshownthatthespintransverse force derived by S.Q. Shen, Phys. Rev. Lett. 95, 187203 (2005) does not exist. This force is an 7 artefact resulting from an approximate form of theemployed Hamiltonian. 0 0 PACSnumbers: 85.75.-d,71.10.Ca,72.20.My 2 n a InarecentpaperShen[1]consideredaforceactingona The force is rarely written this way, but the above for- J relativisticelectroninthepresenceofanexternalelectric mula is correct since p−eA = m(v)v, see for example 6 potential and a magnetic field. Using an approximate Refs. 4,5. We have 1 Hamiltonian, which contained the spin-orbit interaction i] aexnedrttsheaDtararwnsivnetresremfo,ritcewoanssahnowenlecthtraotnthsepienlewcthriecnfitehlde ddpt = ~i[H,p]=−∇V +ec∇(α·A) , (4) c electron is moving. The effects resulting from this force s - in semiconductors were also considered. rl In the following we critically reexamine this problem dA = δA + i[H,A]= δA +c(α·∇)A , (5) t dt δt ~ δt m and show that the spin transverse force does not exist. To begin with, the Hamiltonian used in Ref. 1 is incom- where ∇ is the differentiationvectoracting only on V or at. plete. As is well known, in the (v/c)2 approximation to A, not on the wave function. Hence m the Dirac equation there appear three terms in addition to the Pauli equation: the spin-orbit term, the Darwin dA d- term and the (p−eA)4 term, in which p is the momen- F=e(−∇φ− dt )+ec[∇(α·A)−(α·∇)A] . (6) n tum and A is the vector potential of a magnetic field. o It was recently shown that the last contribution should Since ∇(α·A)−(∇·α)A) = α×(∇×A), we finally c be completed by a spin term, see Ref. 2. The last two obtain [ 2 terms, which are also of the (v/c) order, were omitted F=eE+ev×B , (7) 1 in [1]. Second, the author used for the force the formula v F=m0dv/dt , where m0 is the rest electron mass. This 8 where we have used the above formulas for v and B and is incorrectsince in the relativistic range the correctfor- 7 the standard relation for an electric field E = −∇φ− mula for the force is F=dm(v)v/dt , where m(v) is the 3 δA/δt. . velocity-dependentLorentzmass. However,the essential 1 Thus the classical result for the Lorentz force is ob- 0 point is that, in order to calculate the force, the author tained, apart from the fact that v is an operator. In 7 used an approximation to the Dirac Hamiltonian, while particular, once the force is expressed by the velocity 0 an exact solution to this problem exists and gives a dif- / ferent result, see Strange [3]. Below we reproduce this (as it is in Ref. 1), the effects of the spin-orbit interac- t a derivation with some comments. tion do not appear. It should be mentioned that in the m non-relativistic quantum mechanics one should use the The complete DiracHamiltonianwith the electric and symmetrized product (v×B−B×v)/2 since the ve- - magnetic potentials is, in the standard notation, d locity v=p/m0 does not commute with B(r), see Refs. on H =cα·(p−eA)+βm0c2+V , (1) 6,7. Inthe relativisticcaseconsideredabovethe velocity (2) commutes with B(r) and the symmetrization is not c : where the potential energy is V = eφ(r), the magnetic needed. v field is B = ∇×A(r), and αi are the Dirac matrices. Clearly, the complete Dirac equation contains the ef- i X The velocity is fectsofthespin-orbitinteraction. Forexample,theexact energiesobtainedforthe hydrogenatomexhibitthe spin r a dr i splittingswhichcanbeinterpretedintermsofthis inter- v= = [H,r]=cα . (2) dt ~ action. But these are the energies, not the force. Thus the spin transverse force claimed in Ref. 1 is The force is due toanapproximatecharacterofthe employedHamil- tonian. If one could expand the Dirac Hamiltonian to d F= (p−eA) . (3) high powers of v/c terms, the result for the force would dt asymptotically reduce to Eq. (7). Such an expansion is easy to write for the presence of a magnetic field alone, as outlined in Ref. 2. It is muchmore complicatedwhen ∗Electronicaddress: [email protected] both potentials are present, see [8]. However,there is no 2 need to consider such expansions, the force given by Eq. gradientoftheperiodicandexternalpotentials. Inother (7) is simple, elegant and exact. words,alsointhiscasethespintransverseforcedoesnot Ref. 1 also discusses possible effects related to the appear. spin transverse force in semiconductors. In the standard To summarize, we demonstrated that the spin trans- band theory of such materials one usually employs the verse force claimed in Ref. 1 for a relativistic electron Schroedinger Hamiltonian with a periodic potential plus does not exist. It is an artefact resulting from an ap- the spin-orbit term. But again, in principle one should proximate character of the employed Hamiltonian. usethecompleteDiracHamiltonianandthenthereason- ing presentedaboveholds. Thus one obtains the forcein Acknowledgements: I acknowledge helpful discussions the Lorentz form (7) with the electric field given as a with Dr. P. Pfeffer and Dr. T.M. Rusin. [1] S.Q. Shen,Phys. Rev.Lett. 95, 187203 (2005). New York,1960). [2] W. Zawadzki, Am.J. Phys. 73, 756 (2005). [6] L.I.Schiff,QuantumMechanics (McGraw-Hill,NewYork, [3] P. Strange, Relativistic Quantum Mechanics (Cambridge 1955). Univ.Press, 1998), p.126. [7] D.I. Blokhintsev, Fundamentals of Quantum Mechanics, [4] L.D. Landau and E.M. Lifshitz, Field Theory (Fizmatlit, (VyzshayaShkola, Moscow, 1961). In Russian. Moscow, 2003). InRussian. [8] E. de Vries, Fortsch. Phys. 18, 149 (1970). [5] H.C. Corben and Ph.Stehle,Classical Mechanics (Wiley,