Table Of ContentNonexistence 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,
