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The Thomas Precession Factor in Spin–Orbit Interaction

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The Thomas Precession Factor in Spin–Orbit Interaction The Thomas precession factor in spin–orbit interaction Herbert Kroemera) Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106 ͑Received 6 December 2002; accepted 8 August 2003͒ The origin of the Thomas factor 1/2 in the spin–orbit Hamiltonian can be understood by considering the case of a classical electron moving in crossed electric and magnetic fields chosen such that the electric Coulomb force is balanced by the magnetic Lorentz force. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1615526͔ I. INTRODUCTION: THE PROBLEM istic QM course, the instructor is likely compelled to use an unsatisfactory ‘‘it can be shown that’’ argument, which is, in It is well known that the spin–orbit Hamiltonian differs by fact, the approach taken in most textbooks. a factor 1/2—the Thomas precession factor—from what one In my own 1994 textbook,4 I tried to go beyond that ap- might expect from a naive Lorentz transformation that as- proach by considering the case where the electron is forced sumes a uniform straight-line motion of the electron. The to move along a straight line, by adding a magnetic field in naive argument runs as follows. the rest frame such that the magnetic Lorentz force would When an electron moves with a velocity v through space exactly balance the electric force. In this case, the Thomas in the presence of an electric field E, the electron will see, in factor of 1/2 was indeed obtained, but the argument— its own frame of reference, a Lorentz-transformed magnetic especially its extension to more general cases—lacked rigor. field BЈ given by the familiar expression Moreover, because it was published outside the mainstream physics literature, it has remained largely unknown to poten- Ã ͒ 2 Ã ͒ ͑E v /c ͑E v tially interested readers inside that mainstream. The purpose BЈϭ ! , ͑1͒ 2 2 ͱ1Ϫ͑␯/c͒ ␯Ӷc c of the present note is to put the argument on a more rigorous basis, and to do so in a more readily accessible manner. where c is the speed of light, and the limit ␯Ӷc has been taken. It is this Lorentz-transformed magnetic field that is supposedly seen by the electron magnetic moment. This ar- gument suggests that in the presence of electric fields, the II. CROSSED-FIELD TREATMENT magnetic field B in the spin Hamiltonian should be replaced by Bϩ(EÃvˆ)/c2, where vˆ is the velocity operator of the From our perspective, the central aspect of the standard ͑ ͒ electron. This conclusion does not agree with observed Lorentz transform 1 is the following: The transform ⇒ atomic spectra. Erest Belectron must be linear in E, and E can occur only in Historically, this discrepancy provided a major puzzle,1 the combination EÃv. This combination reflects the fact that until it was pointed out by Thomas2 that this argument over- only the component of E perpendicular to the velocity v can looks a second relativistic effect that is less widely known, play a role, and that the resulting B field must be perpendicu- but is of the same order of magnitude: An electric field with lar to both E and v. a component perpendicular to the electron velocity causes an In the absence of any specific arguments to the contrary, additional acceleration of the electron perpendicular to its we would expect that this proportionality to EÃv carriers instantaneous velocity, leading to a curved electron trajec- over to the case of a curved trajectory. ͑For example, a com- tory. In essence, the electron moves in a rotating frame of ponent of E parallel to v would not contribute to a curved reference, implying an additional precession of the electron, trajectory and to the accompanying rotation of the electron called the Thomas precession. A detailed treatment is given, frame of reference.͒ If one accepts this argument, it follows 3 ϵ for example, by Jackson, where it is shown that this effect that BЈ Belectron should be given by an expression of the changes the interaction of the moving electron spin with the general form electric field, and that the net result is that the spin–orbit ͑EÃv͒ interaction is cut in half, as if the magnetic field seen by the BЈϭ␣ , ͑3͒ electron has only one-half the value in Eq. ͑1͒, c2 1 ͑EÃv͒ with some scalar proportionality factor ␣, which may still BЈϭ . ͑2͒ 2 c2 depend on the velocity, but which cannot depend on either E or on any magnetic field B in the rest frame. If there also is It is this modified result that is in agreement with experi- a magnetic field B present in the rest frame, Eq. ͑3͒ should ment. be generalized to A rigorous derivation3 of this ͑classical͒ result requires knowledge of some aspects of relativistic kinematics, which, ͑EÃv͒ BЈϭ␣ ϩ␤B, ͑4͒ although not difficult, are unfamiliar to most students. In a c2 course on relativistic quantum mechanics, the result ͑2͒ can be derived directly from the Dirac equation, without refer- where ␤ is another constant subject to the same constraints ence to classical relativistic kinematics. But in a nonrelativ- as ␣. 51 Am. J. Phys. 72 ͑1͒, January 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 51 To determine the constants ␣ and ␤, we consider not sim- which is of the required form ͑4͒, with ␣ϭ1/2 and ␤ϭ1. ply the Lorentz transformation of a pure electric field E, but Equation ͑9͒ is the magnetic field that is seen by the in- of a specific combination of electric and magnetic fields such trinsic magnetic moment of the electron. It determines the that potential energy of that moment in the presence of both elec- EϭϪvÃB, ͑5a͒ tric and magnetic fields for our specific combination of crossed fields, up to terms linear in EÃv. The leading B term where v is the velocity of the electron, and B is chosen already is included in the nonrelativistic spin Hamiltonian; perpendicular to v. For this combination, the electric Cou- the remainder is the first-order correction due to the spin– lomb force and the magnetic Lorentz force on the electron orbit interaction. Note that it contains the Thomas factor 1/2, cancel, and the electron will move along a straight line. But in agreement with Eq. ͑2͒. in this case the simple Lorentz transformation for uniform The rest is straightforward. Our argument suggests that the straight-line motion is rigorously applicable, without having magnetic field B in the spin Hamiltonian should be replaced to worry about a rotating frame of reference. by an operator that is equivalent to Eq. ͑9b͒. Without loss of generality, we may choose a Cartesian Following standard textbook arguments, we obtain the fa- coordinate system such that the velocity is in the x direction, miliar spin–orbit contribution to the Hamiltonian, the magnetic field is in the z direction; and the electric field is in the y direction, ␮ e ϭ␯ ͑ ͒ Hˆ ϭ␮ ␴ˆ Bˆ Јϭ ␴ˆ ͑EÃvˆ͒, ͑10͒ Ey xBz . 5b SO e • 2c2 • If this combination of the two fields is Lorentz-transformed into the uniformly moving frame of the electron, the mag- where ␮ is the intrinsic magnetic moment of the electron, Ј e netic field Bz in that frame is and vˆ is now the operator for the velocity. B ϪE ␯ /c2 Јϭ z y x ͑ ͒ Bz . 6 ͱ1Ϫ͑␯ /c͒2 x III. DISCUSSION Inasmuch as we are interested only in the limit ␯Ӷc,we ͑ ͒ ␯ may expand the right-hand side of Eq. 6 in powers of x . The central assumption in our derivation was that the pro- The result may be written as portionality of BЈ to EÃv carries over to a rotating frame of reference. This proportionality is, of course, an exact result. ␯2 ␯ 2 ␯ ␯ 2 Bz x 1 3 x Ey x 1 x BЈϭB ϩ b ϩϩ ͩ ͪ cϪ b1ϩ ͩ ͪ c But in a paper explicitly dedicated to the classroom didactics z z c2 • 2 8 c c2 • 2 c of the Thomas factor without invoking less well-known as- pects of relativistic kinematics, it is probably best to treat it ϩ ͑ ͒ ¯ , 7 as an eminently plausible assumption. ␯2 where the dots represent omitted terms of order higher than I close with a comment on our retention of the Bz x term ␯4 ͑ ͒ ␯2ϭ ␯ ͑ ͒ ␯ x . But under the condition 5b , we have Bz x Ey x , and in Eq. 7 , albeit converted into a Ey x term. This term re- Eq. ͑7͒ may be simplified by combining terms to read places rotating-frame corrections in the case of a pure elec- tric field. Neglecting it would be exactly equivalent to ne- ␯ ␯ 2 Ey x 1 1 x BЈϭB Ϫ b ϩ ͩ ͪ cϩ . ͑8͒ glecting the effects of a rotating frame of reference for a z z c2 • 2 8 c ¯ more general choice of fields. ␯Ӷ In the limit c, we obtain a͒Electronic mail: [email protected] 1 1 E ␯ See, for example, M. Jammer, The Conceptual Development of Quantum Јϭ Ϫ y x ͑ ͒ Mechanics ͑McGraw-Hill, New York, 1966͒, p. 152. B Bz . 9a z 2 c2 2L. H. Thomas, ‘‘The motion of the spinning electron,’’ Nature ͑London͒ 117, 514 ͑1926͒. In three-dimensional vector form, 3J. D. Jackson, Classical Electrodynamics ͑Wiley, New York, 1962͒, Sec. Ã 11.5. 1 E v 4 BЈϭBϩ , ͑9b͒ H. Kroemer, Quantum Mechanics ͑Prentice–Hall, Englewood Cliffs, 2 c2 1994͒, Sec. 12.4.1. 52 Am. J.
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