Schrödinger Pauli equation for spin-3/2 particles

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1 INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 50 3) JUNIO 004 Schrödinger Pauli euation for spin-3/ particles G.F. Torres del Castillo Departamento de Física Matemática, Instituto de Ciencias Universidad Autónoma de Puebla, 7570 Puebla, Pue., México J. Velázuez Castro Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Puebla, Apartado Postal 115, 7001 Puebla, Pue., México Recibido el 4 de septiembre de 003; aceptado el 15 de enero de 004 A non-relativistic euation for spin-3/ particles is proposed and the gyromagnetic ratio for charged spin-3/ particles is determined. Keywords: Spin-3/ particles; gyromagnetic ratio. Se propone una ecuación no relativista para partículas de espín 3/ y se determina la razón giromagnética para partículas de espín 3/ cargadas. Descriptores: Partículas de espín 3/; razón giromagnética. PACS: w 1. Introduction In the uantum-mechanical description of particles, there are various, relativistic or non-relativistic wave euations whose form depends on the spin of the particles. The usual Schrödinger euation applies to the spin-0 particles in the non-relativistic domain, while the Klein Gordon euation is the relativistic euation appropriate for spin-0 particles. The spin-1/ particles are governed by the relativistic Dirac euation which, in the non-relativistic limit, leads to the Schrödinger Pauli euation see, e.g., Refs. 1 3). In the case of particles with spin 1 or higher, only relativistic euations are usually considered see, e.g., Ref. 4). A charged particle with non-zero spin couples to an external magnetic field as if, in addition to its electric charge, it had a magnetic dipole moment. In the case of a spin-1/ charged particle, the relation between the magnitudes of the magnetic dipole moment and of the intrinsic angular momentum given by the Dirac or the Schrödinger Pauli euation does not coincide with that of a uniformly charged rotating body given by classical physics, but somewhat surprisingly it does coincide with that of a rotating charged black hole in the Einstein Maxwell theory see, e.g., Refs. 5, 6). In this paper we propose a non-relativistic wave euation for spin-3/ particles directly by analogy with the Schrödinger Pauli euation, to obtain the gyromagnetic ratio of a charged spin-3/ particle. We find that the relation between the greatest eigenvalue of the magnetic dipole moment, and the charge to mass ratio has a common value for spin-3/ and spin-1/ particles. In the relativistic case, there exist several acceptable wave euations for spin-3/ fields see, e.g., Ref. 7 and the references cited therein), but we are not studying their non-relativistic limits. In Sec. the Schrödinger Pauli euation for spin-1/ particles is written making use of the Pauli matrices and of the two-component spinor notation which is employed in Sec. 3 to write the proposed euation for spin-3/ particles. The notation and conventions used throughout this paper are summarized in Sec. ; further details can be found in Refs. 8, 9.. Spin-1/ particles The usual Schrödinger euation for a spin-0 particle of mass M in a potential V r), M ψ + V r)ψ = i ψ, 1) can be obtained from the classical Hamiltonian H = p /M + V, using the fact that the momentum operator in the coordinate representation is given by i. In the case of a spin-1/ particle, the wave function is not a complex-valued function but a two-component spinor ψr, t) = ψ 1 r, t) ψ r, t) ), ) which under a rotation through an angle α about the axis defined by a unit vector n transforms into see, e.g., Refs. 10, 9) ψ = cos 1 α I i sin 1 α n σ) ψ, 3) where I is the identity matrix and σ = σ 1, σ, σ 3 ) is formed by the Pauli matrices ) 0 1 σ 1 =, 1 0 ) 0 i σ =, i 0 ) 1 0 σ 3 =. 4) 0 1 Thus, for an infinitesimal rotation, ψ ψ iα 1 n σ ψ,

2 SCHRÖDINGER PAULI EQUATION FOR SPIN-3/ PARTICLES 307 which means that n S = 1/) n σ is the operator corresponding to the component of the spin angular momentum operator along n. The Pauli matrices satisfy σ i σ j = δ ij I + iε ijk σ k, 5) where ε ijk is totally antisymmetric with ε 13 = 1, i, j,... = 1,, 3, and there is summation over repeatedindices. Since the entries of the Pauli matrices are constant, making use of E. 5), ) σ ) 0 = σ i σ j = σ i σ j = x i x j x i x j 0, where the x i are Cartesian coordinates; therefore, the euation M σ ) ψ + V r)ψ = i ψ for the two-component spinor ) implies that each component of ψ ψ 1 and ψ ) satisfies the Schrödinger euation 1), and conversely. However, when there is a magnetic field present this euivalence disappears and E. 6) leads to a coupling between the two components of the spinor ψ. The standard procedure to take into account the interaction of a particle of electric charge with an electromagnetic field consists in replacing the partial derivatives /x i and / by /x i ia i /) and / + iφ/, respectively, where A = A 1, A, A 3 ), and φ are potentials of the electromagnetic field. In this manner, for a spin-0 charged particle, from E. 1) one obtains see, e.g., Ref. 10) ψ i i A ψ A) ψ M ) A ψ] +V r)ψ+φψ = i ψ 7) and, similarly, making use of Es. 5), E. 6) yields ψ i i ) A ψ M A)ψ A ψ + ] B σψ +V r)ψ+φψ = i ψ. 8) When B = 0, E. 8) reduces to two independent euations of the form 7), one for each component of the spinor ψ. However, when B 0, the components of ψ are coupled through the term 1 B Mc σψ. Recalling that the energy of a magnetic dipole moment µ in a magnetic field B is eual to µ B, it follows that a charged spin-1/ particle obeying E. 8) behaves as if it had a magnetic dipole moment represented by the operator 6) µ = 1 Mc σ = S. 9) Mc By contrast with the electric charge, which is a c-number, the magnetic dipole moment associated with the particle is an operator.) Euation 9) shows that the ratio of the magnetic dipole moment to the intrinsic angular momentum is eual to Mc. 10) Before considering an analog of E. 6) applicable to spin-3/ particles, it will be convenient to write E. 6) making use of the two-component spinor notation that will be employed in the treatment of spin-3/ particles see also Refs. 8 and 9). The entries of the Pauli matrices 4) will be denoted by σ i A B A, B,... = 1, ), so that σ i A B stands for the entry in the A-th row and B-th column of the matrix σ i. The spinor indices, such as those of the spinor ), and of the Pauli matrices, will be lowered or raised following the convention where φ A = ε AB φ B, φ A = φ B ε BA, 11) ε AB ) ) ε AB ). 1) Thus, φ 1 = φ, φ = φ 1.) Hence, ε A B = δ A B and φ A ψ A = ε AB φ B ψ A = φ B ε BA ψ A = φ B ψ B = φ A ψ A. 13) Any tensor with Cartesian components t ij k has a spinor euivalent defined by t ABCD MN 1 σ i 1 AB σ j CD 1 σ k MN t ij k, 14) where, following the conventions stated above, σ iab = ε AC σ i C B. Since we are considering here Cartesian coordinates only, the tensor indices are lowered or raised by means of the metric tensor δ ij and its inverse δ ij ; hence, σ iab = σ i AB.) An explicit computation shows that σ iab = σ iba. 15) Furthermore, since the Pauli matrices have a vanishing trace, from E. 5) we obtain tr σ i σ j ) = δ ij, i.e., σ i A B σ j B A = δ ij or, euivalently see E. 13)] σ i AB σ jab = δ ij. 16) Hence, from Es. 14) and 16) we find that, if t AB and s AB are the spinor euivalents of t i and s i, respectively t AB s AB = t i s i. 17) Rev. Mex. Fís. 50 3) 004)

3 308 G.F. TORRES DEL CASTILLO AND J. VELÁZQUEZ CASTRO According to the definition 14), we shall write AB = 1 σ i AB i, 18) where i /x i. Thus, the Schrödinger Pauli euation 6) can be expressed as M A C C Bψ B + V r)ψ A = i ψa. 19) We can see that E. 19) is euivalent to two decoupled Schrödinger euations using the fact that if φ AB = φ BA, then φ AB = 1 φr Rε AB. 0) Indeed, any antisymmetric matrix must be proportional to ε AB ) see E. 1)] and, as can be readily seen, φ AB = φ 1 ε AB = 1 φ 1 φ 1 ) ε AB = 1 φ + φ 1 1) ε AB = 1 φr R ε AB. Owing to E. 13), and the fact that AB = BA see Es. 15) and 18)] AC C B= A C CB = CB A C = BC C A, hence see Es. 0), 13), and 17)], AC C B = 1 ε AB R C C R = 1 ε AB RC RC which is euivalent to so that, in effect, E. 19) amounts to = 1 ε AB, A C C B = 1 δa B, 1) M ψ A + V r)ψ A = i ψa. When there is an electromagnetic field present, we replace A B by A B i/)a A B and / by / + i/ )φ in E. 19), and we obtain A C i ) M AA C C B i ) AC B ψ B which is euivalent to M +V r)ψ A + φψ A = i ψa, A C C Bψ B i A CA C B)ψ B i AC B A Cψ B i AA C C Bψ B ] A A C A ) C Bψ B +V r)ψ A +φψ A = i ψa. ) In order to reduce this last expression we begin by noticing that see E. 0)] AC A C B = 1 ACA C B + BC A C A) + 1 ACA C B BC A C A) = A C A C B) + ε AB R CA C R, where the parenthesis denotes symmetrization on the indices enclosed e.g., M AB) = 1/)M AB + M BA )), and the indices between bars are excluded from the symmetrization. The first term in the right-hand side of the last euality is the spinor euivalent of i/ ) A, which follows from the fact that the spinor euivalent of the Levi-Civita symbol ε ijk is ε ABCDEG = i/ )ε AC ε BE ε DG +ε BD ε AG ε CE ) 8,9], while the last term is eual to 1/)ε AB A see Es. 13) and 17)]. Making use of Es. 13) and 0) we find that A C B AC + A AC C B = A C B AC A C A BC = ε BA A CR RC = ε AB A. Finally, by analogy with E. 1), A A CA C B = 1/)δB AA. Thus, E. ) can be also be written as ψ A + M BA Bψ B i A)ψA i ) ] A ψa A ψ A + V r)ψ A +φ ψ A = i ψa, 3) where B AB denotes the spinor euivalent of B, and one can verify that this expression coincides with E. 8). 3. Spin-3/ particles A spin-3/ particle is described by a totally symmetric threeindex spinor field, ψ ABC see E. 5) below], which under rotations transforms according to ψ ABC = U A RU B SU C T ψ RST, where U A B) is the SU) matrix appearing in E. 3), namely U A B = cos 1 α δa B i sin 1 α na B 4) and n AB is the spinor euivalent of the unit vector n. Hence, for an infinitesimal rotation, ψ ABC δr A iα ) n A R δs B iα ) n B S δt C iα ) n C T ψ RST ψ ABC 3i α n A Rψ BC)R, Rev. Mex. Fís. 50 3) 004)

4 SCHRÖDINGER PAULI EQUATION FOR SPIN-3/ PARTICLES 309 which implies that the operator n S given by n S ψ) ABC = 3 n A Rψ BC)R 5) corresponds to the component of the spin along n. By analogy with the Schrödinger Pauli euation 19), for a spin-3/ particle we propose the euation M A R R Sψ S BC) + V r)ψ ABC = i ψabc 6) which, owing to E. 1), means that, in the absence of an electromagnetic field, each component of ψ ABC satisfies the Schrödinger euation 1). It should be noticed that, even in the absence of an electromagnetic field, the components of the spinor field ψ ABC may be coupled among themselves if a non-cartesian basis is employed, in which case the partial derivatives appearing in E. 6) have to be replaced by covariant derivatives 8, 9].) Instead of the symmetrized second derivatives A R R Sψ S BC) appearing in E. 6), one could also consider A Rχ R BC), with χ RBC R Sψ S BC) ; however, in the latter case, each component ψ ABC would not satisfy the Schrödinger euation. By combining E. 6) and its complex conjugate one obtains the continuity euation ψ ABC ψ ABC ) + im RSψ SBC AR ψabc + ψ SBC AR ψ ABC ) = 0, where ψ ABC ψ ABC 9]; hence, ψabc ψ ABC is real and positive. As in the case of the relativistic description of spin-3/ particles, instead of three-index spinors, one can employ wave functions with one spinor index and one tensor index 4]. In the present case, we can define ψ ia 1 σ i BC ψ ABC, then the symmetry of ψ ABC in its three indices is expressed by the condition σ iab ψ ia = 0. When there is an electromagnetic field present, we replace A B by A B i/)a A B and / by / + i/ )φ, in E. 6), which yields A R i ) M AA R R S i ) A R S ψ S BC) +V r)ψ ABC +φψ ABC = i ψabc. 7) Following the same steps as in E. ) we obtain ψ ABC + M BA Sψ BC)S i A)ψABC i A ψabc ] A ) ψ ABC + V r)ψ ABC +φ ψ ABC = i ψabc. 8) Thus, taking into account E. 5), one concludes that in the present case there is an interaction with the magnetic field of the form µ B, with µ = 3Mc S 9) cf. E. 9)] that corresponds to the gyromagnetic ratio 3Mc. 30) Owing to the difference between the gyromagnetic ratios 10) and 30), in both cases, the greatest eigenvalue of the operator µ is given by µ max = Mc. 31) Euations 19) and 6) can be readily generalized for any value of the spin. A spin-s particle would be represented by a totally symmetric s-index spinor field, ψ AB L, satisfying M A R R Sψ S B L) + V r)ψ AB L = i ψab L 3) and if the particle has electric charge, the gyromagnetic ratio would be given by smc. As pointed out above, under the rotation corresponding to the SU) matrix U A B), the Cartesian components of a spinor ψ AB...L transform according to ψ AB...L = U A P U B Q U L Rψ P Q...R. Since each term in E. 3) transforms in the same manner as ψ AB...L, the validity of E. 3) in a given Cartesian frame implies its validity in any Cartesian frame obtained from the original one by means of a rotation. This can be seen as a conseuence of the fact that a contraction of the form A Rψ RB...L transforms as a s-index spinor since U RM U R N = detu A B) ε NM = ε NM Rev. Mex. Fís. 50 3) 004)

5 310 G.F. TORRES DEL CASTILLO AND J. VELÁZQUEZ CASTRO see E. 0)] and therefore A Rψ RB...L = U A P U RM P M ) U R NU B Q U L Sψ NQ...S = U RM U R N U A P U B Q U L S P M ψ NQ...S = ε NM U A P U B Q U L S P M ψ NQ...S = U A P U B Q U L S P N ψ NQ...S. On the other hand, under the Galilean transformation r = r vt, t = t, where v is constant, using the chain rule, one finds that x i = x i, then, assuming that ψ AB...L = ψ AB...L exp = vi x i + i mv r ) + i mv t, a straightforward computation, making use of E. 1), shows that E. 3) is form-invariant under Galilean transformations. 4. Concluding remarks The preceding euations show that a genuine spin-3/ charged particle would behave in a different way than an assembly of three charged spin-1/ particles in a spin-3/ state, since in the latter case one would have a gyromagnetic ratio eual to that of a single spin-1/ particle, which differs from 30). The origin and physical significance of the coincidence of the gyromagnetic ratios of a spin-1/ charged particle, and of a rotating charged black hole, mentioned in the Introduction are not evident; and the fact that the gyromagnetic ratio derived from E. 3) depends on the spin of the particle suggests that this coincidence is not a straightforward conseuence of some basic principle. 1. J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics McGraw-Hill, New York, 1964).. A.S. Davydov, Quantum Mechanics, nd ed., Pergamon, Oxford, 1965) A. Messiah, Quantum Mechanics, Vol. II, Wiley, New York, 1968). 4. V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Quantum Electrodynamics, nd ed., Pergamon, Oxford, 1989). 5. G.C. Debney, R.P. Kerr, and A. Schild, J. Math. Phys ) E.T. Newman, Phys. Rev. D 65 00) G. Silva-Ortigoza, Gen. Rel. Grav ) G.F. Torres del Castillo, Rev. Mex. Fís ) G.F. Torres del Castillo, 3-D Spinors, Spin-Weighted Functions and their Applications, Birkhäuser, Boston, 003). 10. J.J. Sakurai, Modern Quantum Mechanics Addison-Wesley, Reading, Mass., 1994). Rev. Mex. Fís. 50 3) 004)