What is the relativistic spin operator?
WWhat is the relativistic spin operator?
Heiko Bauke, ∗ Sven Ahrens, Christoph H. Keitel, and Rainer Grobe
1, 2 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Intense Laser Physics Theory Unit and Department of Physics,Illinois State University, Normal, IL 61790-4560 USA(Dated: April 14, 2014)
Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spinoperator within the framework of relativistic quantum mechanics. We investigate the properties of di ff erentproposals for a relativistic spin operator. It is shown that most candidates are lacking essential features of properangular momentum operators, leading to spurious Zitterbewegung (quivering motion) or violating the angularmomentum algebra. Only the Foldy-Wouthuysen operator and the Pryce operator qualify as proper relativisticspin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identifya legitimate relativistic spin operator experimentally.Keywords: spin, relativistic quantum mechanics, hydrogen-like ions PACS numbers: 03.65.Pm, 31.15.aj, 31.30.J-
Introduction
Quantum mechanics forms the universallyaccepted theory for the description of physical processes onthe atomic scale. It has been validated by countless experi-ments and it is used in many technical applications. However,quantum mechanics presents physicists with some conceptualdi ffi culties even today. In particular, the concept of spin isrelated to such di ffi culties and myths [1, 2]. Although there isconsensus that elementary particles have some quantum me-chanical property that is called spin, the understanding of thephysical nature of the spin is still incomplete [3].Historically, the concept of spin was introduced in orderto explain some experimental findings such as the emissionspectra of alkali metals and the Stern-Gerlach experiment. Adirect measuring of the spin (or more precisely the electron’smagnetic moment), however, was missing until the pioneeringwork by Dehmelt [4]. Nevertheless, spin measurement exper-iments [5–10] still require sophisticated methods. Pauli andBohr even claimed that the spin of free electrons was impossi-ble to measure for fundamental reasons [11]. Recent renewedinterest in fundamental aspects of the spin arose, for example,from the growing field of (relativistic) quantum information[12–17], quantum spintronics [18], spin e ff ects in graphene[19–21] and in light-matter interaction at relativistic intensities[22–24].According to the formalism of quantum mechanics, eachmeasurable quantity is represented by a Hermitian operator.Taking the experiments that aim to measure bare electron spinsseriously, we have to ask the question: What is the correct(relativistic) spin operator. Although the spin is regarded as afundamental property of the electron, a universally acceptedspin operator for the Dirac theory is still missing. The pivotalquestion we try to tackle is: Which mathematical operator cor-responds to an experimental spin measurement? This questionmay be answered by comparing experimental results with theo-retical predictions originating from di ff erent spin operators andtesting which operator is compatible with experimental data. A relativistic spin operator may be introduced by splittingthe undisputed total angular momentum operator ˆ J into anexternal part ˆ L and an internal part ˆ S , commonly referred to asthe orbital angular momentum and the spin, viz. ˆ J = ˆ L + ˆ S . Thequestion for the right splitting of the total angular momentuminto an orbital part and a spin part is closely related to thequest for the right relativistic position operator [25–27]. Thisbecomes evident by writing ˆ L = ˆ r × ˆ p with the position operatorˆ r and the kinematic momentum operator ˆ p , which is in atomicunits as used in this paper ˆ p = − i ∇ . Thus, di ff erent definitionsof the spin operator ˆ S induce di ff erent relativistic positionoperators ˆ r .Introducing the position vector r and the operator ˆ Σ = ( ˆ Σ , ˆ Σ , ˆ Σ ) T via ˆ Σ i = − i α j α k (1)with ( i , j , k ) being a cyclic permutation of (1 , ,
3) and thematrices ( α , α , α ) T = α obeying the algebra α i = , α i α k + α k α i = δ i , k , (2)the operator of the relativistic total angular momentum is givenby ˆ J = r × ˆ p + ˆ Σ /
2. Thus, the most obvious way of splitting ˆ J is to define the orbital angular momentum operator ˆ L P = r × ˆ p and the spin operator ˆ S P = ˆ Σ /
2, which is a direct generaliza-tion of the orbital angular momentum operator and the spinoperator of the nonrelativistic Pauli theory. This naive splitting,however, su ff ers from several problems, e. g., ˆ L P and ˆ S P do notcommute with the free Dirac Hamiltonian nor with the DiracHamiltonian for central potentials. Thus, in contrast to classi-cal and nonrelativistic quantum theory the angular momentaˆ L P and ˆ S P are not conserved. This has consequences, e. g., forthe labeling of the eigenstates of the hydrogen atom. In nonrel-ativistic theory, bound hydrogen states may be constructed assimultaneous eigenstates of the Pauli-Coulomb Hamiltonian,the squared orbital angular momentum, the z -components ofthe orbital angular momentum and the spin. In the Dirac theory, a r X i v : . [ qu a n t - ph ] A p r however, the squared total angular momentum ˆ J , the total an-gular momentum in z -direction ˆ J , and the so-called spin-orbitoperator ˆ K (or the parity) are utilized [28, 29]. In particular,it is not possible to construct simultaneous eigenstates of theDirac-Coulomb Hamiltonian and some component of ˆ S P . Relativistic spin operators
To overcome conceptualproblems with the naive splitting of ˆ J into ˆ L P and ˆ S P , severalalternatives for a relativistic spin operator have been proposed.However, there is no single commonly accepted relativisticspin operator, leading to the unsatisfactory situation that therelativistic spin operator is not unambiguously defined. Wewill investigate the properties of di ff erent popular definitionsof the spin operator which result from di ff erent splittings of ˆ J with the aim to find means that allow to identify the legitimaterelativistic spin operator by experimental methods.Table 1 summarizes various proposals for a relativistic spinoperator ˆ S . These operators are often motivated by abstractgroup theoretical considerations rather than by experimentalevidence. For example, Wigner showed in his seminal work[54–56] that the spin degree of freedom can be associated withirreducible representations of the sub-group of the inhomoge-neous Lorentz group that leaves the four-momentum invariant.We will denote individual components of ˆ S by ˆ S i with index i ∈ { , , } . The spin operators are defined in terms of theparticle’s rest mass m , the speed of light c , the matrix β suchthat β = , α i β + βα i = , (3)the free particle Dirac Hamiltonianˆ H = c α · ˆ p + m c β , (4)and the operator ˆ p = ( m c + ˆ p ) / . (5)In the nonrelativistic limit, i. e., when the plane wave expansionof a wave packet has only components with momenta which aresmall compared to m c , expectation values for all operators inTab. 1 converge to the same value. Note that the nomenclaturein Tab. 1 is not universally adopted in the literature and otherauthors may utilize di ff erent operator names. Furthermore,the spin operators can be formulated by various di ff erent butalgebraically equivalent expressions. For example, the so-called Gürsey-Ryder operator in [46, 47] is equivalent to theChakrabarti operator of Tab. 1.One may conclude that an operator can not be considered asa relativistic spin operator if it does not inherit the key proper-ties of the nonrelativistic Pauli spin operator. In particular, wedemand from a proper relativistic spin operator the followingfeatures:1. It is required to commute with the free Dirac Hamiltonian.2. A spin operator must feature the two eigenvalues ± / S i , ˆ S j ] = i ε i , j , k ˆ S k (6) with ε i , j , k denoting the Levi-Civita symbol.The first property is required to ensure that the relativistic spinoperator is a constant of motion if forces are absent, such thatspurious Zitterbewegung of the spin is prevented. The secondrequirement is commonly regarded as the fundamental propertyof angular momentum operators of spin-half particles [57]. Thephysical quantity that is represented by the operator ˆ S shouldnot depend on the orientation of the chosen coordinate system.This can be ensured by fulfilling [57][ ˆ J i , ˆ S j ] = i ε i , j , k ˆ S k . (7)The angular momentum algebra (6) and the relation (7) de-termine the properties of the spin and the orbital angular mo-mentum as well as the relationship between them. As a con-sequence of (7), the orbital angular momentum ˆ L = ˆ J − ˆ S that is induced by a particular choice of the spin obeys[ ˆ J i , ˆ L j ] = i ε i , j , k ˆ L k . Thus, ˆ L is a physical vector operator, too.As ˆ L represents an angular momentum operator, it must obeythe angular momentum algebra. Furthermore, we may say thatthe total angular momentum ˆ J is split into an internal part ˆ S and an external part ˆ L only if internal and external angularmomenta can be measured independently, i. e., ˆ S and ˆ L com-mute. Both conditions are fulfilled if, and only if, the spinoperator ˆ S satisfies the angular momentum algebra (6) becausethe commutator relations[ ˆ L i , ˆ L j ] = i ε i , j , k ˆ L k + [ ˆ S i , ˆ S j ] − i ε i , j , k ˆ S k , (8)[ ˆ L i , ˆ S j ] = i ε i , j , k ˆ S k − [ ˆ S i , ˆ S j ] (9)follow from (7). All spin operators in Tab. 1 fulfill (7). TheCzachor spin operator ˆ S Cz , the Frenkel spin operator ˆ S F , andthe Fradkin-Good operator ˆ S FG , however, are disqualified asrelativistic spin operators by violating the angular momentumalgebra (6). Furthermore, the Pauli spin operator ˆ S P and theChakrabarti spin operator ˆ S Ch do not commute with the freeDirac Hamiltonian, ruling them out as meaningful relativis-tic spin operators. According to our criteria, only the Foldy-Wouthuysen spin operator ˆ S FW and the Pryce spin operator ˆ S Pr remain as possible relativistic spin operators. Electron spin of hydrogen-like ions
The question ofwhich of the proposed relativistic spin operators (if any) inTab. 1 provides the correct mathematical description of spincan be answered definitely only by comparing theoretical pre-dictions with experimental results. For this purpose one needsa physical setup that shows strong relativistic e ff ects and isas simple as possible. Such a setup is provided by the boundeigenstates of highly charged hydrogen-like ions, i. e., atomicsystems with an atomic core of Z protons and a single elec-tronic charge. These ions can be produced at storage rings [58]or by utilizing electron beam ion traps [59, 60] up to Z = H C = ˆ H − Z | r | (10) TABLE 1:
Definitions and commutation properties of various relativistic spin operators.[ ˆ H , ˆ S ] [ ˆ S i , ˆ S j ] EigenvaluesOperator name Definition = = i ε i , j , k ˆ S k ? = ± / S P =
12 ˆ Σ no yes yesFoldy-Wouthuysen [33–37] ˆ S FW =
12 ˆ Σ + i β p ˆ p × α − ˆ p × ( ˆ Σ × ˆ p )2 ˆ p ( ˆ p + m c ) yes yes yesCzachor [38] ˆ S Cz = m c p ˆ Σ + i m c β p ˆ p × α + ˆ p · ˆ Σ p ˆ p yes no noFrenkel [39–41] ˆ S F =
12 ˆ Σ + i β m c ˆ p × α yes no noChakrabarti [42–48] ˆ S Ch =
12 ˆ Σ + i2 m c α × ˆ p + ˆ p × ( ˆ Σ × ˆ p )2 m c ( m c + ˆ p ) no yes yesPryce [47, 49–52] ˆ S Pr = β ˆ Σ +
12 ˆ Σ · ˆ p (1 − β ) ˆ p ˆ p yes yes yesFradkin-Good [46, 53] ˆ S FG = β ˆ Σ +
12 ˆ Σ · ˆ p (cid:32) ˆ H c ˆ p − β (cid:33) ˆ p ˆ p yes no yes are commonly expressed as simultaneous eigenstates ψ n , j , m ,κ of ˆ H C , ˆ J , ˆ J , and the so-called spin-orbit operator ˆ K = β { ˆ Σ · [ r × ( − i ∇ ) + } fulfilling the eigenequations [28, 29]ˆ H C ψ n , j , m ,κ = E ( n , j ) ψ n , j , m ,κ n = , , . . . , (11)ˆ J ψ n , j , m ,κ = j ( j + ψ n , j , m ,κ j = , , . . . , n − , (12)ˆ J ψ n , j , m ,κ = m ψ n , j , m ,κ m = − j , ( j − , . . . , j , (13)ˆ K ψ n , j , m ,κ = κψ n , j , m ,κ κ = − j − , j + . (14)The eigenenergies are given with α el denoting the fine structureconstant by E ( n , j ) = m c + α Z n − j − / + (cid:113) ( j − / − α Z − / . (15)In order to establish a close correspondence between thenonrelativistic Schrödinger-Pauli theory and the relativisticDirac theory, one may desire to find a splitting of ˆ J into a sumˆ J = ˆ L + ˆ S of commuting operators such that both ˆ L and ˆ S
1. fulfill the angular momentum algebra, and2. form a complete set of commuting operators that containsˆ H C as well as ˆ S and / or ˆ L .The latter property would ensure that all hydrogenic energyeigenstates are spin eigenstates and / or orbital angular momen-tum eigenstates, too. Such hypothetical eigenstates would besuperpositions of ψ n , j , m ,κ of the same energy. Consequently,these superpositions are eigenstates of ˆ J , too, because theenergy (15) depends on the principal quantum number n aswell as the quantum number j . Thus, any complete set of com-muting operators for specifying hydrogenic quantum statesnecessarily includes ˆ J . As a consequence of the postulatedangular momentum algebra for ˆ L and ˆ S , the operator ˆ J com-mutes with ˆ L as well as with ˆ S , but with neither ˆ S nor ˆ L [61] excluding ˆ S and ˆ L from any complete set of commut-ing operators for specifying relativistic hydrogenic eigenstates.In conclusion, hydrogenic energy eigenstates are generallynot eigenstates of any spin operator that fulfills the angularmomentum algebra.In momentum space, the relativistic spin operators intro-duced in Tab. 1 are simple matrices. Thus, by employ-ing the momentum space representation of ψ n , j , m ,κ , spin ex-pectation values of the degenerate hydrogenic ground states ψ ↑ = ψ , / , / , and ψ ↓ = ψ , / , − / , − can be evaluated [62].For simplicity, we measure spin along the z -direction for the re-minder of this section. The spin expectation value of a generalsuperposition ψ = cos( η/ ψ ↑ + sin( η/ i ζ ψ ↓ of the hydro-genic ground states ψ ↑ and ψ ↓ is given by (cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ S (cid:12)(cid:12)(cid:12) ψ (cid:69) = cos η (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) + sin η (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) + η η ζ Re (cid:104) ψ ↑ | ˆ S | ψ ↓ (cid:105) . (16)For all spin operators introduced in Tab. 1, the mixingterm Re (cid:104) ψ ↑ | ˆ S | ψ ↓ (cid:105) vanishes and, furthermore, (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) = − (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) >
0. Thus, the expectation value (16) is maximalfor η = η = π and the inequality (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) ≤ (cid:104) ψ | ˆ S | ψ (cid:105) ≤ (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) (17)holds for all hydrogenic ground states ψ .For every proposed spin operator in Tab. 1 we get di ff erentvalues for the upper and lower bounds in (17). The spin ex-pectation values (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) and implicitly (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) for theoperators of Tab. 1 are displayed as a function of the atomicnumber Z in Fig. 1. None of the spin operators in Tab. 1 com-mutes with ˆ H C . Thus, the expectation values (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) and (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) generally do not equal one of the eigenvalues of ˆ S .For small atomic numbers ( Z < ± /
2. For larger Z , however, expectation values di ff ersignificantly from each other. In particular, spin expectation Z . . . . . . . h ψ ↑ | ˆ S | ψ ↑ i ˆ S P , ˆ S FW , ˆ S Cz , ˆ S F , ˆ S Ch , ˆ S Pr , ˆ S FG , FIG. 1:
Spin expectation values (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) of various relativistic spinoperators for the hydrogenic ground state ψ ↑ as a function of theatomic number Z measured in z -direction. Spin expectation valuesfor ψ ↓ follow by symmetry via (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) = − (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) . values di ff er from ± / ± /
2. This means, it is possible to discriminate betweendi ff erent relativistic spin operator candidates. The spin expec-tation value’s magnitude decreases with growing Z when thePauli, the Fouldy-Wouthuysen, the Czachor, the Chakrabarti,or the Fradkin-Good spin operator is applied. The Frenkel spinoperator yields spin expectation values with modulus exceed-ing 1 / ± / Z . In fact, calculations show that allhydrogenic states ψ n , j , m ,κ with m = ± j are eigenstates of thePryce spin operator but not those states with m (cid:44) ± j . An experimental test for relativistic spin operators
Theoretical considerations led to several proposals for a rela-tivistic spin operator as illustrated in Tab. 1. The identificationof the correct relativistic spin operator, however, demands anexperimental test. The inequality (17) may serve as a basis forsuch an experimental test. More precisely, the inequality (17)allows falsification of the hypothesis that the spin measure-ment procedure is an experimental realization of some operatorˆ S , where ˆ S is one of the operators in Tab. 1. In this test theelectron of a highly charged hydrogen-like ion is prepared inits ground state ψ ↑ first, e. g., by exposing the ion to a strongmagnetic field in z -direction and turning it o ff adiabatically.(Preparing a superposition of ψ ↑ and ψ ↓ will reduce the sen-sitivity of the experimental test.) Afterwards, the spin willbe measured along z -direction, e. g., by a Stern-Gerlach-likeexperiment, yielding the experimental expection value s . Com-paring this experimental value to each of the seven boundsshown in Fig. 1 will allow exclusion of some of the proposedspin operators. The hypothesis that the spin measurementprocedure is an experimental realization of the operator ˆ S iscompatible with the experimental result s if, and only if, theinequality (cid:104) ψ ↓ | ˆ S | ψ ↓ (cid:105) ≤ s ≤ (cid:104) ψ ↑ | ˆ S | ψ ↑ (cid:105) is fulfilled. Otherwise,this operator is excluded as a relativistic spin operator by exper- imental evidence. In particular, realizing full spin-polarization,i. e., s = ± /
2, eliminates all operators in Tab. 1 except thePryce operator.
Conclusions
We investigated the properties of variousproposals for a relativistic spin operator. Only the Fouldy-Wouthuysen operator and the Pryce operator fulfill the angularmomentum algebra and are constants of motion in the absenceof forces. While di ff erent theoretical considerations led to dif-ferent spin operators, the definite relativistic spin operator hasto be justified by experimental evidence. Energy eigenstatesof highly charged hydrogen-like ions, in particular the groundstates, can be utilized to exclude candidates for a relativisticspin operator experimentally. The proposed spin operatorspredict di ff erent maximal degrees of spin polarization. Onlythe Pryce spin operator allows for a complete polarization ofspin in the hydrogenic ground state.We have enjoyed helpful discussions with Prof. C. Müller,S. Meuren, Prof. Q. Su and E. Yakaboylu. R. G. acknowledgesthe warm hospitality during his sabbatical leave in Heidelberg.This work was supported by the NSF. ∗ [email protected][1] H. Nikoli´c, Foundations of Physics , 1563 (2007).[2] D. Giulini, Studies In History and Philosophy of Science Part B:Studies In History and Philosophy of Modern Physics , 557(2008).[3] M. Morrison, Studies In History and Philosophy of Science PartB: Studies In History and Philosophy of Modern Physics , 529(2007).[4] H. Dehmelt, Science , 539 (1990).[5] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev. Lett. ,120801 (2008).[6] P. Neumann, J. Beck, M. Steiner, F. Rempp, H. Fedder, P. R.Hemmer, J. Wrachtrup, and F. Jelezko, Science , 542 (2010).[7] B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom,Science , 1212 (2010).[8] T. Close, F. Fadugba, S. C. Benjamin, J. Fitzsimons, and B. W.Lovett, Phys. Rev. Lett. , 167204 (2011).[9] S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman,W. Quint, G. Werth, C. H. Keitel, and K. Blaum, Phys. Rev. Lett. , 023002 (2011).[10] J. DiSciacca and G. Gabrielse, Phys. Rev. Lett. , 153001(2012).[11] N. F. Mott, Proc. R. Soc. Lond. A , 425 (1929).[12] A. Peres and D. R. Terno, Rev. Mod. Phys. , 93 (2004).[13] W. T. Kim and E. J. Son, Phys. Rev. A , 014102 (2005).[14] R. Wiesendanger, Rev. Mod. Phys. , 1495 (2009).[15] N. Friis, R. A. Bertlmann, M. Huber, and B. C. Hiesmayr, Phys.Rev. A , 042114 (2010).[16] K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M.Taylor, A. A. Houck, and J. R. Petta, Nature , 380 (2012).[17] P. L. Saldanha and V. Vedral, New J. Phys. , 023041 (2012).[18] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu, andJ. R. Petta, Science , 1174 (2013).[19] D. A. Abanin, R. V. Gorbachev, K. S. Novoselov, A. K. Geim,and L. S. Levitov, Phys. Rev. Lett. , 096601 (2011).[20] M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. , 116803(2011). [21] J. Güttinger, T. Frey, C. Stampfer, T. Ihn, and K. Ensslin, Phys.Rev. Lett. , 116801 (2010).[22] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel,Rev. Mod. Phys. , 1177 (2012).[23] S. Ahrens, H. Bauke, C. H. Keitel, and C. Müller, Phys. Rev.Lett. , 043601 (2012).[24] S. Ahrens, H. Bauke, C. H. Keitel, and R. Grobe, “Electron-spindynamics induced by photon spins,” (2014), arXiv:1401.5976.[25] T. D. Newton and E. P. Wigner, Rev. Mod. Phys. , 400 (1949).[26] T. F. Jordan and N. Mukunda, Phys. Rev. , 1842 (1963).[27] R. O’Connell and E. Wigner, Phys. Lett. A , 319 (1978).[28] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- andTwo-Electron Atoms (Dover, Mineola, 2008).[29] B. Thaller,
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