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Larmor Diamagnetism and Van Vleck Paramagnetism in Relativistic Quantum Theory: the Gordon Decomposition Approach

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Larmor Diamagnetism and Van Vleck Paramagnetism in Relativistic Quantum Theory: the Gordon Decomposition Approach PHYSICAL REVIEW A, VOLUME 65, 032112 Larmor diamagnetism and Van Vleck paramagnetism in relativistic quantum theory: The Gordon decomposition approach Radosław Szmytkowski* Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Technical University of Gdan´sk, Narutowicza 11/12, PL 80-952 Gdan´sk, Poland ͑Received 18 June 2001; published 20 February 2002͒ We consider a charged Dirac particle bound in a scalar potential perturbed by a classical magnetic field derivable from a vector potential A(r). Using a procedure based on the Gordon decomposition of a field- induced current, we identify diamagnetic and paramagnetic contributions to the second-order perturbation- theory correction to the particle’s energy. In contradiction to earlier findings, based on the sum-over-states E(2)ϭ 2 ͗⌿(0)͉␤ 2⌿(0)͘ ⌿(0) approach, it is found that the resulting diamagnetic term is d (q /2m) A , where (r)isan unperturbed eigenstate and ␤ is the matrix associated with the rest-energy term in the Dirac Hamiltonian. DOI: 10.1103/PhysRevA.65.032112 PACS number͑s͒: 03.65.Pm, 75.20.Ϫg, 31.30.Jv I. INTRODUCTION (2) The contribution Ed is clearly positive and is called a dia- magnetic ͑or Larmor͒ component of E(2).IfE(0) is the A nonrelativistic quantum-mechanical problem of a spin- ground-state energy of Hˆ (0), the contribution E(2) is nega- 1 p 2 particle of electric charge q bound in a potential V(r) tive; it is called a paramagnetic component of E(2) and is perturbed by a classical static magnetic field derivable from a usually associated with the name of Van Vleck ͓1,3͔.Itisto vector potential A(r) is frequently encountered in physics. It be mentioned that since the vector potential A(r) is gauge is well known ͓1–3͔ that if the perturbing magnetic field is (2) (2) (2) dependent, the splitting of E into Ed and Ep is not weak and the potential A(r) has no singularities, the first- unique. Still, E(2) and E(1) are gauge invariant. order perturbation-theory correction to an unperturbed en- ͓ ͔ ͓ ͔͑ (0) Feneuille 4 and, more recently, Aucar et al. 5 cf. also ergy level E associated with an unperturbed wave function Ref. ͓6͔͒ attempted to explain the origin of diamagnetism ␺(0) (r) may be expressed in the form within the framework of the Dirac relativistic quantum me- chanics. Considerations based on the sum-over-states ap- (1)ϭ͗␺(0)͉ ˆ (1)␺(0)͘ ͑ ͒ E H , 1.1 proach led these authors to the conclusion that diamagnetism had to be attributed to the ‘‘redressing’’ of a relativistic par- and the second-order one in the form ticle by a perturbing magnetic field. It was claimed that in E(2)ϭE(2)ϩE(2), ͑1.2͒ the relativistic theory the diamagnetic contribution to the d p second-order energy expression was due to the negative- with the contributions given by energy Dirac sea and was given by the following formula: E(2)ϭ͗␺(0)͉Hˆ (2)␺(0)͘ ͑1.3͒ q2 d E (2)ϭ ͗⌿(0)͉A2⌿(0)͘, ͑1.7͒ d 2m and formally identical with its nonrelativistic counterpart ͑1.3͒ E(2)ϭϪ͗␺(0)͉Hˆ (1)Gˆ (0)Hˆ (1)␺(0)͘. ͑1.4͒ p ͓7͔. Here ⌿(0)(r) is a four-component eigenfunction of the zeroth-order ͑i.e., with the magnetic field switched off͒ Dirac Here Gˆ (0) is the generalized Green operator of the unper- Hamiltonian. turbed energy eigenproblem, associated with the energy level In the present paper, we reconsider the problem of deter- (0) ˆ (1) E , and the particle-field interaction operators H and mining diamagnetic and paramagnetic contributions to the Hˆ (2) are second-order energy correction within the framework of the single-particle Dirac theory. Our approach is entirely differ- ប ប iq q ent from that proposed in Refs. ͓4,5͔. Instead of utilizing the Hˆ (1)ϭ ͓ A͑r͒ϩA͑r͒ ͔Ϫ ␴ B͑r͒ ͑1.5͒ 2m “• •“ 2m • sum-over-states scheme, we find a field-induced electric cur- rent in a linear-response approximation ͓8–12͔ and identify ͓with B(r)ϭcurl A(r) and ␴ denoting the vector composed its diamagnetic and paramagnetic components. This is of the Pauli matrices͔ and achieved using an ingenious ͑but, regrettably, not appreciated enough͒ procedure devised by Gordon ͓13–15͔ in the very q2 Hˆ (2)ϭ A2͑r͒. ͑1.6͒ early days of relativistic quantum mechanics. Subsequent use 2m of the decomposed current in a formula linking the second- order energy correction E (2) with a vector potential and the induced current allows us to identify diamagnetic and para- *Electronic address: [email protected] magnetic contributions to the former. Somewhat surprisingly, 1050-2947/2002/65͑3͒/032112͑8͒/$20.0065 032112-1 ©2002 The American Physical Society RADOSL”AW SZMYTKOWSKI PHYSICAL REVIEW A 65 032112 E (2) (0) (0) (0) our procedure leads us to the conclusion that d is not ͓Hˆ ϪE ͔⌿ ͑r͒ϭ0 ͑2.6͒ given by Eq. ͑1.7͒ but rather has the form ͑for the sake of brevity, henceforth we shall assume that the q2 eigenvalue E (0) is nondegenerate͒, we may seek correspond- E (2)ϭ ͗⌿(0)͉␤A2⌿(0)͘, ͑1.8͒ d 2m ing eigensolutions to the problem ͑2.1͒ in the form ⌿͑ ͒ϭ⌿(0)͑ ͒ϩ⌿(1)͑ ͒ϩ⌿(2)͑ ͒ϩ ͑ ͒ where ␤ is the matrix associated with the rest-energy term in r r r r •••, 2.7 the Dirac Hamiltonian ͓notice that both expressions ͑1.7͒ and EϭE (0)ϩE (1)ϩE (2)ϩ . ͑2.8͒ ͑1.8͒ have the same nonrelativistic limit͔. ••• Searching through the literature of the subject, we found From this, in the standard way we obtain equations that our idea of applying the Gordon decomposition to deter- mine diamagnetic and paramagnetic contributions to mag- ͓Hˆ (0)ϪE (0)͔⌿(1)͑r͒ϭϪ͓Hˆ (1)ϪE (1)͔⌿(0)͑r͒, ͑2.9͒ netic properties was anticipated by a similar idea of Pyper who exploited it in the context of the theories of nuclear ͓Hˆ (0)ϪE (0)͔⌿(2)͑r͒ϭϪ͓Hˆ (1)ϪE (1)͔⌿(1)͑r͒ shielding ͓16–19͔ and hyperfine interaction ͓20͔. There are, however, two main differences between Pyper’s and our ϩE (2)⌿(0)͑r͒, ͑2.10͒ work. First, somewhat different physical problems are con- sidered. Second, different procedures are used: while Pyper which are to be supplemented by pertinent boundary condi- used the sum-over-states approach performing decomposi- tions. Imposing the constraints tions of virtual transition currents between unperturbed ͗⌿(0)͉⌿(0)͘ϭ1, ͗⌿(0)͉⌿(1)͘ϭ0, ͑2.11͒ eigenstates of the zeroth-order Dirac Hamiltonian, we apply the linear-response approach decomposing the field-induced from Eqs. ͑2.9͒ and ͑2.10͒ we infer current. E (1)ϭ͗⌿(0)͉Hˆ (1)⌿(0)͘, ͑2.12͒ II. RELATIVISTIC THEORY OF FIRST- AND SECOND- ORDER MAGNETIC CORRECTIONS TO ENERGY E (2)ϭ͗⌿(0)͉Hˆ (1)⌿(1)͘. ͑2.13͒ A. Generalities Formally, a solution to Eq. ͑2.9͒ is Consider a relativistic Dirac particle of electric charge q ͑for an electron qϭϪe) bound in a field of force derivable ⌿(1)͑r͒ϭϪGˆ (0)Hˆ (1)⌿(0)͑r͒. ͑2.14͒ from a potential V(r) ͑not necessarily of an electromagnetic origin͒ and perturbed by a classical static magnetic field where Gˆ (0) ͓possessing the property Gˆ (0)⌿(0)(r)ϭ0͔ is the characterized by a real nonsingular vector potential A(r). generalized Green operator for the zeroth-order Dirac Hamil- The time-independent wave equation for the particle has the tonian Hˆ (0), associated with the eigenvalue E (0) of the latter. form The operator Gˆ (0) has an integral kernel G (0)(r,rЈ) which is a solution to the equation ͓Hˆ ϪE͔⌿͑r͒ϭ0 ͑2.1͒ ͓Hˆ (0)ϪE (0)͔G (0)͑r,rЈ͒ϭI ␦͑rϪrЈ͒Ϫ⌿(0)͑r͒⌿(0)†͑rЈ͒ with the Hamiltonian ͑2.15͒ Hˆ ϭ ␣ ͓Ϫ ប Ϫ ͑ ͔͒ϩ␤ 2ϩ ͑ ͒ ͑ ͒ ͑here I is the 4ϫ4 unit matrix͒ supplemented by the same c • i “ qA r mc V r . 2.2 boundary conditions that have been imposed on ⌿(0)(r). On Here ␣ and ␤ are standard Dirac matrices. The Hamiltonian substituting the solution ͑2.14͒ into Eq. ͑2.13͒,wefind Hˆ may be conveniently written as E (2)ϭϪ͗⌿(0)͉Hˆ (1)Gˆ (0)Hˆ (1)⌿(0)͘. ͑2.16͒ Hˆ ϭHˆ (0)ϩHˆ (1), ͑2.3͒ In principle, at this stage the problem of finding the first- E (0) where and the second-order corrections to may be considered as solved since Eq. ͑2.12͒ and, depending on the particular computational technique used, either Eq. ͑2.13͒ or Eq. ͑2.16͒ Hˆ (0)ϭϪicប␣ ϩ␤mc2ϩV͑r͒, ͑2.4͒ •“ are suitable for computational purposes. Still, on a purely aesthetic ground, one may be slightly dissatisfied with the Hˆ (1)ϭϪ ␣ ͑ ͒ ͑ ͒ qc •A r . 2.5 above results. First, Eqs. ͑1.1͒ and ͑2.12͒ are only seemingly similar since the operators Hˆ (1) and Hˆ (1) have completely Henceforth, we shall assume that the magnetic field is weak, different structures ͓cf. Eqs. ͑1.5͒ and ͑2.5͔͒. Second, while (1) which implies that the operator Hˆ may be treated as a in the nonrelativistic theory the second-order correction is small perturbation of Hˆ (0). Then, if ⌿(0)(r) and E (0) are the sum of the diamagnetic and paramagnetic parts ͓cf. Eqs. particular eigensolutions to the zeroth-order Dirac eigen- ͑1.2͒–͑1.4͔͒, in the relativistic case the second-order correc- problem tion has the compact form ͑2.16͒ which is, but only superfi- 032112-2 LARMOR DIAMAGNETISM AND VAN VLECK .
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