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Physical Review A

Larmor diamagnetism and Van Vleck paramagnetism in relativistic quantum theory: The Gordon decomposition approach

Radosław Szmytkowski

Published Feb 20, 2002• 38 citations
Publisher

Abstract

We consider a charged Dirac particle bound in a scalar potential perturbed by a classical magnetic field derivable from a vector potential $\mathbf{A}(\mathbf{r}).$ 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 approach, it is found that the resulting diamagnetic term is ${\mathcal{E}}_{\mathrm{d}}^{(2)}{=(q}^{2}/2m)〈{\ensuremath{\Psi}}^{(0)}|\ensuremath{\beta}{\mathbf{A}}^{2}{\ensuremath{\Psi}}^{(0)}〉,$ where ${\ensuremath{\Psi}}^{(0)}(\mathbf{r})$ is an unperturbed eigenstate and $\ensuremath{\beta}$ is the matrix associated with the rest-energy term in the Dirac Hamiltonian.

DOI
10.1103/physreva.65.032112

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•Dec 16, 2011•0 citations
We consider a charged Dirac particle bound in a scalar potential perturbed by a classical magnetic field derivable from a vector potential A(r).\mathbf{A}(\mathbf{r}).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 approach, it is found that the resulting diamagnetic term is Ed(2)=(q2/2m)〈\ensuremathΨ(0)∣\ensuremathβA2\ensuremathΨ(0)〉,{\mathcal{E}}_{\mathrm{d}}^{(2)}{=(q}^{2}/2m)〈{\ensuremath{\Psi}}^{(0)}|\ensuremath{\beta}{\mathbf{A}}^{2}{\ensuremath{\Psi}}^{(0)}〉,Ed(2)​=(q2/2m)〈\ensuremathΨ(0)∣\ensuremathβA2\ensuremathΨ(0)〉, where \ensuremathΨ(0)(r){\ensuremath{\Psi}}^{(0)}(\mathbf{r})\ensuremathΨ(0)(r) is an unperturbed eigenstate and \ensuremathβ\ensuremath{\beta}\ensuremathβ is the matrix associated with the rest-energy term in the Dirac Hamiltonian.
A unitary transformation of the Dirac operator in a magnetic field is presented, which leads to a reformulation of the interaction of a Dirac particle with a magnetic field, in which, as in nonrelativistic theory, diamagnetic and paramagnetic contributions appear naturally, but at a four-component-spinor level. The diamagnetic contribution to the magnetic susceptibility consists of two terms, each of which is evaluated as a simple expectation value with the unperturbed relativistic wave function. One of the two terms closely resembles its nonrelativistic counterpart. The proposed formalism is analyzed in the context of the direct perturbation theory of relativistic effects. It is compared with the more traditional sum-over-states approach including negative-energy states, as well as with a Fock-space formulation. In the latter, the vacuum energy depends on the external magnetic field. The creation of a particle (electron or positron) is accompanied by a change of the vacuum energy via a kind of exclusion effect. This change can be identified with the diamagnetism of the particle. The access to diamagnetism and paramagnetism based on the Gordon decomposition of the induced current density is to some extent, but not entirely, equivalent to that which results from the unitary transformation. For a physically meaningful decomposition of the current density a combination of the Gordon approach with the unitary transformation is recommended. Neither the interpretation nor the computation of diamagnetic contributions in terms of negative-energy states is encouraged.
The Dirac Hamiltonian H(D)H^{\left(D\right)}H(D) for relativistic charged fermions minimally coupled to (possibly time-dependent) electromagnetic fields is transformed with a purpose-built flow equation method, so that the result of that transformation is unitary equivalent to H(D)H^{\left(D\right)}H(D) and granted to strive towards a limiting value H(NW)H^{\left(NW\right)}H(NW) commuting with the Dirac βββ-matrix. Upon expansion of H(NW)H^{\left(NW\right)}H(NW) to order v2c2\frac{v^2}{c^2}c2v2​ the nonrelativistic Hamiltonian H(SP)H^{\left(SP\right)}H(SP) of Schrödinger-Pauli quantum mechanics emerges as the leading order term adding to the rest energy mc2mc^2mc2. All the relativistic corrections to H(SP)H^{\left(SP\right)}H(SP) are explicitly taken into account in the guise of a Magnus type series expansion, the series coefficients generated to order (v2c2)n\left(\frac{v^{2}}{c^{2}}\right)^{n}(c2v2​)n for n≥2n\geq2n≥2 comprising partial sums of iterated commutators only. In the special case of static fields the equivalence of the flow equation method with the well known energy-separating unitary transformation of Eriksen is established on the basis of an exact solution of a reverse flow equation transforming the βββ-matrix into the energy-sign operator associated with H(D)H^{\left(D\right)}H(D). That way the identity H(NW)=βH(NW)H(NW)H^{\left(NW\right)}=β\sqrt{H^{\left(NW\right)}H^{\left(NW\right)}}H(NW)=βH(NW)H(NW)​ is established implying H(NW)H^{\left(NW\right)}H(NW) being determined unambiguously.
Up to this point in our study of quantum mechanics, we have ignored relativistic effects and taken spin as an extra observed property of particles with no specific model. In formulating an appropriate theory for the extension of quantum mechanics to include relativistic effects, it is logical to begin with the Hamiltonian based on our knowledge of relativity, where we have Hˆ = c[(pˆ− eA)2 +m2c2]1/2 + eφ , (11.1) where the first portion portion comes from W 2 = p2c2 + m2c4 using the canonical momentum pˆ = γmv + eA when there is a magnetic field present represented by the vector potential A, and then we have added the scalar electrostatic potential. This form for the Hamiltonian is very formidable, since it includes a square root with an operator inside (pˆ → (~/i)∇). By rearranging and squaring, Equation (11.1) can be put into a more symmetric form that avoids this difficulty, so that (Hˆ − eφ)2 − (pˆ− eA)2c2 = m2c4 . (11.2) One possible interpretation of this equation is that we should make the usual substitutions pˆ→ ~ i ∇ and Hˆ → −~ i ∂ ∂t , which leads to[( −~ i ∂ ∂t − eφ )2 − ( ~ i ∇− eA )2 c2 ] ψ = m2c4ψ . (11.3) This equation is called the Klein-Gordon equation, and represents one possible way to generalize the Schro¨dinger equation to include relativistic effects.
Abstract We present summary results of a bound‐state perturbation theory for a one‐space and one‐time dimension nonrelativistic spinless (Schrödinger) particle, a relativistic spinless (Klein‐Gordon) particle, and a relativistic spin‐half (Dirac) particle in central fields due to scalar or fourth‐component vector‐type interactions for an arbitrary bound state. This is accomplished by the reduction of the wave equations to Ricatti form. This enables a decoupling between the pair of coupled first order differential equations on the large and small component Dirac wave functions or a decoupling of the second order differential equation in the Schrödinger or Klein‐Gordon equations. All corrections to the energies and wave functions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchial scheme, without the use of either the Green's function or the sum over intermediate states. For the ground states of a Schrödinger particle, it is possible to extend this technique to multidimension in the case where the perturbation is due to noncentral fields, for example, in the problem of a nonrelativistic hydrogen atom in a linear combination of multiple fields.
This work presents the theoretical background for the computation of nuclear magnetic shielding, nuclear hyperfine and magnetizability tensors of paramagnetic molecules, using a magnetically induced current density framework to account for both orbital and spin contributions. The resulting magnetizability tensor is fully consistent with the general Van Vleck formulation, recovering the temperature-dependent Curie contribution through the explicit integration of the magnetically induced spin current density. The methodology proposed herein provides a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians. While the theoretical framework is general, we present applications rooted on physically motivated approximations where scalar relativistic effects are incorporated through corrections based on the Zeroth-Order Regular Approximation (ZORA) Hamiltonian within the ground-state spin density. This approach combines a relativistic self-consistent field (SCF) calculation for the ground-state spin density with a non-relativistic, origin-independent current density calculation for the orbital contribution. This hybrid strategy is shown to capture the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings, particularly in paramagnetic molecular systems containing transition metals up to the 3d series. By restricting the relativistic treatment to the spin density, where scalar relativistic effects are dominant, and neglecting such effects on the orbital contribution of light atoms, this method offers a good compromise between computational efficiency and accuracy for the characterization of large open-shell molecular systems.
Neste trabalho, investigamos a dinâmica de uma partícula relativística de spin 1/2 submetida a um potencial generalizado, constituído por um termo Coulombiano mais um termo escalar de Lorentz. A supersimetria é utilizada como recurso algébrico, bem como a atuação dos operadores escada, obtemos assim, um maior aperfeiçoamento matemático e entendimento físico do problema envolvido. Este formalismo e aplicado para o problema de Dirac-Coulomb generalizado que e um potencial exatamente solúvel em mecânica quântica relativística. As simetrias do problema coulombiano relativístico são pesquisadas a partir de um ponto de vista conceitual explorando analogias entre o caso clássico e o caso da mecânica quântica. A simetria do problema não-relativístico descrito pela álgebra de Lie 50(4) e usado como ideia norteadora. As propriedades do problema Dirac-Coulomb são discutidas em detalhe e relações entre as várias abordagens algébricas para estes problemas são apontadas. A relação natural entre a simetria dinâmica e o hamiltoniano de Dirac via o operador de Johnson-Lippmann generalizado no problema Dirac-Coulomb e investigado na nossa dissertação.