Relativistic Landau quantization for a neutral particle

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 80, Issue 3, 2009, Pages

  Bakke, K ^a   Furtado, C ^a  

^a FEDERAL UNIVERSITY OF PARAÍBA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EIGEN FUNCTION; EIGENVALUES; EXTERNAL ELECTRIC FIELD; LANDAU LEVELS; LANDAU QUANTIZATION; MAGNETIC DIPOLE; NEUTRAL PARTICLES; PERMANENT MAGNETIC DIPOLE; QUANTUM DYNAMICS;

ELECTRIC FIELDS; QUANTUM THEORY;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 70249103454     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.032106     Document Type: Article

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42. When we use the direct coordinate transformation of Eq. 4 to cylindrical coordinates we obtain the following Dirac equation: i γ0 ∂ψc/∂t + i γ̄ 1 ∂ ψc / ∂ρ + i γ̄ 2 /ρ ∂ ψc / ∂φ + i γ3 ∂ ψc / ∂z -m ψc =0, where γ̄ 0 = γ0, γ̄ 1 =cos φ γ1 + sin φ γ2, γ̄ 2 =-sin φ γ1 + cos φ γ2, and γ̄ 3 = γ3 are space-dependent Dirac matrices in cylindrical coordinates. Clearly, the spinor ψ of Eq. 11 and the spinor ψc are not identical but are connected by following transformation ψ=S ψc with S=exp [i 1 2 φ Σ3], which is a unitary transformation. In this way, we can use the Dirac equation in form 11 with the spinorial connection or make use of the above Dirac equation obtained by a transformation of coordinates where the Dirac matrices are space dependent. The relation between the spinors is a unitary matrix transforming ψ into ψc. Notice that an observable Ǒ c in Cartesian representation is transformed according to Ǒ c → Ǒ = S-1 Ǒ c S. In this way the energy in both representations is the same due to the fact that the unitary transformations are time independent.
43. In it is well discussed that the solution of the second-order equation, originated from the Dirac equation, includes redundant solutions which do not satisfy the original first-order equation. The appropriate choice of solutions requires a specific procedure. The customary procedure is that, if φ is solution of a second-order equation, the correct solution of the first-order equation is ψ= [i γμ ∂μ -i γi Ai AC +m] φ. If we multiply this equation by [i γμ ∂μ -i γi Ai AC - m], we find that the right-hand side vanishes if φ satisfies Dirac equation 7.
44. Due to the choice of Cartesian coordinates in the Landau gauge, the dependence on I does not appear explicitly in expression 46 for the eigenvalues. Notice that if we compare expression 28 46 we can find that the relation between v and n is v = n + l ζ sl / 2 + s ζ s/2.