Relativistic Invariance of Dirac's Equation Revisited

Hamiltonian operators are gauge dependent. For overcome this difficulty we reexamined the effect of a gauge transformation on Schrödinger and Dirac equations. We show that the gauge invariance of the operator $H-i\hbar\frac{\partial}{\partial t}$ provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].Hamiltonian operators are gauge dependent. For overcome this difficulty we reexamined the effect of a gauge transformation on Schrödinger and Dirac equations. We show that the gauge invariance of the operator $H-i\hbar\frac{\partial}{\partial t}$ provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].

arXiv:1208.1425Aug 7, 20121 citation

Gauge invariance and the dirac equation

Donald H. Kobe

International Journal of Theoretical PhysicsAug 1, 19828 citations

Algebra of the spinor invariants and the relativistic hydrogen atom

A. A. Eremko, Larissa S. Brizhik, В. М. Локтев

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry SU(2) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry SO(3)⊗SU(2). The generator of the SO(3) symmetry group is given by the total momentum operator, and the generator of SU(2) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson–Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.

arXiv:2211.01857Annals of PhysicsMar 2, 20236 citations

Relativistic Invariance of Quantum-Mechanical Equations

Eugene Paul Wigner

Springer eBooksJan 1, 19972 citations

H-i\hbar\frac{\partial}{\partial t}

provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].