Note on the Energy Eigenfunctions of the Dirac Hamiltonian

E. K. Broch

Abdulaziz D Alhaidari

Tore Berggren

Hans Albrecht Bethe

H. O. Girotti, F. Fonseca Romero

The formulation of Berry for the Aharonov-Bohm effect is generalized to the relativistic regime. Then, the problem of finding the self-adjoint extensions of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background potential, is solved in a novel way. The same treatment also solves the problem of finding the self-adjoint extensions of the Dirac Hamiltonian in a background Aharonov-Casher. PACS: 03.65.Bz, 03.65.Pm Typeset using REVTEX 1 A novel systematics for the computation of the energy eigenfunctions of a spinless electrically charged nonrelativistic particle in the presence of an Aharonov-Bohm [1] external potential, was put forward by Berry [2] sometime ago. We propose here a generalization of this systematics to the case where the particle is a relativistic electron. As will be shown, the set of eigenfunctions found through this generalization serves as a basis for writing any function in the domain of self-adjointness of the corresponding Dirac Hamiltonian. This corroborates the correctness of our proposal, which can also be used to obtain the selfadjoint extensions of the Dirac Hamiltonian in an Aharonov-Casher [3] background field. The extension of Berry’s ideas to the relativistic regime is the original contribution in this work. To make this paper self contained, we start by reviewing the main features of Berry’s formulation [2]. As known, the Hamiltonian operator H(⃗r, ⃗p) describing the dynamics of a spinless two-dimensional nonrelativistic particle (mass m and electric charge e) in the presence of an external static magnetic field B (B = ⃗ ∇ × ⃗ ( A) is H(⃗r, ⃗p) = H0 ⃗r, ⃗p − e ⃗ A(⃗r)) where ⃗r ≡ (x1, x2) are the particle coordinates, ⃗p ≡ (p1, p2) are the corresponding canonically conjugate momenta and H0(⃗r, ⃗p) is the free particle Hamiltonian. The energy eigenfunctions of the operator H can formally be constructed in terms of those of H0 as follows [4] ψ(⃗r) = ψ0(⃗r) exp ie ⃗r

Weimin Sun, Xiang-Song Chen, Xiaofu Lü +1 more

For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle was recently provided by us [X.-S. Chen et al., Phys. Rev. Lett. 100, 232002 (2008)]. Based on the separation of the electromagnetic potential into pure-gauge and gauge-invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen-atom problem: Starting from the Hamiltonian of the coupled electron, proton, and electromagnetic field, under the infinite proton mass approximation, we derive the gauge-invariant hydrogen-atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge-invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.