The relativistically corrected Schr\"odinger (RS) equation [( ${m}^{2}$ ${c}^{4}$ + ${p}^{2}$ ${c}^{2}$ ${)}^{1/2}$ +V-E]\ensuremath{\Psi}=0 is sometimes used to describe charged fermions, which are subject to strong Zitterbewegung. Since, however, equations of this type describe bosons with weak Zitterbewegung effects, it is mandatory to screen the Coulomb potential V in an r range of order h/mc, and also to add a screened spin-orbit potential. Thereby the singularity problems with Coulomb potentials are avoided. While the Pauli Hamiltonian must not be used in variational or eigenvalue equations, the RS Hamiltonian may be. An effective potential for the RS equation is derived, which yields relativistically corrected energies and wave functions for fermion systems with Coulomb potentials. Accurate numerical solutions for an unshielded single-center Coulomb potential are presented.The relativistically corrected Schr\"odinger (RS) equation [(${m}^{2}$${c}^{4}$+${p}^{2}$${c}^{2}$${)}^{1/2}$ +V-E]\ensuremath{\Psi}=0 is sometimes used to describe charged fermions, which are subject to strong Zitterbewegung. Since, however, equations of this type describe bosons with weak Zitterbewegung effects, it is mandatory to screen the Coulomb potential V in an r range of order h/mc, and also to add a screened spin-orbit potential. Thereby the singularity problems with Coulomb potentials are avoided. While the Pauli Hamiltonian must not be used in variational or eigenvalue equations, the RS Hamiltonian may be. An effective potential for the RS equation is derived, which yields relativistically corrected energies and wave functions for fermion systems with Coulomb potentials. Accurate numerical solutions for an unshielded single-center Coulomb potential are presented.

Physical Review ADec 1, 198812 citations

Many-particle relativistic equations for fermions

V. E. Rochev

Theoretical and Mathematical PhysicsApr 1, 19825 citations

Relativistic quantum mechanics of fermions

Egon Marx

International Journal of Theoretical PhysicsOct 1, 197019 citations

Relativistic Aharonov-Bohm effect in the presence of planar Coulomb potentials

V. R. Khalilov

Exact analytic solutions are found to the Dirac equation in $2+1$ dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.Exact analytic solutions are found to the Dirac equation in $2+1$ dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.

arXiv:quant-ph/0406033Physical Review AJan 19, 200547 citations

RELATIVISTIC CORRECTIONS IN SYSTEMS OF FERMIONS

F. Sanchez-Martin, F. Lopez-Gordo

Mar 1, 19640 citations

Relativistic saturation of Coulomb-induced electron decoherence: from eikonal phase noise to Bethe--Salpeter kinetic theory

Yury A. Budkov

We develop a microscopic theory of the mutual coherence and small-angle scattering of relativistic electron matter waves propagating through a Coulomb-fluctuating medium. Starting from the Dirac equation, we derive a relativistic paraxial wave equation and then obtain, rather than assume, the Bethe--Salpeter and Wigner--Boltzmann kinetic equations for the disorder-averaged two-point coherence. The Coulomb environment enters through the eikonal phase-coupling factor $A_{\rm rel}=1/(\hbar v)$ , which saturates at $1/(\hbar c)$ for ultra-relativistic particles. This saturation is the central physical result: increasing the beam voltage suppresses Coulomb phase noise only up to a finite relativistic floor. We first formulate the eikonal approximation, in which the transmitted electron wave accumulates a random longitudinal phase. For a one-component Coulomb medium the electrostatic potential correlator retains a Coulomb tail, producing a logarithmic phase structure function and an algebraic decay of the transverse mutual coherence. We then go beyond the eikonal construction by deriving the Bethe--Salpeter equation for the Cooperon-like two-point coherence propagator directly from the paraxial wave equation. After a longitudinal Markov projection this equation reduces, after a Wigner transform, to a kinetic equation with a Coulomb small-angle scattering kernel. The coordinate-space solution of the same Bethe--Salpeter equation recovers the eikonal mutual coherence function, while the Wigner form additionally describes angular diffusion and transverse broadening caused by multiple small-angle scattering. The resulting theory separates three effects which are often conflated in charged-particle wave propagation: attenuation of the coherent amplitude, decay of mutual coherence, and conservative redistribution of intensity in transverse phase space.We develop a microscopic theory of the mutual coherence and small-angle scattering of relativistic electron matter waves propagating through a Coulomb-fluctuating medium. Starting from the Dirac equation, we derive a relativistic paraxial wave equation and then obtain, rather than assume, the Bethe--Salpeter and Wigner--Boltzmann kinetic equations for the disorder-averaged two-point coherence. The Coulomb environment enters through the eikonal phase-coupling factor $A_{\rm rel}=1/(\hbar v)$, which saturates at $1/(\hbar c)$ for ultra-relativistic particles. This saturation is the central physical result: increasing the beam voltage suppresses Coulomb phase noise only up to a finite relativistic floor. We first formulate the eikonal approximation, in which the transmitted electron wave accumulates a random longitudinal phase. For a one-component Coulomb medium the electrostatic potential correlator retains a Coulomb tail, producing a logarithmic phase structure function and an algebraic decay of the transverse mutual coherence. We then go beyond the eikonal construction by deriving the Bethe--Salpeter equation for the Cooperon-like two-point coherence propagator directly from the paraxial wave equation. After a longitudinal Markov projection this equation reduces, after a Wigner transform, to a kinetic equation with a Coulomb small-angle scattering kernel. The coordinate-space solution of the same Bethe--Salpeter equation recovers the eikonal mutual coherence function, while the Wigner form additionally describes angular diffusion and transverse broadening caused by multiple small-angle scattering. The resulting theory separates three effects which are often conflated in charged-particle wave propagation: attenuation of the coherent amplitude, decay of mutual coherence, and conservative redistribution of intensity in transverse phase space.

arXiv:2605.20141May 19, 20260 citations

The relativistically corrected Schr\"odinger (RS) equation [(

{m}^{2}

{c}^{4}

{p}^{2}

{c}^{2}

{)}^{1/2}

+V-E]\ensuremath{\Psi}=0 is sometimes used to describe charged fermions, which are subject to strong Zitterbewegung. Since, however, equations of this type describe bosons with weak Zitterbewegung effects, it is mandatory to screen the Coulomb potential V in an r range of order h/mc, and also to add a screened spin-orbit potential. Thereby the singularity problems with Coulomb potentials are avoided. While the Pauli Hamiltonian must not be used in variational or eigenvalue equations, the RS Hamiltonian may be. An effective potential for the RS equation is derived, which yields relativistically corrected energies and wave functions for fermion systems with Coulomb potentials. Accurate numerical solutions for an unshielded single-center Coulomb potential are presented.

Exact analytic solutions are found to the Dirac equation in

2+1

dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.

A_{\rm rel}=1/(\hbar v)

, which saturates at

1/(\hbar c)

for ultra-relativistic particles. This saturation is the central physical result: increasing the beam voltage suppresses Coulomb phase noise only up to a finite relativistic floor. We first formulate the eikonal approximation, in which the transmitted electron wave accumulates a random longitudinal phase. For a one-component Coulomb medium the electrostatic potential correlator retains a Coulomb tail, producing a logarithmic phase structure function and an algebraic decay of the transverse mutual coherence. We then go beyond the eikonal construction by deriving the Bethe--Salpeter equation for the Cooperon-like two-point coherence propagator directly from the paraxial wave equation. After a longitudinal Markov projection this equation reduces, after a Wigner transform, to a kinetic equation with a Coulomb small-angle scattering kernel. The coordinate-space solution of the same Bethe--Salpeter equation recovers the eikonal mutual coherence function, while the Wigner form additionally describes angular diffusion and transverse broadening caused by multiple small-angle scattering. The resulting theory separates three effects which are often conflated in charged-particle wave propagation: attenuation of the coherent amplitude, decay of mutual coherence, and conservative redistribution of intensity in transverse phase space.