On the quaternionic form of the Pauli–Schrödinger equation

Abstract

A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e. division algebras. Quaternions, with their commutation rules isomorphic to SU(2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli–Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli–Schrödinger equation is then written using left-chirality quaternions, i.e. by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.

Quaternionic elastic scattering

“…By way of example, we quote several results of this kind . We also point out the existence of quaternic applications in quantum mechanics that cannot be considered HQM, such as [28][29][30][31][32][33][34][35][36][37][38].…”

Square-well potential in quaternionic quantum mechanics

“…where A n , B p ∈ C, n, p ∈ N and ψ n are complex wave functions that satisfy (3), (4). Thus, (22) comprises two non-degenerate solutions of CQM, where n = p, jointed in a single quaternionic structure. We name such functions the combined quaternionic solution.…”