Modifications of Schrödinger's Equation Invariant under Galilean Transformations

Morton Hamermesh

R. Huegele, Z. E. Musielak, John L. Fry

A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Levy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Levy-Leblond and Schrodinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrodinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrodinger equation are discussed.

P. L. Torres

R. Huegele, Z. E. Musielak, John L. Fry

A search for fundamental (Galilean invariant) dynamical equations for two and four-component spinor wave functions is conducted in Galilean space-time. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wave functions. A method to derive the Lévy-Leblond equation for a four-component spinor wave function is presented. It is formally proved that the Lévy-Leblond and Schrödinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrödinger phase factor. Physical implications of the obtained results and their relationships to the Pauli-Schrödinger equation are discussed.

Ching-Chuan Su

In a Galilean theory recently developed, the Lorentz force law has been modified in such a way that the electromagnetic force is invariant when observed in different inertial frames, as expected intuitively. In this investigation, based on the local-ether model of wave propagation, a Galilean-invariant wave equation is proposed. This wave equation incorporates a natural frequency and the electric scalar potential. A unique feature is that this potential connects to the augmentation operator which in turn is associated with the momentum operator and the velocity of the source particle. From this local-ether wave equation, a first-order time evolution equation is derived, which presents Galilean modifications of Schrodinger's equation. This modified equation in turn leads to the Galilean-invariant electromagnetic force law given in terms of the augmented potentials.

Yu. N. Demkov, N. V. Semenova