Galilean invariance and the Schrödinger equation

Ching-Chuan Su

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.

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.

Jacques Voisin

P. L. Torres

Alejandro López‐Castillo

In the present work we propose a method for solving the Schr\"odinger equation in the adiabatic approximation within the molecular context. We employ this method to solve problems with non-Galilean invariance, and an attempt is made to explain some theoretical aspects of it. Simple applications are also considered. This method, which we could call the central-field method (or linear combination of central fields), enables a better understanding of the problem of non-Galilean invariance. \textcopyright{} 1996 The American Physical Society.