Order of Time Derivatives in Quantum-Mechanical Equations

Abstract

The problem is rather old, almost as quantum mechanics itself. The first, primary idea of Schrodinger was the relativistic one, with the d’Alembert operator on the left-hand side of quantum-mechanical equation, so, with the second-order time derivatives. Unfortunately, it turned out that the following results were in a rather clear contradiction with experimental data, although some kind of compatibility did exist. Schrodinger felt disappointed and at least temporarily he rejected his primary equation. Later on, basic on the idea of Lagrange-Hamilton optical-mechanical analogy and on certain de Broglie ideas, he in a sense derived his famous equation which seemed to remain in a beautiful agreement with spectroscopic data and was approximately compatible with the Bohr-Sommerfeld quantization rules. Nevertheless, it was of course drastically incompatible with the relativistic idea of Poincare symmetry. But, after the fall of the primary substantial interpretation by Schrodinger, it was compatible with the Born statistical interpretation of his formalism, and with the corresponding continuity equation for the probabilistic density (Veltman, 2003). Later on history was rather complicated. Dirac formulated his relativistic quantum theory of electrons based on first-order space-time derivatives of multicomponent wave functions. The multicomponent character of waves had to do obviously with the particle spin. It was also understood that the relativistic velocity-dependence of the electron mass and the spin phenomena act in opposite directions, and because of this non-relativistic Schrodinger equation seemed to be better than his second order equation, rediscovered later on by Klein and Gordon. The formalism of quantum field theory rehabilitated the Klein-Gordon equation, i.e., primaevally the relativistic Schrodinger equation, as one describing some physics. And, let us also mention that the field-theoretic approach based on the Pauli exclusion principle removed certain problems with the quantum-mechanical Dirac equation for a single electron. And certain inconsistencies on one-particle relativistic theory were resolved. The only, and fundamental inadequacy which remained, was one connected with the essential non-linearity of the quantumfield-theoretic equations for the field operators and the resulting interpretation difficulties. Nevertheless, they were in a sense solvable on the basis of renormalization procedure. But in spite of everything said, the problem is still alive. There are certain not completely clear facts within the framework of field theory based on the Dirac-Clifford paradigm of first-order differential equations of quantum mechanics with h/2-spin. One can show that they become more clear when we assume that in a sense some second-order equations are primary and the first-order ones are some approximations valid for slowly-varying fields. There are also 3