Dirac’s purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. Dirac himself had not developed sufficiently to treat this problem. Although Dirac’s original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics. These matrices and the form of the wave function have introductory applied quantum and statistical mechanics pdf deep mathematical significance.
The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics. The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Space and time derivatives both enter to second order.
This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The continuity equation is as before. Thus, we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.
Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. Most physicists had little faith in such a process, even if it were technically possible. Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.