Schrödinger-Like Relativistic Wave Equation of Motion for the Lorentz-Scalar Potential

Young‐Sea Huang

Ingram Bloch, Horace W. Crater

We discuss a relativistic particle affected simultaneously by two static potentials, a Lorentz scalar and the time component of a four-vector. We show that the scalar potential has effects on the motion of a particle that are distinctively different from those of a four-vector potential. With the aid of an example we show that, in both classical and quantum mechanics, there are two inequivalent but compatible criteria of nonrelativistic behavior: (mean) kinetic energy much smaller than rest energy and nonrelativistic potential energy independent of total energy. The second of these is the correct condition for use of nonrelativistic methods, but it is the first that is usually invoked to justify their use. The two criteria are equivalent only when the scalar potential has a (mean) value much less in magnitude than the particle’s rest energy. If this condition is not satisfied, nonrelativistic formalisms such as ordinary Newtonian mechanics and the Schrödinger equation may be inadequate even if v/c is small. On the other hand, it is possible that a large S may permit relativistic energy relations to be written in a nonrelativistic form even if v/c is not small. Such an influence of the size of the scalar potential on what is considered nonrelativistic behavior is unfamiliar and is rarely mentioned in textbooks and other publications. In our opinion, it should be better known.

Wolfgang Ernst Friederich Pauli, Victor F. Weisskopf

E. Trübenbacher

Abstract Using the concept of distributions, the square root of the operator - Δ + m2 is taken in a mathematically well defined way for one component wave functions. A new representation of proper Lorentz transformations for one component wave functions makes it possible to construct a relativistic quantum mechanics for spin 0, comprising a Lorentz invariant wave equation, a scalar product, and a positive definite density satisfying, together with a current, a continuity equation, and coupling of scalar and vector potentials. Some interesting consequences of the theory concerning the concept of particle trajectory and velocity of propagation of the probability amplitude are discussed in detail. As an example of practical application a perturbation theory for discrete states is set up.

F. Masafumi Toyama, Yukihisa Nogami, Zhongxin Zhao

For the Dirac equation in one space dimension with a potential of the Lorentz scalar type, we present a complete solution for the problem of constructing a transparent potential. This is a relativistic extension of the Kay-Moses method which was developed for the nonrelativistic Schr\"odinger equation. There is an infinite family of transparent potentials. The potentials are all related to solutions of a class of coupled, nonlinear Dirac equations. In addition, it is argued that an admixture of a Lorentz vector component in the potential impairs perfect transparency.

Peter Hawkes, Erwin K. Kasper