Abstract
Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli’s equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which is the more conservative Copenhagen’s interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through the Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli’s equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered.
Keywords:
Introduction
Quantum mechanics is usually interpreted by the Copenhagen approach. This approach objects to the physical reality of the quantum wave function and declares it to be epistemological (a tool for estimating probability of measurements) in accordance with the Kantian [1] depiction of reality, and its denial of the human ability to grasp any thing in its reality (ontology). However, we also see the development of another approach of prominent scholars that think about quantum mechanics differently. This school believes in the ontological existence of the wave function. According to this approach the wave function is an element of reality much like an electromagnetic field. This was supported by Einstein and Bohm [2,3,4] has resulted in different understandings of quantum mechanics among them the fluid realization championed by Madelung [5,6] which stated that the modulus square of the wave function is a fluid density and the phase is a potential of the velocity field of the fluid.
A non-relativistic quantum equation for a spinor was first introduced by Wolfgang Pauli in 1927 [7], this was motivated by the need to explain the Stern–Gerlach experiments. Later it was shown that the Pauli equation is a low-velocity limit of the relativistic Dirac equation (see for example [8] and references therein). This equation is based on a two dimensional operator matrix Hamiltonian. Two-dimensional operator matrix Hamiltonians are common in the literature [9,10,11,12,13,14,15,16,17,18,19,20,21,22] and describe many types of quantum systems. A Bohmian analysis of the Pauli equation was given by Holland and others [3,4], however, the analogy of the Pauli theory to fluid dynamics and the notion of spin vorticity were not considered. In [23] spin fluid dynamics was introduces for a single electron with a spin. One thus must contemplate where do those internal energies originate? The answer to this question seems to come from measurement theory [24,25]. Fisher information is a basic notion of measurement theory, and is a figure of merit of a measurement quality of any quantity. It was shown [25] that this notion is the internal energy of a spin less electron (up to a proportionality constant) and can be used to partially interpret the internal energy of an electron with spin. An attempt to derive most physical theories from Fisher information is due to Frieden [26]. It was suggested [27] that there exists a velocity field such that the Fisher information will give a complete explanation for the spin fluid internal energy. It was also suggested that one may define comoving scalar fields as in ideal fluid mechanics, however, this was only demonstrated implicitly but not explicitly. A common feature of previous work on the fluid and Fisher information interpretation of quantum mechanics, is the negligence of electromagnetic interaction thus setting the vector potential to zero. This was recently corrected in [28].
Ehrenfest [29] published his paper in 1927 as well with the title: “Remark on the approximate validity of classical mechanics within quantum mechanics”. Using this approach we can accept the orthodox Copenhagen’s interpretation denying a trajectory of the electron but at the same time accept the existence of a trajectory of the electron’s position vector expectation value through Ehrenfest theorem. The Ehrenfest approach is thus independent of interpretation, and can be applied according to both the Copenhagen and Bohm schools. However, only in the Bohm approach may one compare the trajectory of the electron to that of its expectation value.
We will begin this paper by reminding the reader of the basic equation describing the motion of a classical electron. This will be followed by a discussion of Schrödinger equation with a non trivial vector potential and its interpretation in terms of Bohmian equation of motion with a quantum force correction. Then we introduce Pauli’s equation with a vector potential and interpret it in terms a Bohmian equation of motion with a quantum force correction which is different from the Schrödinger case. Finally we derive an equation for Pauli’s electron position vector expectation value using Ehrenfest theorem and compare the result to the results obtained in Bohm’s approach, similarities and differences will arise, a concluding section will follow discussing the Stern–Gerlach experiment.
