Philip Du Toit and Dr. Jean Francois Van Huele
Introduction
Quantum mechanics attributes to matter the properties of waves. The postulate that matter behaves as a wave and can be described by a wave function provides a mathematical description or model of many previously unexplained physical phenomena such as the diffraction pattern of an electron beam. Although the wave function itself bears no physical significance, information about all physically observable quantities is contained within the wave function. The wave functions are solutions to a quantum mechanical wave equation.
The simplest of these wave equations is the Schr`dinger equation, a nonrelativistic second order differential equation. When the Schr`dinger equation is solved for a specific potential, the solutions reveal that the energy is quantized in discrete values, or energy levels. Zeeman noticed that each of the energy levels for the electron in a hydrogen atom split into two levels in the presence of a magnetic field. The property that causes this degeneracy in the energy levels is called spin. Since this property is absent in the Schr`dinger equation, the equation fails to predict the splitting of energy levels. Pauli found that he could recover the data of the Zeeman experiment if he appended an extra spin term of the form (e over 2m, σ.B) to the Schr`dinger equation.
This extra term introduced non-commuting objects, σ, or spin operators in a space of spinors. This amendment to the Schr`dinger equation results in two solutions of different energies, corresponding to the case of spin up or spin down. The addition of the extra term in Pauli’s equation provides the desired results, yet appears as a forceful addition of the spin property by a hand-placed constraint. A more appealing derivation of the spin property followed later with Dirac’s relativistic approach.
Dirac incorporated the principles of relativity into quantum mechanics by deriving a relativistic wave equation that was first order both in time and position. This approach requires the noncommuting properties of the spin operators. Hence Dirac was able to derive spin from the equation rather than hand-placing it in the equation a la Pauli. The Dirac equation is a relativistic wave equation for particles of spin 1/2. A non-relativistic reduction of the Dirac equation leads to the Pauli equation.
Dirac’s serendipitous derivation of spin within a relativistic context has led many to believe that spin is a purely relativistic effect. In 1969, Levy-Leblond [2] showed that the spin property can be derived without appealing to special relativity, but by assuming Galilean invariance. The Levy Leblond equation consists of two first order differential equations obtained by factorizing the Schrodinger equation.
Statement of the Problem
Quantum mechanics defines the probability of finding a particle at a particular location, rather than its exact position. Instead of calculating actual velocities of particles, we must rather consider the flow of the probability density. The flow of probability density is referred to as the probability current. The usual method for deriving the probability current is to manipulate the wave equation to a form in which we can compare it with the continuity equation
This allows us to read off the probability current J.
An ambiguity arises when we compare the probability current for the Pauli equation, with the
non-relativistic limit of the probability current for the Dirac equation. The latter contains an extra
term of the form
Notice that this extra term contains the spin matrices, σ, and is thus an additional contribution due to the spin of the particle. The inability of the Pauli equation to produce this term in the probability current suggests that the Pauli equation is not a complete equation for non-relativistic particles with spin and that we must use another equation to calculate the probability current.
Results
During the course of my research, I have determined that this extra term is not a relativistic effect, but rather a consequence of the incompleteness of information on spin within the Pauli equation. I have researched the nature and properties of this extra term. Specifically, I have studied its behavior and contributions for the specific problem of an electron in a homogeneous magnetic field.
I have derived the expressions for the probability currents for the Schrodinger, Pauli, and Dirac equations for an electron both without and within an electromagnetic field. In order to conclude whether the extra term is a relativistic effect we have appealed to the nonrelativistic Levy- Leblond equation. When we calculate the probability current for the Levy-Leblond equation the extra term appears. Thus the extra term can be derived from an equation that contains information about spin but not relativity.
During the research we have also applied these findings to a specific case of a homogeneous magnetic field [3]. We have solved each of the wave equations discussed for this potential, and used the resulting wave functions to calculate the probability currents. Comparing the currents, we note that for this particular potential the spin current is non-zero in the y-component. Whereas the current for the Pauli equation is identical for both spin up and spin down, the spin current differentiates between the two cases. It is interesting that when we add the currents for these two spin cases we retrieve the current predicted by the Pauli equation. It appears then that the Pauli equation yields only generalized information about the probability current, and that the spin current term is required to include the contributions of spin to the current.
We conclude that we may not follow the same method for calculating the probability current for the Pauli equation as is used in the spin-0 Schrodinger case. The Pauli equation contains insufficient information about spin. Using the more complete Levy-Leblond equation will include the extra contribution of the spin current. The spin current gives a real contribution measurable by experiment.
References
- I. Rabi, “Das freie Elektron im homogenen Magnetfeld nach der Dirachsen Theorie” Zeitschrift fur Physik bd. 507 1928
- J. Levy-Leblond, “Nonrelativistic Particles and Wave Equations,” Commun. Math. Phys. 6, 286-311 (1969).
- L.Landau, “Diamagnetismus der Metalle,” Zeitschrift fur Physik bd. 64 1930
- M. Nowakowski, “The Quantum Mechanical Current of the Pauli Equation,” Am. J. Phys. 67, 916-919 (1999).
- I. Bialynick-Birula, Theory of Quanta, (Oxford University Press, New York, 1992).
- L. Ballentine, Quantum Mechanics, (Prentice Hall, New Jersey, 1990). 108