Philip Du Toit and Dr. Jean-Francois Van Huele, Physics and Astronomy
Introductory quantum mechanics texts seldom treat the quantum mechanical probability current for nonrelativistic particles with spin. The procedure for calculating probability current is most often presented in relation to particles described by the Schrödinger equation, that is particles without spin.1 In later chapters, after having introduced readers to the Pauli equation and its description of particles with spin, the authors rarely return to the topic of probability current. This omission may cause readers to incorrectly infer that the calculation of the probability current for particles with spin follows the same line of argument as is used for particles without spin.
A closer inspection of the derivation of the probability current for particles with spin reveals that adopting the same procedure as is used for particles without spin gives rise to a nontrivial ambiguity. Indeed, we discover that applying the standard procedure to the Pauli equation fails to determine the probability current uniquely. We may add an extra term to the resultant expression for the probability current with impunity, and still satisfy the continuity equation. This result is intolerable since the probability current is physically measurable and must be uniquely defined. We must conclude that the standard approach used for calculating the probability current of the Schrödinger equation is incapable of deriving the unique probability current for particles with spin.
In a recent paper submitted to American Journal of Physics, Marek Nowakowski addresses this ambiguity.2 Nowakowski argues that a nonrelativistic reduction of the probability current for relativistic particles with spin reveals the correct and unique expression for the probability current of nonrelativistic particles with spin. His calculations show that the spin of the particle does indeed introduce an extra term to the probability current. This extra term is called the spin current: the contribution to the probability current that arises due to the spin of the particle.
Nowakowski’s method for deriving the extra spin current term requires the use of relativistic quantum mechanics. Indeed, he regards this approach as imperative in order to resolve the ambiguity:
This ambiguity cannot be resolved by means of nonrelativistic quantum mechanics alone. Or, to put it differently, this ambiguity only appears from the point of view of nonrelativistic quantum mechanics. (…) Hence, there is a priori no way to decide from the point of view of nonrelativistic quantum mechanics whether a term … should be added to the [probability current] or not. Nowakowski 99
The purpose of the present research is to derive the unique probability current, including the spin current term, for nonrelativistic particles with spin without appealing to a relativistic theory. Such a derivation highlights the fact that spin current is not a relativistic effect. Having established the nonrelativistic nature of spin current, the research investigated the properties and behavior of spin current. Specifically, we investigated the contributions of the extra spin current term for the case of an electron in a homogeneous magnetic field.
During the course of the research we derived the probability current for an electron using the Levy-Leblond equation. The Levy-Leblond equation describes the wave functions of nonrelativistic particles with spin. It differs, however, from the Pauli equation in that spin is an intrinsic property of the equation and not a hand-placed constraint. The Levy-Leblond equation is obtained by factorizing the Schrödinger equation.
The resultant probability current for the Levy-Leblond equation indicates, without appealing to relativistic theory, that the unique expression for the probability current of nonrelativistic particles with spin must indeed include an extra spin current term.
The derivation of the probability current using the Levy-Leblond equation demonstrates that spin current is a non-relativistic effect, and can be derived without appealing to relativistic theory.
An analysis of the spin current for an electron in a homogeneous magnetic field reveals that the spin property, although it cannot contribute to the momentum of the particle, produces interesting interaction effects with the Pauli current. The swirl of the spin current introduces a nontrivial contribution that is potentially measurable by experiment. Specifically, the spin current contribution is dependent on the spin state of the electron.
References
- See for example a text by D. Griffiths titled Introduction to Quantum Mechanics.
- M. Nowakowski, “The quantum mechanical current of the Pauli equation,” Am. J. Phys. 67, 916-919 (1999).