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Cambridge University Press eBooks

How Pauli's rule became the exclusion principle: from Fermi–Dirac statistics to the spin–statistics theorem

Michela Massimi

Published Aug 4, 2005• 0 citations
Publisher

Abstract

Shortly after their introduction, the electron's Zweideutigkeit and Pauli's Ausschliessungsregel were embedded into a growing theoretical framework: from Fermi–Dirac statistics in 1926 (Section 4.2), to the non-relativistic quantum mechanics of the magnetic electron with Pauli's spin matrices in 1927 (Section 4.3); from Wigner and von Neumann's group theoretical derivation of the spin matrices in 1927 (Section 4.4), to Jordan's reinterpretation of Pauli's exclusion principle in terms of anticommutation relations for particle creation and annihilation operators in 1928 (Section 4.5), which paved the way for quantum field theory. The most important step in building up this theoretical framework was the transition from non-relativistic to relativistic quantum mechanics, with Dirac's equation for the electron in 1928 (Section 4.6), which finally allowed the derivation of the electron's spin with its anomalous magnetic moment and hence clarified in a conclusive way the origin of spectroscopic anomalies. The negative energy solutions of the Dirac equation, and Dirac's attempt to accommodate them via the hole theory in 1930, anticipated the experimental discovery of the antiparticle of the electron, the positron. Sections 4.7-4.9 reconstruct Pauli's sustained criticism of Dirac's hole theory. The history of this debate is intertwined with Pauli's search for a spin–statistics connection, which finally culminated in the spin–statistics theorem in 1940. With this theorem, the nomological shift from the status of a phenomenological rule to that of a fundamental scientific principle is finally completed. Introduction The year 1925 was the annus mirabilis for quantum mechanics.

DOI
10.1017/cbo9780511535352.008

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The origins of the exclusion principle: an extremely natural prescriptive rule

Michela Massimi

The exclusion principle was the final outcome of Pauli's struggle to understand some spectroscopic anomalies in the early 1920s: doublets were observed in the spectra of alkali metals, singlets and triplets in the spectra of the alkaline earths, and even more anomalous patterns were observed when chemical elements were placed in an external magnetic field (anomalous Zeeman effect and Paschen–Back effect). These anomalous spectra challenged the old quantum theory, and prompted a radical theoretical change (Section 2.1). From 1920 to 1924 Alfred Landé, Werner Heisenberg, and Niels Bohr were all engaged in trying to save the traditional spectroscopic model (the so-called atomic core model) and to reconcile it with the observed anomalies. The impasse was solved only with Pauli's introduction of a fourth degree of freedom for the electron, and the consequent demise of the atomic core model (Section 2.2). What Pauli called the ‘twofoldness’ [ Zweideutigkeit ] of the electron's angular momentum was soon reinterpreted as the electron's spin (Section 2.3). Pauli's exclusion rule was announced in this semi-classical spectroscopic context that characterized the revolutionary transition from the old quantum theory to the new quantum theory around 1925. The prehistory of Pauli's exclusion principle Atomic spectra and the Bohr–Sommerfeld theory of atomic structure The existence of spectral lines had been known to scientists since the beginning of the nineteenth century when Wollaston and Fraunhofer first observed the dark absorption lines in the spectrum of the Sun.

Cambridge University Press eBooks•Aug 4, 2005•1 citation

History of Spin and Statistics

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These lectures were given in the framework of the ``Dixième séminaire rhodanien de physique'' entitled ``Le spin en physique'', given at Villa Gualino, Turin, March 2002. We have shown how the difficulties of interpretation of atomic spectra led to the Pauli exclusion principle and to the notion of spin, and then described the following steps: the Pauli spin with 2$\times$2 matrices after the birth of "new" quantum mechanics, the Dirac equation and the magnetic moment of the electron, the spins and magnetic moments of other particles, proton, neutron and hyperons. Finally, we show the crucial role of statistics in the stability of the world.

arXiv:hep-ph/0209068•Sep 6, 2002•4 citations

Introduction

Michela Massimi

The history of the exclusion principle is already an old one, but its conclusion has not yet been written. It is now eighty years since Wolfgang Pauli introduced an ‘extremely natural’ prescriptive rule, while dealing with some spectroscopic anomalies that beset physicists in the heyday of the old quantum theory. The rule excluded the possibility that any two bound electrons in an atom were in the same dynamic state, identified by a set of four quantum numbers. Hence the name of Ausschlieβungsregel (exclusion rule), or Pauli's Verbot (Pauli's veto) as Werner Heisenberg nicknamed it. The far-reaching physical significance of this rule became clear only later. From spectroscopy to atomic physics, from quantum field theory to high-energy physics, there is hardly another scientific principle that has more far-reaching implications than Pauli's exclusion principle. It is thanks to Pauli's principle that one obtains the electronic configurations underlying the classification of chemical elements in Mendeleev's periodic table as well as atomic spectra. To this same principle we credit the statistical behaviour of any half-integral spin particles (protons, neutrons, among many others) and the stability of matter. Shifting to high-energy physics, it is the exclusion principle that fixes the crucial constraint for binding quarks in hadrons, which together with leptons compose our physical world. This book advances a philosophical analysis of the enduring and far-reaching validity of Pauli's principle. It does not aim to address what a scientific principle is.

Cambridge University Press eBooks•Aug 4, 2005•2 citations

The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure

Alex De Giuseppe

The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure For nearly a century, the Pauli exclusion principle (PEP) and the spin-statistics theorem have stood as foundational postulates of quantum mechanics and relativistic quantum field theory. The PEP dictates that no two identical fermions can occupy the same quantum state, determining the structure of atoms, the stability of matter, and the properties of neutron stars and white dwarfs. The spin-statistics theorem connects the intrinsic spin of a particle to its statistical behavior: half-integer spin particles (fermions) obey Fermi-Dirac statistics, while integer spin particles (bosons) obey Bose-Einstein statistics. Despite their universal success, these principles have never been derived from a more primitive geometric or topological principle — until now. This paper demonstrates that both the Pauli exclusion principle and the spin-statistics theorem emerge naturally and rigorously from a single geometric fact: the **non-injectivity of a timelike worldline** for Lorentz factors exceeding a critical threshold, \(\gamma > \gamma_{\text{crit}}\). In this regime, a single worldline intersects a constant-time hypersurface \(\Sigma_t\) in \(N > 1\) distinct spatial points, generating a multi‑sheet spacetime structure. This phenomenon, known as **worldline non-injectivity**, is the foundation of the TPST–DGQ framework. **Key results and logical flow:** 1. **Multi‑sheet structure for a single particle:** For \(\gamma > \gamma_{\text{crit}}\), a particle's worldline produces \(N(\epsilon) \sim \epsilon^{-(d-2)}\) topological sheets, each carrying the same physical properties (mass, charge, spin) via the Ontological Identity Principle. 2. **Composite worldline for two identical particles:** By combining two identical particles into a composite worldline in configuration space, we obtain a two‑particle sheet structure. Identical particles share isomorphic sheet structures, reducing the \(N \times N\) sheet pairs to \(N\) independent pairs. 3. **Exchange as a permutation of sheet blocks:** The exchange operator \(P_{12}\) swaps the sheet blocks associated with the two particles. This permutation is a topological operation in the two‑particle configuration space. 4. **Exchange phase from winding number:** The exchange path corresponds to half of a complete winding cycle around the fold structure of the composite worldline. The accumulated topological phase is \(\Phi_{\text{exchange}} = \pi w\), where \(w\) is the **winding number** of the worldline. Hence the exchange phase factor is \(\eta = e^{i\pi w} = (-1)^w\). 5. **Spin from winding number:** The winding number \(w\) is directly related to the spin \(s\) by the requirement that a \(4\pi\) rotation returns the physical state to itself (single‑valuedness). This yields \(w = 2s\). Therefore \(\eta = (-1)^{2s}\): half‑integer spin gives \(\eta = -1\) (fermions, antisymmetric), integer spin gives \(\eta = +1\) (bosons, symmetric). This is the spin‑statistics theorem derived from geometry. 6. **Pauli exclusion principle:** For two identical fermions (\(\eta = -1\)) at spatial coincidence, the two‑particle amplitude satisfies \(\mathcal{A} = \eta\mathcal{A}\), forcing \(\mathcal{A} = 0\). Thus no two fermions can occupy the same quantum state. The PEP is no longer a postulate; it is a theorem. 7. **Rapidity as inter‑sheet phase measure:** The rapidity \(\zeta = \operatorname{artanh}(v/c)\) is shown to be a monotonic function of the inter‑sheet phase difference \(\Phi_n\). The non‑injectivity threshold \(\gamma > \gamma_{\text{crit}}\) corresponds to \(\zeta > \zeta_{\text{crit}}\). The speed of light \(c\) emerges as the asymptotic limit where the inter‑sheet phase diverges, making \(v = c\) unreachable for massive particles. 8. **Universal cancellation identity:** The same algebraic identity \(N(\epsilon)\cdot\epsilon^{d-2} = O(1)\) that regularizes holographic entropy, Coulomb self‑energy, wavefunction normalization, electromagnetic fields, and the cosmological constant also regularizes the two‑particle exchange amplitude — zero for fermions, finite for bosons. Quantum statistics thus joins the list of phenomena unified under the principle of worldline non‑injectivity. **Implications:** This work completes the derivation of the foundations of quantum mechanics within the TPST–DGQ framework, following previous derivations of the Born rule, the Schrödinger equation, Planck’s constant, and wavefunction collapse. It also establishes a geometric origin for the spin‑statistics connection, a result that in standard physics relies on the heavy machinery of relativistic quantum field theory (unitarity, causality, positivity of energy). Here, it follows from the topology of worldline folds. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created.

•Apr 1, 2026•0 citations

Toward an understanding of the spin-statistics theorem

Ian Duck, E. C. G. Sudarshan

We respond to a recent request from Neuenschwander for an elementary proof of the Spin-Statistics Theorem. First, we present a pedagogical discussion of the results for the spin-0 Klein–Gordon field quantized according to Bose–Einstein statistics; and for the spin-12 Dirac field quantized according to Fermi–Dirac statistics and the Pauli Exclusion Principle. This discussion is intended to make our paper accessible to students familiar with the matrix solution of the quantum harmonic oscillator. Next, we discuss a number of candidate intuitive proofs and conclude that none of them pass muster. The reasons for their shortcomings are fully discussed. Then we discuss an argument, originally suggested by Sudarshan, which proves the theorem with a minimal set of requirements. Although we use Lorentz invariance in a specific and limited part of the argument, we do not need the full complexity of relativistic quantum field theory. Motivated by our particular use of Lorentz invariance, if we are permitted to elevate the conclusion of flavor symmetry (which we explain in the text) to the status of a postulate, one could recast our proof without any explicit relativistic assumptions, and thus make it applicable even in the nonrelativistic context. Such an argument, presented in the text, sheds some light on why it is that the ordinary Schrödinger field, considered strictly in the nonrelativistic context, seems to be quantizable with either statistics. Finally, an argument starting with ordinary-number valued (commuting), and with Grassmann-valued (anticommuting), oscillators shows in a natural way that these must relativistically embed into Klein–Gordon spin-0 and Dirac spin-12 fields, respectively. In this way, the Spin-Statistics Theorem is understood at the expense of admitting the existence of the simplest Grassmann-valued field.

American Journal of Physics•Apr 1, 1998•107 citations

Electron exchange-correlation in quantum mechanics

Burke Ritchie

It is shown that Fermi-Dirac statistics is guaranteed by the Dirac current, from which spin-dependent quantum velocity fields and spin-dependent quantum trajectories can be inferred. Pauli's exclusion principle is demonstrated using the spin-dependent quantum trajectories. The Dirac current, unlike the Schroedinger current, is nonzero for stationary bound states due to the permanent magnetic moment of the electron. It is of order c{sup 0} in agreement with observation that Fermi-Dirac statistics is independent of electronic velocity. In summary the physical basis for exchange-correlation is found in Dirac's equation, although Schroedinger's equation may be used to evaluate the Dirac current in the nonrelativistic regime of electronic velocity.

•Jan 30, 2009•0 citations
Shortly after their introduction, the electron's Zweideutigkeit and Pauli's Ausschliessungsregel were embedded into a growing theoretical framework: from Fermi–Dirac statistics in 1926 (Section 4.2), to the non-relativistic quantum mechanics of the magnetic electron with Pauli's spin matrices in 1927 (Section 4.3); from Wigner and von Neumann's group theoretical derivation of the spin matrices in 1927 (Section 4.4), to Jordan's reinterpretation of Pauli's exclusion principle in terms of anticommutation relations for particle creation and annihilation operators in 1928 (Section 4.5), which paved the way for quantum field theory. The most important step in building up this theoretical framework was the transition from non-relativistic to relativistic quantum mechanics, with Dirac's equation for the electron in 1928 (Section 4.6), which finally allowed the derivation of the electron's spin with its anomalous magnetic moment and hence clarified in a conclusive way the origin of spectroscopic anomalies. The negative energy solutions of the Dirac equation, and Dirac's attempt to accommodate them via the hole theory in 1930, anticipated the experimental discovery of the antiparticle of the electron, the positron. Sections 4.7-4.9 reconstruct Pauli's sustained criticism of Dirac's hole theory. The history of this debate is intertwined with Pauli's search for a spin–statistics connection, which finally culminated in the spin–statistics theorem in 1940. With this theorem, the nomological shift from the status of a phenomenological rule to that of a fundamental scientific principle is finally completed. Introduction The year 1925 was the annus mirabilis for quantum mechanics.
The exclusion principle was the final outcome of Pauli's struggle to understand some spectroscopic anomalies in the early 1920s: doublets were observed in the spectra of alkali metals, singlets and triplets in the spectra of the alkaline earths, and even more anomalous patterns were observed when chemical elements were placed in an external magnetic field (anomalous Zeeman effect and Paschen–Back effect). These anomalous spectra challenged the old quantum theory, and prompted a radical theoretical change (Section 2.1). From 1920 to 1924 Alfred Landé, Werner Heisenberg, and Niels Bohr were all engaged in trying to save the traditional spectroscopic model (the so-called atomic core model) and to reconcile it with the observed anomalies. The impasse was solved only with Pauli's introduction of a fourth degree of freedom for the electron, and the consequent demise of the atomic core model (Section 2.2). What Pauli called the ‘twofoldness’ [ Zweideutigkeit ] of the electron's angular momentum was soon reinterpreted as the electron's spin (Section 2.3). Pauli's exclusion rule was announced in this semi-classical spectroscopic context that characterized the revolutionary transition from the old quantum theory to the new quantum theory around 1925. The prehistory of Pauli's exclusion principle Atomic spectra and the Bohr–Sommerfeld theory of atomic structure The existence of spectral lines had been known to scientists since the beginning of the nineteenth century when Wollaston and Fraunhofer first observed the dark absorption lines in the spectrum of the Sun.
These lectures were given in the framework of the ``Dixième séminaire rhodanien de physique'' entitled ``Le spin en physique'', given at Villa Gualino, Turin, March 2002. We have shown how the difficulties of interpretation of atomic spectra led to the Pauli exclusion principle and to the notion of spin, and then described the following steps: the Pauli spin with 2$\times$2 matrices after the birth of "new" quantum mechanics, the Dirac equation and the magnetic moment of the electron, the spins and magnetic moments of other particles, proton, neutron and hyperons. Finally, we show the crucial role of statistics in the stability of the world.
The history of the exclusion principle is already an old one, but its conclusion has not yet been written. It is now eighty years since Wolfgang Pauli introduced an ‘extremely natural’ prescriptive rule, while dealing with some spectroscopic anomalies that beset physicists in the heyday of the old quantum theory. The rule excluded the possibility that any two bound electrons in an atom were in the same dynamic state, identified by a set of four quantum numbers. Hence the name of Ausschlieβungsregel (exclusion rule), or Pauli's Verbot (Pauli's veto) as Werner Heisenberg nicknamed it. The far-reaching physical significance of this rule became clear only later. From spectroscopy to atomic physics, from quantum field theory to high-energy physics, there is hardly another scientific principle that has more far-reaching implications than Pauli's exclusion principle. It is thanks to Pauli's principle that one obtains the electronic configurations underlying the classification of chemical elements in Mendeleev's periodic table as well as atomic spectra. To this same principle we credit the statistical behaviour of any half-integral spin particles (protons, neutrons, among many others) and the stability of matter. Shifting to high-energy physics, it is the exclusion principle that fixes the crucial constraint for binding quarks in hadrons, which together with leptons compose our physical world. This book advances a philosophical analysis of the enduring and far-reaching validity of Pauli's principle. It does not aim to address what a scientific principle is.
The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure For nearly a century, the Pauli exclusion principle (PEP) and the spin-statistics theorem have stood as foundational postulates of quantum mechanics and relativistic quantum field theory. The PEP dictates that no two identical fermions can occupy the same quantum state, determining the structure of atoms, the stability of matter, and the properties of neutron stars and white dwarfs. The spin-statistics theorem connects the intrinsic spin of a particle to its statistical behavior: half-integer spin particles (fermions) obey Fermi-Dirac statistics, while integer spin particles (bosons) obey Bose-Einstein statistics. Despite their universal success, these principles have never been derived from a more primitive geometric or topological principle — until now. This paper demonstrates that both the Pauli exclusion principle and the spin-statistics theorem emerge naturally and rigorously from a single geometric fact: the **non-injectivity of a timelike worldline** for Lorentz factors exceeding a critical threshold, γ>γcrit\gamma > \gamma_{\text{crit}}γ>γcrit​. In this regime, a single worldline intersects a constant-time hypersurface Σt\Sigma_tΣt​ in N>1N > 1N>1 distinct spatial points, generating a multi‑sheet spacetime structure. This phenomenon, known as **worldline non-injectivity**, is the foundation of the TPST–DGQ framework. **Key results and logical flow:** 1. **Multi‑sheet structure for a single particle:** For γ>γcrit\gamma > \gamma_{\text{crit}}γ>γcrit​, a particle's worldline produces N(ϵ)∼ϵ−(d−2)N(\epsilon) \sim \epsilon^{-(d-2)}N(ϵ)∼ϵ−(d−2) topological sheets, each carrying the same physical properties (mass, charge, spin) via the Ontological Identity Principle. 2. **Composite worldline for two identical particles:** By combining two identical particles into a composite worldline in configuration space, we obtain a two‑particle sheet structure. Identical particles share isomorphic sheet structures, reducing the N×NN \times NN×N sheet pairs to NNN independent pairs. 3. **Exchange as a permutation of sheet blocks:** The exchange operator P12P_{12}P12​ swaps the sheet blocks associated with the two particles. This permutation is a topological operation in the two‑particle configuration space. 4. **Exchange phase from winding number:** The exchange path corresponds to half of a complete winding cycle around the fold structure of the composite worldline. The accumulated topological phase is Φexchange=πw\Phi_{\text{exchange}} = \pi wΦexchange​=πw, where www is the **winding number** of the worldline. Hence the exchange phase factor is η=eiπw=(−1)w\eta = e^{i\pi w} = (-1)^wη=eiπw=(−1)w. 5. **Spin from winding number:** The winding number www is directly related to the spin sss by the requirement that a 4π4\pi4π rotation returns the physical state to itself (single‑valuedness). This yields w=2sw = 2sw=2s. Therefore η=(−1)2s\eta = (-1)^{2s}η=(−1)2s: half‑integer spin gives η=−1\eta = -1η=−1 (fermions, antisymmetric), integer spin gives η=+1\eta = +1η=+1 (bosons, symmetric). This is the spin‑statistics theorem derived from geometry. 6. **Pauli exclusion principle:** For two identical fermions (η=−1\eta = -1η=−1) at spatial coincidence, the two‑particle amplitude satisfies A=ηA\mathcal{A} = \eta\mathcal{A}A=ηA, forcing A=0\mathcal{A} = 0A=0. Thus no two fermions can occupy the same quantum state. The PEP is no longer a postulate; it is a theorem. 7. **Rapidity as inter‑sheet phase measure:** The rapidity ζ=artanh⁡(v/c)\zeta = \operatorname{artanh}(v/c)ζ=artanh(v/c) is shown to be a monotonic function of the inter‑sheet phase difference Φn\Phi_nΦn​. The non‑injectivity threshold γ>γcrit\gamma > \gamma_{\text{crit}}γ>γcrit​ corresponds to ζ>ζcrit\zeta > \zeta_{\text{crit}}ζ>ζcrit​. The speed of light ccc emerges as the asymptotic limit where the inter‑sheet phase diverges, making v=cv = cv=c unreachable for massive particles. 8. **Universal cancellation identity:** The same algebraic identity N(ϵ)⋅ϵd−2=O(1)N(\epsilon)\cdot\epsilon^{d-2} = O(1)N(ϵ)⋅ϵd−2=O(1) that regularizes holographic entropy, Coulomb self‑energy, wavefunction normalization, electromagnetic fields, and the cosmological constant also regularizes the two‑particle exchange amplitude — zero for fermions, finite for bosons. Quantum statistics thus joins the list of phenomena unified under the principle of worldline non‑injectivity. **Implications:** This work completes the derivation of the foundations of quantum mechanics within the TPST–DGQ framework, following previous derivations of the Born rule, the Schrödinger equation, Planck’s constant, and wavefunction collapse. It also establishes a geometric origin for the spin‑statistics connection, a result that in standard physics relies on the heavy machinery of relativistic quantum field theory (unitarity, causality, positivity of energy). Here, it follows from the topology of worldline folds. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created.
We respond to a recent request from Neuenschwander for an elementary proof of the Spin-Statistics Theorem. First, we present a pedagogical discussion of the results for the spin-0 Klein–Gordon field quantized according to Bose–Einstein statistics; and for the spin-12 Dirac field quantized according to Fermi–Dirac statistics and the Pauli Exclusion Principle. This discussion is intended to make our paper accessible to students familiar with the matrix solution of the quantum harmonic oscillator. Next, we discuss a number of candidate intuitive proofs and conclude that none of them pass muster. The reasons for their shortcomings are fully discussed. Then we discuss an argument, originally suggested by Sudarshan, which proves the theorem with a minimal set of requirements. Although we use Lorentz invariance in a specific and limited part of the argument, we do not need the full complexity of relativistic quantum field theory. Motivated by our particular use of Lorentz invariance, if we are permitted to elevate the conclusion of flavor symmetry (which we explain in the text) to the status of a postulate, one could recast our proof without any explicit relativistic assumptions, and thus make it applicable even in the nonrelativistic context. Such an argument, presented in the text, sheds some light on why it is that the ordinary Schrödinger field, considered strictly in the nonrelativistic context, seems to be quantizable with either statistics. Finally, an argument starting with ordinary-number valued (commuting), and with Grassmann-valued (anticommuting), oscillators shows in a natural way that these must relativistically embed into Klein–Gordon spin-0 and Dirac spin-12 fields, respectively. In this way, the Spin-Statistics Theorem is understood at the expense of admitting the existence of the simplest Grassmann-valued field.
It is shown that Fermi-Dirac statistics is guaranteed by the Dirac current, from which spin-dependent quantum velocity fields and spin-dependent quantum trajectories can be inferred. Pauli's exclusion principle is demonstrated using the spin-dependent quantum trajectories. The Dirac current, unlike the Schroedinger current, is nonzero for stationary bound states due to the permanent magnetic moment of the electron. It is of order c{sup 0} in agreement with observation that Fermi-Dirac statistics is independent of electronic velocity. In summary the physical basis for exchange-correlation is found in Dirac's equation, although Schroedinger's equation may be used to evaluate the Dirac current in the nonrelativistic regime of electronic velocity.