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American Journal of Physics

Terms Obtained from Configurations of Equivalent Electrons

E. R. Tuttle

Published Jan 1, 1967• 2 citations
Publisher

Abstract

A simple method is given for finding the terms obtained from a configuration containing a number of equivalent electrons. This method is based entirely upon two facts: (1) the Pauli principle prevents any two electrons from having the same quantum numbers; and, (2) one and only one completely antisymmetric wavefunction can be constructed from any given set of allowed quantum numbers. Because of its simplicity, both in concept and in application, this system is well suited to presentation to undergraduate students of modern physics.

DOI
10.1119/1.1973853

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A simple method is given for finding the terms obtained from a configuration containing a number of equivalent electrons. This method is based entirely upon two facts: (1) the Pauli principle prevents any two electrons from having the same quantum numbers; and, (2) one and only one completely antisymmetric wavefunction can be constructed from any given set of allowed quantum numbers. Because of its simplicity, both in concept and in application, this system is well suited to presentation to undergraduate students of modern physics.
A simple method is given for finding the values of the total angular momentum J for configurations of equivalent electrons in jj coupling. It is also shown how a knowledge of both the LS and jj terms can contribute to the student’s understanding of the role of different representations in elementary quantum mechanics. Because the principles and the computations required are elementary, this material is particularly suitable for presentation to undergraduates and to graduate students taking a first course in quantum mechanics.
The vector model is examined in the light of the quantum theory of wave fields. First the distributions of electrons among the electronic states are limited to correspond to the atomic states arising from one electronic configuration. A set of independent angular momentum vectors 11{1}_{1}11​, 12{1}_{2}12​,...S1{\mathrm{S}}_{1}S1​, S2{\mathrm{S}}_{2}S2​... is found, in which one orbital and one spin vector are associated with each group of electronic states, rather than individual electrons, having given values of the quantum numbers nnn and lll. Any other vector T may be decomposed into a sum of independent vectors ti{\mathrm{t}}_{i}ti​, one associated with each of the same group of electronic states. The usual commutation relations obtain among the vectors ti{\mathrm{t}}_{i}ti​, li{\mathrm{l}}_{i}li​ and si{\mathrm{s}}_{i}si​. On further limiting the distributions to correspond to atomic states arising from electronic configurations containing no equivalent electrons, it is shown that the matrix of ${\ensuremath{\Sigma}}_{k}f({{\mathrm{l}}_{k}}^{0}, {{\mathrm{s}}_{k}}^{0})$ (an operator in coordinate space in which fff is any algebraic function of the components of 1 and s) can be constructed from the vectors 11{1}_{1}11​, 12{1}_{2}12​...S1{\mathrm{S}}_{1}S1​, S2{\mathrm{S}}_{2}S2​,... according to the usual laws of matrix mechanics.
The Pauli exclusion principle is a key postulate of the quantum theory and informs much of what we know about matter. In terms of electronic structure, the lone, deceptively simple mathematical requirement is that the total wave function be antisymmetric with respect to the exchange of any two electrons. However, visualizing the effect on the electron distribution presents a formidable challenge to students because of the complexity and number of variables involved. Approaches to gaining insight developed here include a quantitative analysis of two-dimensional configuration spaces as well as a qualitative assessment of interference patterns. For 2p x 1 2p y 1 states as representative examples, the analysis reveals that the Pauli principle directs electrons with opposing spins to codistribute along axes passing through the nucleus in the plane of the occupied orbitals. On the other hand, electrons with the same spin are never simultaneously in the same place or on the same axis; instead, they preferentially distribute in perpendicular directions.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.
A general method of obtaining many-electron wave functions which are eigenfunctions of ${\stackrel{^}{S}}^{2}$ and which also satisfy Pauli's principle is developed. This method is based on the properties of the irreducible units of the group algebra of the symmetric group. Using this method, expansions of the exact wave function are found which will in succeeding articles form the basis of a powerful method of obtaining accurate and quite useful many-electron wave functions for atoms, molecules, and solids.
Abstract Quantum mechanics (QM) forms the most crucial ingredient of modern-era physical science curricula at undergraduate level. The abstract ideas involved in QM related concepts pose a challenge towards appropriate visualization as a consequence of their counter-intuitive nature and lack of experiment-assisted visualization tools. At the heart of the quantum mechanical formulation lies the concept of ‘wavefunction’, which forms the basis for understanding the behavior of physical systems. At undergraduate level, the concept of ‘wavefunction’ is introduced in an abstract framework using mathematical tools and therefore opens up an enormous scope for alternative conceptions and erroneous visualization. The present work is an attempt towards exploring the visualization models constructed by undergraduate students for appreciating the concept of ‘wavefunction’. We present a qualitative analysis of the data obtained from administering a questionnaire containing four visualization based questions on the topic of ‘wavefunction’ to a group of ten undergraduate-level students at an institute in India which excels in teaching and research of basic sciences. Based on the written responses, all ten students were interviewed in detail to unravel the exact areas of difficulty in visualization of ‘wavefunction’. The outcome of present study not only reveals the gray areas in students’ conceptualization, but also provides a plausible route to address the issues at the pedagogical level within the classroom.