Atoms in Crystals and Correlation Diagrams

  • Richard L. Liboff
Part of the Undergraduate Texts in Contemporary Physics book series (UTCP)


Consider an atom with Z protons in its nucleus and with Z outer electrons (the atomic number of the atom is Z). In the central-field approximation, atomic electrons are assumed to be independent of one another. The Hamiltonian of the ith atomic electron is given by
$${H_{i}}({r_{i}}) = \frac{{p_{i}^{2}}}{{2m}} - \frac{{Z{e^{2}}}}{{{r_{i}}}}$$
which has the eigenstates
$${\Psi _{nlm}}({r_i}) = {R_{nl}}({r_i}){Y_{lm}}({\theta _i},{\phi _i}){\xi _i}$$


Point Group Orbital Angular Momentum Young Diagram Principal Quantum Number Symmetry Operation 
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  1. 1.
    See, for example, R.L. Liboff, Introductory Quantum Mechanics, 4th ed. Addison-Wesley, San Francisco, CA (2002), Table 10.3.Google Scholar
  2. 2.
    See, for instance, D.M. Bishop, Group Theory and Chemistry (see Bibliography).Google Scholar
  3. 3.
    For further discussion see, A.F. Cotton, Chemical Applications of Group Theory, 3rd ed., ipid (see Bibliography); D.M. Bishop, Group Theory and Chemistry (see Bibliography); B.N. Figgis, Introduction to Ligand Fields (see Bibliography).Google Scholar
  4. 4.
    For further discussion, see J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York (1999).MATHGoogle Scholar
  5. 5.
    A similar situation occurs in quantum mechanics. The coupled spin states of three electrons are combinations of the tensor forms F ijk = α i β j γ k where α i is, say, the spin state of the ‘α’ electron and (i, j, k) = 1,2. So F ijk is a tensor with r = 3 and n = 2. It is known that antisymmetric coupled spin states of three or more electrons do not exist. [See, R.L. Liboff, Am. J. Physics 52, 561 (1984).]ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2004

Authors and Affiliations

  • Richard L. Liboff
    • 1
  1. 1.School of Electrical EngineeringCornell UniversityIthacaUSA

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