Operator Techniques in Atomic Spectroscopy

Operator Techniques in Atomic Spectroscopy

BRIAN R. JUDD
Copyright Date: 1998
Pages: 254
https://www.jstor.org/stable/j.ctt7zvhcb
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    Operator Techniques in Atomic Spectroscopy
    Book Description:

    In the 1920s, when quantum mechanics was in its infancy, chemists and solid state physicists had little choice but to manipulate unwieldy equations to determine the properties of even the simplest molecules. When mathematicians turned their attention to the equations of quantum mechanics, they discovered that these could be expressed in terms of group theory, and from group theory it was a short step to operator methods.

    This important development lay largely dormant until this book was originally published in 1963. In this pathbreaking publication, Brian Judd made the operator techniques of mathematicians comprehensible to physicists and chemists. He extended the existing methods so that they could handle heavier, more complex molecules and calculate their energy levels, and from there, it was another short step to the mathematical analysis of spectra. This book provides a first-class introduction to continuous groups for physicists and chemists. Although first written from the perspective of atomic spectroscopy, its major topics and methods will appeal to anyone who has an interest in understanding particle theories of nuclear physics.

    Originally published in 1998.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-6477-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. PREFACE TO THE PAPERBACK EDITION
    (pp. ix-x)
    Brian R. Judd
  4. PREFACE
    (pp. xi-xii)
    Brian R. Judd
  5. 1 CLASSICAL METHODS
    (pp. 1-24)

    All but the lightest atoms are dynamic systems of great complexity. To analyze the properties of such systems, certain assumptions and approximations must first be made. The immediate aim is to simplify the mathematics, but it is important to be guided by physical considerations; for the purpose of the analysis is not just to account for the properties of a particular atom as closely as possible but also to gain insight into its structure and to discern features that are shared by other atomic systems. After the assumptions and approximations have been made, their implications must be rigorously worked out....

  6. 2 CRYSTAL FIELDS
    (pp. 25-53)

    It often happens that the electrons of an ion that is situated in a crystal lattice are sufficiently localized to allow the effect of the environment of the ion to be treated as a perturbation on the system of configurations of the free ion. The absorption and fluorescence spectra of salts for which this situation obtains vary considerably from salt to salt: sometimes, as for crystals containing ions of the rare-earth series, the lines are quite sharp, and the energy level structures of the free ions can be deduced from the spectra. It is fairly simple to arrange matters so...

  7. 3 THE n-j SYMBOLS
    (pp. 54-75)

    We have seen that the coefficient of |j1m1j2m2) in the expansion of |j1j2jm) can be expressed as an algebraic function ofj1,m1,j2,m2,j, andm. This gives a general solution to the problem of passing fromSLJMJtoSMSLMLquantization. It is clear from Sec. 1-6, however, that in general the transition tosmslmlquantization involves the study of coupling schemes of several angular momenta. As a first step toward putting the many-electron problem on an algebraic footing we therefore investigate the coupling of three angular momentaj1,j2, andj3to give a resultant,j. It...

  8. 4 CONFIGURATIONS OF TWO ELECTRONS
    (pp. 76-94)

    The formulas derived in Sec. 3-6, particularly Eq. (3-35) and its three special cases, can be applied to a large number of problems in atomic spectroscopy. To illustrate their use, the configurationf2will be considered in detail, though it will be clear that most of the methods could be illustrated equally well by other configurations comprising two electrons outside closed shells. For many years, 4f2was known only as an excited configuration of LaII (see Condon and Shortley¹), but later work40showed that it also occurs as the ground configuration of PrIV. The ion Pr3+can be studied in...

  9. 5 CONTINUOUS GROUPS
    (pp. 95-153)

    The methods described in the previous chapter, which avoid the explicit introduction of determinantal product states, can be easily extended to any configuration comprising two electrons. However, when systems possessing more than two electrons are examined, the possibility of the occurrence of more than one term with a givenSandLraises the problem of defining a state. Sometimes it is convenient to introduce a particular coupling scheme; forf2d, for example, the state\[\left| {{f}^{2}}SL,sd,{S}'{L}'{{{{M}'}}_{S}}{{{{M}'}}_{L}} \right)\]is completely defined. A description of a state in these terms makes it easy to apply the theory of tensor operators; moreoverS′ and...

  10. 6 SENIORITY
    (pp. 154-165)

    In the preceding chapter, the spin and orbital parts of a many-electron eigenfunction have been treated as far as possible as two separate entities, impinging on each other only at the point where their respective tableaux are constructed to be adjoint. As a preliminary step toward studying the relationship between spin and orbit in greater detail, we introduce a type of operator having well-defined properties with respect to these quantities.

    The operator T(κk), comprising the (2κ+ 1)(2k+ 1) componentsTπq(κk), is said to be a double tensor if it behaves as a tensor of rankκwith respect...

  11. 7 FRACTIONAL PARENTAGE COEFFICIENTS
    (pp. 166-192)

    In the group-theoretical classification of states, the set of quantum numberslnSMSLMLis augmented by the irreducible representations of certain groups, rather than by the nondescript symbolγ. The immediate advantage is that many states can be uniquely specified without going to the lengths of calculating the particular linear combinations of determinantal product states to which they correspond; however, if we are obliged to construct these linear combinations in order to evaluate the matrix elements of the various operators in the Hamiltonian, nothing is gained. The problem of circumventing the explicit introduction of determinantal product states has been given an...

  12. 8 CONFIGURATIONS OF MORE THAN TWO EQUIVALENT ELECTRONS
    (pp. 193-224)

    The coefficients of fractional parentage allow the matrix elements of an operator to be evaluated for any configurationln. In spite of its power, this method does not expose many simple properties that the matrix elements possess; on the contrary, it frequently obscures them. For example, the reduced matrix elements of V(1), being proportional to those of L, are given by a very simple closed expression [see Eq. (7-58)]; yet the general equation for the reduced matrix elements of V(k)[namely, Eq. (7-52)] does not immediately give this expression if we setk= 1. In fact, the lack of...

  13. APPENDIX 1. RADIAL INTEGRALS FOR HYDROGENIC EIGENFUNCTIONS
    (pp. 225-226)
  14. APPENDIX 2. THE COEFFICIENTS (UL|U′L′ + f) AND (WU|W′U′ + f)
    (pp. 227-232)
  15. REFERENCES
    (pp. 233-236)
  16. INDEX
    (pp. 237-242)