NMR pp 475-492 | Cite as

Product Operators for Spins 1/2

  • Rainer Kimmich


The evolution of spin coherences, longitudinal spin polarizations, and spin order in the course of pulse sequences can generally be treated on the basis of the density operator formalism (see Chap. 47). After several pulses and evolution periods, however, the result tends to be complex, and it is difficult to pursue the coherence pathways leading to the final signal contributions. Depending on the spin system and the pulse sequence, all sorts of multiple-quantum coherences, longitudinal spin polarization, and spin order states may appear together at a time. What is of interest for the experimenter is that only the single-quantum coherences can be detected as a signal. It is therefore often more favorable to trace the individual coherence pathways from the beginning. All contributions which do not lead to single-quantum coherences in the detection interval can then be omitted from further treatment as soon as this becomes obvious.


Product Operator Density Operator Evolution Rule Reduce Density Operator Coherence Pathway 
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  1. 1.
    For a definition of a “direct product” see Sect. 51.3.Google Scholar
  2. 2.
    As will become obvious below, it is somewhat more convenient to use half the unity operator and to multiply all product operators by a factor of 2 for formal reasons. This of course does not touch the principle of the product operator representation and has no physical meaning. Besides, the unity operator is not explicitly written in practical treatments, but is tacitly assumed with each single operator occurring in expressions for a two-spin system, for instance.Google Scholar
  3. 3.
    Another simplification is the disregard of relaxation which must be distinguished from the evolution processes considered here. The latter refer to spin coherences, longitudinal spin polarizations, and spin order which are affected by rf pulses, chemical shifts, offset fields, and spin couplings.Google Scholar
  4. 4.
    There is an analogy to wave mechanics where it is known that arbitrary wavefunctions can be expressed by linear combinations of orthogonal basis functions provided that these form a complete set (e.g., all eigenfunctions of the system). The counterpart to the Hilbert space in wave mechanics is the Liouville space considered in density operator formalisms. The dimensionality of the Liouville space is, however, equal to the square of that of the Hilbert space for a given system.Google Scholar
  5. 5.
    This is in contrast to the spherical product operators to be described below.Google Scholar
  6. 6.
    The unity operator ε and the direct-product symbol ⊗ have been introduced for completeness and with respect to the didactic matrix representation. In practical product operator treatments these symbols are usually omitted. Note also that the spin-operator products are multiplied by a factor of 2 as usual.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Rainer Kimmich
    • 1
  1. 1.Sektion KernresonanzspektroskopieUniversität UlmUlmGermany

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