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Lie Algebraic Methods and Their Applications to Simple Quantum Systems

Elsevier Science & Technology
DOI: 10.1016/s0065-3276(08)60613-9
  • Mathematics
  • Physics


Publisher Summary The algebraic methods were first introduced in the context of the new matrix mechanics around 1925. The importance of the concept of angular momentum in quantum mechanics, in contrast to classical mechanics was soon recognized and the necessary formalism was developed principally by Wigner, Weyl, and Racah. The algebraic treatment of the angular momentum can be found nowdays in almost every textbook on quantum mechanics, often in parallel with the differential equation approach. In contrast, this cannot be said about another basic quantum mechanical system, namely the hydrogen atom, which is almost universally treated by the latter approach only. In the midst of various attempts to solve this difficult problem, the elementary particle physicists examined several noncompact Lie algebras in the mid-l950s, hoping that these physicists can provide a clue to the classification of elementary particles. A close relationship of the angular momentum and the so(3) algebra dates certainly to the prequantum mechanics era, while the realization that SO(4) is the symmetry group of the Kepler problem was first demonstrated by Fock. Soon afterward Bargmann showed that the generators of Fock's SO(4) transformations were the angular momentum and Runge–Lenz vectors.

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