Quantum Mechanics of the Hydrogen Atom

  • Hermann Haken
  • Hans Christoph Wolf

Abstract

In this chapter, we shall solve the Schrödinger equation of the hydrogen atom. For our calculations, we will not initially restrict ourselves to the Coulomb potential of the electron in the field of the nucleus of charge Z, V(r) = − Ze 2/(4πε0 r), but rather will use a general potential V(r), which is symmetric with respect to a centre. As the reader may know from the study of classical mechanics, the angular momentum of a particle in a spherically symmetric potential field is conserved; this fact is expressed, for example, in Kepler’s law of areas for the motion of the planets in the solar system. In other words, we know that in classical physics, the angular momentum of a motion in a central potential is a constant as a function of time. This tempts us to ask whether in quantum mechanics the angular momentum is simultaneously measurable with the energy. As a criterion for simultaneous measurability, we know that the angular momentum operators must commute with the Hamiltonian. As we have already noted, the components l x , l y , and l z of the angular momentum l are not simultaneously measurable; on the other hand, l z and l 2, for example, are simultaneously measurable. A long but straightforward calculation reveals that these two operators also commute with the Hamiltonian for a central-potential problem. Since the details of this calculation do not provide any new physical insights, we shall not repeat it here.

Keywords

Dinates sinO 

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Bibliography

  1. J. Avery: Quantum Theory of Atoms, Molecules, and Photons ( McGraw-Hill, New York 1972 )Google Scholar
  2. H. A. Bethe, E. F. Salpeter: The Quantum Mechanics of One-and Two-Electron Atoms ( Plenum, New York 1977 )CrossRefGoogle Scholar
  3. W. A. Blanpied: Modern Physics: An Introduction to Its Mathematical Language (Holt, Rinehart Winston, New York 1971 )Google Scholar
  4. C. Cohen-Tannoudji, B. Diu, F. Laloë: Quantum Mechanics I and II, 2nd ed. ( Wiley, New York 1977 )Google Scholar
  5. K. Gottfried: Quantum Mechanics, Vol. I ( Benjamin, New York and Amsterdam 1966 )Google Scholar
  6. L. D. Landau, E. M. Lifshitz: Quantum Mechanics, 3rd ed. ( Pergamon, Oxford 1982 )Google Scholar
  7. J. L. Martin: Basic Quantum Mechanics (Oxford University Press, Oxford 1981 )MATHGoogle Scholar
  8. E. Merzbacher: Quantum Mechanics, 2nd ed. ( Wiley, New York 1969 )Google Scholar
  9. A. Messiah: Quantum Mechanics I and II (Halsted, New York 1961 and 1962 )Google Scholar
  10. L. I. Schiff: Quantum Mechancis, 3rd ed. ( McGraw-Hill, New York 1968 )Google Scholar
  11. F. Schwabl: Quantum Mechanics (Springer, Berlin, Heidelberg 1992 )Google Scholar
  12. R. Shankar: Principles of Quantum Mechanics ( Plenum, New York 1980 )Google Scholar
  13. E. H. Wichmann: Quantum Physics (Vol. IV of Berkeley Physics Course) ( McGraw-Hill, New York 1971 )Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Hermann Haken
    • 1
  • Hans Christoph Wolf
    • 2
  1. 1.Institut für Theoretische PhysikUniversität StuttgartStuttgartGermany
  2. 2.Physikalisches InstitutUniversität StuttgartStuttgartGermany

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