Playing with the Schrödinger Equation

  • Tamás Veszprémi
  • Miklós Fehér


In order to determine the state of a microsystem computationally, we must solve its Schrödinger equation. The first step in this process is to devise the appropriate Hamiltonian operator for the problem at hand. The kinetic energy operator is universal; it depends only on the number of particles in the system. The potential energy function is more characteristic of the system, and the difficulty of solution depends on its form. It is only possible to solve the Schrödinger equation analytically in the case of the simplest potential functions. We will see three such examples in this chapter. Our first example will be the motion of a free particle in space, the simplest of all cases. The second will be that of the harmonic oscillator. In the third case, the particle is in a central Coulomb field, which is the case in the hydrogen atom. This will be discussed in greatest detail because the conclusions from this example are far-reaching for the rest of the book. It is not our intent to present a rigorous derivation of the solutions in these examples; for that, the reader is referred to specialist books on quantum mechanics.


Angular Momentum Quantum Chemistry Harmonic Oscillator Atomic Orbital Eigenvalue Equation 
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Suggested Reading

  1. Eisberg, R., and R. Resnick, Quantum Physics of Atoms, Molecules, Solids,Nuclei and Particles. 2nd ed. Wiley, New York, 1985. Chapters 6–8 are recommended.Google Scholar
  2. Gombás, P. and D. Kisdi, Wave Mechanics and Its Applications. Akademia, Budapest, 1973. This has the detailed solutions of several simple quantum systems.Google Scholar
  3. King, R. B., J. Phys. Chem. 101, 4653–4656 (1997). This is an interesting description of atomic g-and h-orbitals.CrossRefGoogle Scholar
  4. Liboff, R. L., Introductory Quantum Mechanics, 3rd ed. Addison-Wesley, New York, 1998. Detailed examples are given in Chapters 7–11.Google Scholar
  5. Norwood, J. Jr., Twentieth Century Physics. Prentice-Hall, London, 1976. This contains the detailed solutions of harmonic oscillator and the hydrogen atom (Chapters 8 and 9).Google Scholar
  6. Pilar F. L., Elementary Quantum Chemistry. McGraw-Hill, New York, 1990. Contains the solution of the Schödinger equation for some simple systems, such as a free particle, particle in a box, tunnelling, harmonic oscillator, and the rigid rotator (Chapters 5 and 7).Google Scholar
  7. Zare R. N., Angular Momentum: Understanding the Spatial Aspects in Chemistry and Physics. Wiley, New York, 1988. High level text, contains everything on angular momentum.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Tamás Veszprémi
    • 1
  • Miklós Fehér
    • 2
  1. 1.Department of Inorganic ChemistryTechnical University of BudapestBudapestHungary
  2. 2.Research Park CentreNanodesign, Inc.GuelphCanada

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