Harmonic Oscillator

Chapter
Part of the UNITEXT for Physics book series (UNITEXTPH)

Abstract

The quantum harmonic oscillator is an important, explicitly solvable model for a particle subject to a confining potential. Using the algebraic method based on creation and annihilation operators, we construct the Hamiltonian and describe the spectrum made only of positive eigenvalues. We also discuss the main properties of the time evolution and, for an initial Gaussian state, we explicitly show that the solution of the Schrödinger equation is again a Gaussian state centered on the classical motion, with mean square deviation of position and momentum given by bounded and periodic functions of the time. Finally, we briefly discuss the dynamics of a particle subject to a constant magnetic field.

References

  1. 1.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, 3rd edn. Pergamon Press, Oxford (1977)MATHGoogle Scholar
  2. 2.
    Teschl, G.: Mathematical Methods in Quantum Mechanics. American Mathematical Society, Providence (2009)Google Scholar
  3. 3.
    Thaller, B.: Visual Quantum Mechanics. Springer, New York (2000)MATHGoogle Scholar
  4. 4.
    Lebedev, N.N.: Special Functions and Their Applications. Dover Publications, New York (1972)MATHGoogle Scholar
  5. 5.
    Hagedorn, G.: Raising and lowering operators for semiclassical wave packets. Ann. Phys. 269, 77–104 (1998)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoUniversità degli Studi di Roma “La Sapienza”RomeItaly

Personalised recommendations