Harmonic Oscillator

  • Alessandro TetaEmail author
Part of the UNITEXT for Physics book series (UNITEXTPH)


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.


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© 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

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