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.
- 2.Teschl, G.: Mathematical Methods in Quantum Mechanics. American Mathematical Society, Providence (2009)Google Scholar