Harmonic Oscillator Solution Using Operator Methods

Jeong-Ryeol Choi

Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.

Tapan Kumar Das

Fuzi Marati Sholihah, Suparmi Suparmi, Viska Inda Variani

Solution of the harmonic oscillator equation has a goal to get the energy levels of particles moving harmonic. The energy spectrums of one dimensional harmonic oscillator are analyzed by 3 methods: path integral, hypergeometry and operator. Analysis of the energy spectrum by path integral method is examined with Schrodinger equation. Analysis of the energy spectrum by operator method is examined by Hamiltonian in operator. Analysis of harmonic oscillator energy by 3 methods: path integral, hypergeometry and operator are getting same results 𝐸 = ℏ𝜔 (𝑛 + 1 2)

K. T. Hecht

Reinhold A. Bertlmann, Nicolai Friis

Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product