Green’s Function Approach to the Solution of the Linear Quantum Harmonic Oscillator

Luca Nanni

Yusuf Ziya Umul

Leon F. Landovitz, Alfred M. Levine, E. Ozizmir +1 more

The solution is obtained to the quantum mechanical linearly damped harmonic oscillator subjected to a time dependent driving force. This solution allows the calculation of interesting quantum quantities such as the Green’s function and the transition amplitudes between harmonic oscillator states. The specific results for a harmonic driving force are given.

Chun-Khiang Chua, Yu-Tsai Liu, Gwo-Guang Wong

The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (SHO) system ( ) can be obtained by solving the equation directly. It has a compact expression, which gives correct eigenvalues and eigenfunctions easily. The Green's function G with an additional delta-function potential can be obtained readily. The same technics of solving the Green's function G 0 can be used to solve the eigenvalue problem of the SHO with an generic delta-function potential at an arbitrary site, i.e. . The Wronskians play an important and interesting role in the above studies. Furthermore, the approach can be easily generalized to solve the quantum system of a SHO with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration.

Diógenes Campos

S. I. Pavlik, D.I. Kulik, Yu. S. Oseledchik

Using the nonequilibrium Green's function method, the problem of thermal relaxation of systems interacting with a heat bath is investigated. We apply this method to the harmonic oscillator interacting with a heat bath which consists of many harmonic oscillators. Using with the Keldysh nonequilibrium Green function technique we derive expressions for the the absorption and emission spectra. We also establish a connection to the the solution of the Heisenberg equation based on a fully quantum-mechanical Hamiltonian.