Time Evolution Operator for the Harmonic Oscillator

Wensen Liu, Ji‐Suo Wang

The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su (1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators ( a , a †), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.

B. B. Varga, S. Ø. Aks

Sunil Kumar, C. L. Mehta

We discuss time evolution operator for N interacting quantum harmonic oscillators and determine certain conditions under which an explicit closed-form expression can be found for this time-ordered operator. The special case of N=2 is discussed, as an example.

Ettore Salusti B, Francesco Zirilli

Francisco Marcelo Fernández

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.