Harmonic oscillator

This simulation shows the evolution of a harmonic oscillator when the initial state is a linear superposition of several stationary states. You can change the relative weights of the different levels directly by dragging the horizontal bars with the mouse. The program then shows the average energy (horizontal yellow line), the average position (blue dot), and the probability density. According to the Ehrenfest theorem, the blue dot undergoes a classical harmonic motion, corresponding here to a sinusoidal oscillation in an harmonic potential.

You can also directly change the average energy by dragging the horizontal line to the target value. The simulator then puts the system in the quasi-classical state, or coherent state, with the energy you have set. Such a state, usually called |a >, exhibits the following properties:

  • This is an eigenstate of the annihilation operator: a|a >=a > a >.
  • The product Dp Dx = hbar/2, i.e. the smallest possible value according to the Heisenberg uncertainty principle.
  • As a consequence, and as can be seen above, the probability density has always a gaussian shape. In the limit of large energy values, this state corresponds to the classical motion of an harmonic oscillator.