They also obey the eigenvalue equation , where is the number operator that gives the number of quanta added to the ground state (GS). The Hamiltonian for the harmonic oscillator is given by and the raising and lowering operators are related to the position and momentum operators by ) and ), with and . The raising and lowering operators are also called ladder operators, because they move up and down the equally spaced energy levels as if on a ladder.

In this Demonstration you can do this by setting the slider to a particular starting energy level (by default, gives the ground state energy) and clicking the corresponding buttons, "raise: " and "lower: ". To go back to the beginning, click the "reset " button. When you reach the vacuum state, , annihilating the state.