This Demonstration shows the application of the second quantization formalism for understanding the quantized energy levels of a 1D harmonic oscillator The raising creation and lowering destruction or annihilation operators respectively add and subtract quanta to the ground state or any other state In this way one can move up and down the energy scale of allowed eigenvalues with the eigenfunctions given by the Hermite polynomials since the following recursion relations hold from quantum mechanics with and for the definition of a vacuum All these relations can be deduced from the ground state by the relation 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