# Creation and annihilation operators

Make a coordinate substitution to nondimensionalize the differential equation

and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,

This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.

Using the commutation relations given above, the Hamiltonian operator can be expressed as

These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.

The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is

We can now describe the occupation of particles on the lattice as a `ket' of the form

note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation

This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.

The commutation relations of creation and annihilation operators in a multiple-boson system are,