N Operator

The Hamiltonian for the harmonic oscillator is given by setting in the expression forr the Hamiltonian giving

where K is the spring constant. is the operator for the x-component of momentum. The time-independent Schrodinger equation (for one-dimension)

becomes

Substituting and into Eq. (2) gives

The potential is shown in the diagram below

As seen in the above diagram the classical region is given when the kinetic energy is positive, i.e. for E > V. This corresponds to the interval -x₀ < x < x₀.

Creation and Annihilation Operators

Let where , i.e. . The operators

are called the annihilation and creation operators respectively. The relation will prove useful and is derived here.

Substituting the relation results in

Note also that

The inverses of Eq. (5) are found to be

The Hamiltonian in Eq. (1) can be expressed in terms of the creation and annihilation operators as follows. Squaring both and gives

Adding Eqs. (10) and (11) gives

This can be simplified by substituting in Eq. (8)

Substituting yields the final result