Raising and Lowering Operators

The non-Hermitian operators a+ and a are defined in terms of the operators for position, q and momentum, p:

The position and momentum can in turn be expressed in terms of the operators:

When these expressions are substituted into the vibrational hamiltonian,

we obtain the expressions

The last step follows from the commutation relation

which follows from

where the bracket represents the commutator. To show this last step we expand the commutator.

and substitute in the definitions for the raising and lowering operators

From the above definition of the hamiltonian the quantity a⁺a is known as the number operator, N. If N operates on the vth vibrational wave function |vñ, it returns the quantum number of that wave function, N|vñ = v|vñ.

This property of the operator N can be used to demonstrate why a and a⁺ are known as the lowering and raising operators.

Consider

= a(a⁺a - 1)|vñ = (v - 1)a|vñ

This shows that |fñ = a|vñ is an eigenfunction of N with eigenvalue v-1. The operator a has lowered the eigenvalue by one. The ket |f ñ must therefore be proportional to the ket |v-1ñ.

|fñ = a|vñ = C|v-1ñ

If we assume that both kets are normalized

áf|f ñ = áv|a⁺a|vñ = C*C = v. Therefore, C = Öv

and a|vñ = Öv|v-1ñ (lowering)

Similarly one can show that

a⁺|vñ = Ö(v+1)|v+1ñ (raising)

Combining these properties leads to the recursion relation