Dirac saw that the oscillator Hamiltonian, a sum of two squares, almost factorises: $a^{2}+b^{2}=(a-ib)(a+ib)$ for numbers. For operators the factorisation leaves something over, because $\hat{x}$ and $\hat{p}$ do not commute, and what is left over is exactly half a quantum. The two factors are operators that step down and up the ladder of levels, one quantum at a time, and with them the whole spectrum is found by algebra alone.

A staircase with identical steps can be explored in the dark: find one step and a rule for going up or down by one, and every step can be reached. The ladder operators are that rule.

The ladder of levels, one quantum apart. The raising operator $\hat{a}^{\dagger}$ takes a state one rung up, adding $\hbar\omega$; the lowering operator $\hat{a}$ takes it one rung down. On the bottom rung the lowering operator gives zero, and the ladder ends there.

Two operators built from $\hat{x}$ and $\hat{p}$, adjoints of each other. Their commutator is one, the Hamiltonian is their product plus half a quantum, and their commutators with the Hamiltonian make them step the energy down and up.

• $[\hat{a},\hat{a}^{\dagger}]=1$: expanding the product, the cross terms give $\frac{}{}$.

Multiply the two definitions, keeping the order of $\hat{x}$ and $\hat{p}$. The squares give the two energies; the cross terms do not cancel, because the operators do not commute, and leave their commutator, which the canonical relation turns into minus half a quantum.

Multiply the definitions in this order.

The cross terms are $\frac{i}{m\omega}(\hat{x}\hat{p}-\hat{p}\hat{x})$: a commutator, not zero.

The canonical commutation relation.

Multiply through by $\hbar\omega$.

The first two terms are the Hamiltonian; move the half quantum across.

The commutator of the Hamiltonian with the raising operator is $\hbar\omega$ times the raising operator. So the raising operator turns a state of energy $E$ into one of energy $E+\hbar\omega$.

Everything about the ladder operators follows from their commutator, which is the number one: a consequence of the canonical commutation relation between $\hat{x}$ and $\hat{p}$.

The lowering operator is not Hermitian: its adjoint is the raising operator, a different operator. Neither stands for a measurable quantity, but their sum and difference give back $\hat{x}$ and $\hat{p}$.

Adding the two ladder operators gives position; subtracting them gives momentum.

Great work! You now know how to:

• define the raising and lowering operators
• factorise the oscillator Hamiltonian and find the half quantum
• use their commutators with $\hat{H}$ to step between levels