Bring the chapter together: the oscillator’s scales, the ladder operators and their commutators, number states and their square roots, the Gaussian ground state and its zero-point energy, Hermite functions, spreads in number states, motion in the Heisenberg picture and coherent states. No worked example sits above the answers.

A pianist practises scales not for their own sake but because every piece is made of them. The oscillator is quantum mechanics’ scale: light, vibrations and fields are all built from it.

Everything follows from $\hat{H}=\hbar\omega\bigl(\hat{a}^{\dagger}\hat{a}+\tfrac12\bigr)$ and $[\hat{a},\hat{a}^{\dagger}]=1$: the whole-number spectrum, the square roots of the ladder, the Gaussian ground state and the classical motion of the averages.

• $\hat{a}^{\dagger}\ket{n}=\sqrt{n+1}\,\ket{n+1}$ and $\ket{n}=(\hat{a}^{\dagger})^{n}\ket{0}/\sqrt{n!}$.
• $\psi_{n}\propto H_{n}(\xi)e^{-\xi^{2}/2}$, with nodes, even or odd as is.
• $(\Delta x)^{2}=x_{0}^{2}\bigl(n+\tfrac12\bigr)$ and in .
• Coherent states: $\hat{a}\ket{\alpha}=\alpha\ket{\alpha}$, Poisson counts with mean $|\alpha|^{2}$, and .