The Harmonic Oscillator

The one-dimensional harmonic oscillator is the single most important solvable system in quantum mechanics. Every potential looks harmonic near a stable equilibrium (Taylor-expand to second order), so the oscillator governs molecular vibrations, phonons, and — most importantly for what follows — the normal modes of a quantum field. Its algebraic solution via ladder operators is the direct prototype for the creation/annihilation operators of QFT Fock space.

The Hamiltonian is

$$\hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2,$$

with $[\hat{x},\hat{p}]=i\hslash$ (see wave mechanics).

1. Ladder Operators

Define the dimensionless annihilation and creation operators

$$\hat{a}=\sqrt{\frac{m\omega }{2\hslash }}(\hat{x}+\frac{i}{m\omega }\hat{p}),{\hat{a}}^{†}=\sqrt{\frac{m\omega }{2\hslash }}(\hat{x}-\frac{i}{m\omega }\hat{p}).$$

From the canonical commutator these satisfy the defining algebra

$$\boxed{[\hat{a},{\hat{a}}^{†}]=1.}$$

Inverting, $\hat{x}=\sqrt{\hslash /2m\omega }(\hat{a}+{\hat{a}}^{†})$ and $\hat{p}=i\sqrt{\hslash m\omega /2}({\hat{a}}^{†}-\hat{a})$, the Hamiltonian becomes

$$\hat{H}=\hslash \omega ({\hat{a}}^{†}\hat{a}+\frac{1}{2})=\hslash \omega (\hat{N}+\frac{1}{2}),\hat{N}\equiv {\hat{a}}^{†}\hat{a},$$

with $\hat{N}$ the number operator (Hermitian, $\ge 0$).

2. The Spectrum from the Algebra

The commutators $[\hat{N},\hat{a}]=-\hat{a}$ and $[\hat{N},{\hat{a}}^{†}]=+{\hat{a}}^{†}$ mean the ladder operators shift $\hat{N}$-eigenvalues by $\mp 1$: if $\hat{N}|n⟩=n|n⟩$, then $\hat{a}|n⟩$ has eigenvalue $n-1$ and ${\hat{a}}^{†}|n⟩$ eigenvalue $n+1$. Since $\hat{N}\ge 0$, the lowering must terminate at a ground state $|0⟩$ with $\hat{a}|0⟩=0$, hence $n=0$. Applying ${\hat{a}}^{†}$ repeatedly generates the whole tower:

$$\hat{a}|n⟩=\sqrt{n}|n-1⟩,{\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩,|n⟩=\frac{({\hat{a}}^{†}{)}^n}{\sqrt{n!}}|0⟩.$$

The energy spectrum is therefore evenly spaced:

$$\boxed{E_n=\hslash \omega (n+\frac{1}{2}),n=0,1,2,\dots }$$

Two features have no classical analog: the ground-state (zero-point) energy $E_0=\frac{1}{2}\hslash \omega >0$, forced by the uncertainty principle (the particle cannot sit still at the minimum), and the exact equal spacing $\hslash \omega$, which is what makes the modes of a free field behave as independent quanta of energy $\hslash \omega$.

3. Position-Space Wavefunctions

The ground state follows from the first-order equation $\hat{a}|0⟩=0$, i.e. $(x+\frac{\hslash }{m\omega }{\partial }_x){\psi }_0=0$, giving a Gaussian:

$${\psi }_0(x)=\left(\frac{m\omega }{\pi \hslash }{)}^{1/4}\exp \right(-\frac{m\omega x^2}{2\hslash }).$$

Excited states are obtained by applying ${\hat{a}}^{†}$, producing Hermite polynomials $H_n$:

$${\psi }_n(x)=\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega }{\pi \hslash }{)}^{1/4}{H}_n\right(\sqrt{\frac{m\omega }{\hslash }}x\left)\exp \right(-\frac{m\omega x^2}{2\hslash }).$$

The Gaussian ground state saturates $\Delta x\Delta p=\hslash /2$ — it is a minimum-uncertainty state.

4. Coherent States

The eigenstates of the (non-Hermitian) annihilation operator,

$$\hat{a}|\alpha ⟩=\alpha |\alpha ⟩,|\alpha ⟩=e^{-|\alpha {|}^2/2}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}|n⟩,\alpha \in \mathbb{C},$$

are the coherent states. They are minimum-uncertainty Gaussians whose centroid $⟨\hat{x}⟩,⟨\hat{p}⟩$ traces the classical oscillation without spreading, giving the closest quantum analog of a classical oscillator. The photon number follows a Poisson distribution $|⟨n|\alpha ⟩{|}^2=e^{-|\alpha {|}^2}|\alpha {|}^{2n}/n!$ with $⟨\hat{N}⟩=|\alpha {|}^2$. Squeezed states redistribute the uncertainty unequally between $\hat{x}$ and $\hat{p}$ while still saturating the product — the basis of sub-shot-noise metrology.

5. Forward Link: Fields as Oscillators

A free quantum field is a continuum of decoupled harmonic oscillators, one per mode $\mathbf{k}$. The ladder operators become the mode creation/annihilation operators, $|0⟩$ the field vacuum, and $|n⟩$ a state of $n$ quanta (particles) in that mode. The zero-point energy $\frac{1}{2}\hslash \omega$ summed over modes is the vacuum energy; the equal spacing is why particles of a given mode are identical. Everything on this page reappears there almost verbatim.

See also

• Uncertainty relations — zero-point energy and coherent states.
• Time-independent perturbation theory — anharmonic corrections.
• QFT: Fock Space Inventory — the field-theoretic sequel.