Recap of the Oscillator

The quantum harmonic oscillator is the single most important solvable model in physics. Any smooth potential, expanded near a stable minimum, looks like a parabola to lowest order — so the oscillator describes vibrating molecules, lattice phonons, and the modes of the electromagnetic field alike. This module rebuilds its entire spectrum using nothing but operator algebra. Before that, we recall the setup.

The classical starting point

A particle of mass $m$ on a spring with stiffness constant $k$ feels a restoring force $F = -kx$ and oscillates at angular frequency

$\omega = \sqrt{\frac{k}{m}}.$

Its total energy is the sum of kinetic and potential parts,

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

where we have written $k = m\omega^2$ . Classically, $E$ can take any non-negative value, and a particle at rest at the bottom ( $x = 0,\ p = 0$ ) has $E = 0$ .

The quantum Hamiltonian

In quantum mechanics, position and momentum become operators $\hat{x}$ and $\hat{p}$ obeying the canonical commutation relation

$[\hat{x}, \hat{p}] = i\hbar.$

Promoting the classical energy to an operator gives the harmonic-oscillator Hamiltonian

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

Solving the time-independent Schrödinger equation $\hat{H}\psi_n = E_n\psi_n$ as a differential equation (the Hermite-polynomial method) yields the famous evenly spaced spectrum

$E_n = \hbar\omega\left(n + \tfrac{1}{2}\right), \qquad n = 0, 1, 2, \ldots$

We will not reuse that differential-equation machinery here. Instead, the next lessons factor the Hamiltonian into two first-order operators — the ladder operators — and read off the same spectrum purely algebraically.

What the spectrum already tells us

Two features of $E_n = \hbar\omega(n + \tfrac{1}{2})$ deserve emphasis, because the algebraic derivation will reproduce them exactly:

Equal spacing. Adjacent levels differ by a fixed quantum of energy, $E_{n+1} - E_n = \hbar\omega$ . This single quantum is what a ladder operator will add or remove.
Nonzero ground state. The lowest energy is $E_0 = \tfrac{1}{2}\hbar\omega \gt 0$ , not zero. This zero-point energy is forced by the uncertainty principle: a state localized in both $x$ and $p$ would violate $\Delta x\,\Delta p \ge \hbar/2$ , so the particle can never sit perfectly still at the bottom of the well.

Where we are headed

The plan for the module is to define operators $a$ (annihilation) and $a^\dagger$ (creation) as specific linear combinations of $\hat{x}$ and $\hat{p}$ , show that they satisfy the simple algebra $[a, a^\dagger] = 1$ , and then prove that the Hamiltonian becomes $\hat{H} = \hbar\omega(a^\dagger a + \tfrac{1}{2})$ . From that one identity the entire spectrum, the ground state, and all matrix elements of $\hat{x}$ and $\hat{p}$ follow with almost no calculus. This is the power of the operator method: it trades differential equations for bookkeeping with a single commutator.