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The quantum harmonic oscillator has discrete energy levels E_n = (n + ½)ℏω. Unlike classically, the ground state has nonzero zero-point energy ½ℏω due to uncertainty.
Classically, a harmonic oscillator is any system with a restoring force proportional to displacement — a mass on a spring, a pendulum for small angles, a ball at the bottom of a curved bowl. The energy is E = p²/2m + mω²x²/2, a sum of kinetic and potential energy, and it can take any continuous value. The particle oscillates between turning points where all the energy is potential, and passes through the center where all of it is kinetic. At zero energy, the particle simply sits motionless at the equilibrium position.
Quantum mechanics changes this picture in two essential ways, both rooted in the Heisenberg uncertainty principle — your prerequisite from canonical commutation relations. If the particle were at rest at the bottom (x = 0, p = 0), both position and momentum would be simultaneously sharp, violating Δx·Δp ≥ ℏ/2. So the quantum particle cannot sit still. It must spread over some range in position and retain some spread in momentum, and this zero-point motion costs energy. The minimum energy is not zero but ½ℏω — the zero-point energy. This is a purely quantum effect with no classical analogue, and it has real physical consequences: liquid helium remains liquid at atmospheric pressure even at absolute zero, because quantum zero-point motion is large enough to prevent solidification.
The energy levels above the ground state are E_n = (n + ½)ℏω for n = 0, 1, 2, .... The spacing between adjacent levels is always exactly ℏω — uniform, unlike the hydrogen atom where levels crowd together at higher energies. You can think of each unit of ℏω as one quantum of excitation, and this idea generalizes enormously. In quantum field theory, a field at each point in space is treated as a harmonic oscillator; the "quanta" of excitation are the particles themselves. A photon is a single quantum of excitation of the electromagnetic field oscillator at a given frequency. An atom in its excited state that spontaneously emits has dropped from n = 1 to n = 0 in that field mode.
Solving the Schrödinger equation for the harmonic oscillator potential V = ½mω²x² yields Hermite polynomial wavefunctions multiplied by a Gaussian envelope. But the most powerful approach — which you will encounter shortly — uses ladder operators a and a†, built from x and p via the commutation relation you already know. These operators raise or lower the quantum number n by one unit, making it almost trivial to generate the full spectrum and compute matrix elements. The quantum harmonic oscillator is not just a pedagogical exercise; it is the single most useful exactly-solvable model in all of physics, underpinning molecular vibrations, phonons in solids, cavity QED, and quantum information.