Quantum Mechanics · Chapter 07

The harmonic oscillator

The one system met everywhere, from molecules to light, solved twice: by the algebra of ladder operators, which gives the whole spectrum without solving a differential equation, and by the wavefunctions with their Hermite polynomials. Zero-point energy, uncertainty in the number states, motion in time, and the coherent states that move exactly as a classical oscillator does.

H^= ℏ ω (a^^{†} a^+ \frac{1}{2})

E_{n} = ℏ ω (n + \frac{1}{2})

[a^, a^^{†}] = 1

a^∣ n ⟩ = n ∣ n - 1 ⟩

Lessons

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05Excited states and Hermite polynomialsRaising the Gaussian once multiplies it by $\xi$; raising again gives a quadratic times the same Gaussian; each rung adds one degree to the polynomial in front. The polynomials are Hermite’s, known long before quantum mechanics. They carry everything visible about the excited states: the $n$-th has $n$ nodes, and the states alternate between even and odd because the potential looks the same from both sides.Read lesson →

06Uncertainty in the number statesIn a number state the particle is on average at the centre and at rest, yet its position and momentum spreads grow with every quantum added, both as $\sqrt{n+\tfrac12}$. The energy is shared equally between kinetic and potential, as it is for a classical oscillator averaged over a swing. And for large $n$ the density itself begins to look classical: it piles up near the ends of the swing, where a classical particle slows down and lingers.Read lesson →

07Time evolution in the oscillatorIn the Heisenberg picture the oscillator is solved in one line: the lowering operator just turns with a phase, and position and momentum then move exactly like the classical coordinates. So every average swings at the frequency $\omega$, whatever the state. A number state has nothing to swing — its averages are zero and stay zero — but a superposition of neighbouring levels sways from side to side at exactly the classical frequency.Read lesson →

08Coherent statesOf all the oscillator’s states, which moves most like a classical particle? The answer is a Gaussian with the ground state’s width, displaced from the centre: it swings back and forth at the classical frequency and never spreads. These coherent states are eigenstates of the lowering operator, their numbers of quanta are Poisson-distributed, and the light of a laser is one.Read lesson →

09Chapter practiceBring 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.Read lesson →

Mapping the lesson.

The harmonic oscillator

Build the foundation