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§ Phase vii · Wavepacket Dynamics · Chapter 03

The harmonic trap

Every potential well, looked at closely enough near its bottom, is a parabola. That single fact turns the quantum harmonic oscillator into the most-used model in all of physics. Schrödinger solved it in his first paper on wave mechanics. Forty years later Roy Glauber found, hiding inside the same equations, a wavepacket that bounces back and forth in the well for all eternity without losing its shape, the closest thing the quantum world has to a marble in a bowl.

If you walk into any physics building in the world, sooner or later you will see a parabola chalked on a blackboard. The teacher will not always say what it is; often they will just draw the cup, drop a horizontal line across it, label the line E, and start computing. That parabola is the harmonic potential, V(x) = ½ m ω² x². It is the single most-used picture in physics because of a small piece of high-school calculus. Near the bottom of any smooth well, the second-order Taylor expansion of V(x) is exactly a parabola. The linear term vanishes (we are at a minimum), so the leading deviation from constant is quadratic. Whether the well in question is the dip in an atomic lattice that holds an ion, the bond between two atoms in a molecule, or the field of a laser inside a mirror cavity, near equilibrium it looks the same. The harmonic oscillator is not one problem. It is the first non-trivial term in a Taylor series, and that is why physicists carry it everywhere.

Erwin Schrödinger understood this from the start. In the first paper of his “Quantisation as a Problem of Proper Values” series, written in early 1926 in the snowed-in chalet at Arosa where he had finally pinned down his wave equation, the second worked example (after the hydrogen atom) is the harmonic oscillator. Schrödinger does not explain why he chose it. He does not need to. Anyone who has done classical mechanics has spent half their life with parabolic potentials. He plugs V = ½ m ω² x² into his time-independent wave equation, expects the usual Hermite polynomials, and gets them. The spectrum drops out almost too easily. The allowed energies are

E_n = (n + ½) ℏω,    n = 0, 1, 2, 3, …

It is the simplest non-trivial spectrum in quantum mechanics. The levels are evenly spaced, each one a step of ℏω above the last. No square roots, no inverse squares, no Rydberg constant. Just a ladder.

The simplicity is deceptive. Pause on the lowest rung. The classical ground state of a marble in a bowl is to sit at the bottom with zero energy and zero motion. The quantum ground state has energy E₀ = ½ ℏω. The marble cannot sit still. It must, even in its quietest possible state, jiggle. We call ½ ℏω the zero-point energy, and it is one of those quantum facts that sounds like a clerical detail and turns out to dominate whole branches of physics. The zero-point motion of atoms in a crystal sets the lattice spacing. The zero-point fluctuations of the electromagnetic field cause the Casimir force between two metal plates and the Lamb shift in hydrogen. The ground state of the universe is not still; it hums.

Why a ladder? The argument is short enough to follow even before you have seen the algebra. The harmonic potential is symmetric: V(x) = V(-x). It is also smooth and unbounded above. Confine any particle in a smooth, symmetric well and you get discrete bound states with alternating even and odd parity. Make the well parabolic and a beautiful coincidence happens. The spacing between successive bound states turns out to be perfectly uniform. There is no other smooth potential for which the spacing is exactly constant. The hydrogen atom, with its Rydberg ladder bunching toward the ionization limit, gets denser at the top. The infinite square well, with E_n proportional to n², gets sparser. Only the parabola gives even rungs. That is also why the harmonic oscillator is the cleanest possible model for any system that radiates at a single sharp frequency: photons of energy ℏω can hop the ladder one rung at a time, never landing in between, always in tune.

The wavefunctions on the ladder are the Hermite functions. The ground state ψ₀(x) is a pure Gaussian, the bell curve, centred on the origin with width

σ₀ = √( ℏ / (2 m ω) ).

This is the smallest packet that quantum mechanics allows. Heisenberg’s uncertainty relation says Δx · Δp ≥ ℏ / 2, and the Gaussian ground state of the harmonic oscillator saturates the bound: Δx · Δp = ℏ / 2 exactly. No other state is so compact. As you climb to higher n, the wavefunctions sprout nodes (zeros where ψ_n = 0) and the bumps push outward toward the classical turning points. By the time you reach n = 30 or so, the probability density |ψ_n(x)|² is concentrated near the edges of the classically allowed region, just as a classical particle would spend more time near its turning points where it moves slowest. The quantum and classical pictures merge at high n; this is Bohr’s correspondence principle in pure form.

Now look hard at any one of those rungs. Each |ψ_n(x)|² is fixed: a stationary photograph of where the particle is likely to be found. Run time forward and it does not move. That is what “stationary state” means. The phase of ψ_n(x, t) winds around at a uniform rate exp(-i E_n t / ℏ), but the modulus, the actual probability density, is frozen. Build any single eigenstate and the particle is everywhere at once forever, smeared over the rung pattern, going nowhere in particular. It is the opposite of a classical marble bouncing back and forth.

So how does the quantum harmonic oscillator ever reproduce the classical limit? How do we get from a stack of frozen patterns to a particle that actually sloshes from one wall to the other and back? The answer waited from 1926 until 1963, when a 38-year-old American theorist named Roy Glauber working on the new field of quantum optics found it.

Glauber’s question was almost ordinary. He had spent the late 1950s as a young Harvard theorist trying to understand the new optical maser, the device Theodore Maiman would soon make famous as the laser. The maser’s puzzle: classical electromagnetism describes a beam of light as a coherent wave with a definite amplitude and phase. Quantum mechanics describes photons as Fock states |n⟩ with definite photon number but completely random phase. Both pictures cannot be right. Glauber asked: what is the quantum state of light that comes closest to a classical sinusoid? He found that it is a particular superposition over every Fock state, with weights drawn from a Poisson distribution:

|α⟩ = exp(-|α|² / 2) Σ_n (α^n / √n!) |n⟩

He called these coherent states. The complex number α encodes both the average amplitude and the phase. The remarkable thing, Glauber showed, is that if you take such a state and let it evolve under the harmonic-oscillator Hamiltonian, the centre of the wavepacket follows the classical orbit exactly. It bounces back and forth between the turning points at angular frequency ω, just like a marble in a bowl. The width of the packet, meanwhile, stays equal to σ₀ = √(ℏ / (2mω)), the same minimum-uncertainty width as the ground state. Forever. It never spreads.

This is unlike anything that happens to a free Gaussian. In Chapter 1 of this phase we watched a free wavepacket smear out: high-frequency components moving faster, low-frequency ones lagging, until the packet became a wide messy puddle. The harmonic potential refuses to let that happen. Every component, every Fock state in the superposition, has an energy E_n = (n + ½) ℏω that is evenly spaced from its neighbours. So every term in the time evolution picks up a phase exp(-i (n + ½) ω t) that rotates around together in lockstep. Plug those phases back into Glauber’s sum and you find, after some algebra, that the whole packet is just |α(t)⟩, the same shape, with a new α(t) = α(0) exp(-i ω t) that rotates around the origin in phase space at angular velocity ω. The marble of phase space orbits. The wavefunction follows.

â |α⟩ = α |α⟩

|α⟩ = exp(-|α|² / 2) Σ_{n=0}^∞ (α^n / √(n!)) |n⟩.

|α(t)⟩ = exp(-i ω t / 2) · exp(-|α|² / 2) Σ_n ((α e^{-i ω t})^n / √(n!)) |n⟩
       = exp(-i ω t / 2) · |α e^{-i ω t}⟩.

⟨x⟩(t) = √(2 ℏ / (m ω)) · |α₀| · cos(ω t − φ)

The implications were enormous. Glauber’s framework, published in three papers in 1963, became the foundation of quantum optics. Coherent states explained what laser light is (a beam of coherent photons rather than a stack of definite photon numbers) and gave a clean operational definition for “coherence” in the quantum world. They also clarified what was new about the laser as compared to a thermal source like the Sun: not the photons themselves but the joint statistics, the way every mode of the field shares the same phase. For this work, and for the decades of follow-up papers that built quantum optics into a field, Glauber shared the 2005 Nobel Prize in Physics with John Hall and Theodor Hänsch. He was 80 years old and still teaching graduate students at Harvard. When the journalist asked him at the press conference how it felt, he answered, “Well, I had thought maybe they had forgotten.”

There is a way to see what the harmonic oscillator and the coherent state are doing without any of the algebra. Imagine the phase space of the particle, position on one axis and momentum on the other. Classically the trajectory of a harmonic oscillator is a perfect circle around the origin, traced at angular velocity ω; that is the rotating phasor that Hamilton’s equations produce when you draw them for V = ½ k x². Now sketch the quantum state on the same picture. A Fock eigenstate |n⟩ is a thin ring of radius √(2n + 1), smeared around the whole circle because it has no preferred phase. A coherent state |α⟩ is a small disc of radius 1 / √2 centred at the point (Re α, Im α) of the phasor; the disc has just enough fuzz to satisfy Heisenberg. As time runs forward, the disc orbits around the origin at angular velocity ω, never changing size. That is the picture Glauber asked us to keep in our heads. The harmonic oscillator is the place where Hamilton’s classical phase space and Heisenberg’s quantum fuzziness fit together most cleanly. The classical orbit is the centre of the quantum disc.

The next chapter takes the same kind of clean mathematics and applies it to two parabolas instead of one, and asks what happens when two atomic potential wells overlap. The answer is a chemical bond. Hooke’s law for atoms, sitting at the heart of every molecule, will look surprisingly like the chapter you have just read.