The Quantum Harmonic Oscillator

Today the flat-topped barriers of chapter 12 give way to a smooth valley whose sides curve upward forever. Its classical prototype is a mass on a spring. Robert Hooke published the rule in 1678 as ut tensio, sic vis, as the stretch, so the force: displace the mass by x and the spring pulls back with force F = −kx, where the spring constant k measures stiffness in newtons per metre. The potential energy, V(x) = ½kx², is a parabola, and the released mass swings with angular frequency ω = √(k/m), in radians per second, whatever the size of the swing. This is simple harmonic motion, and the system is a harmonic oscillator.

Why should a course about atoms care about springs? Because almost every stable equilibrium is a spring up close. Take any smooth potential energy curve with a minimum at x₀. At the bottom the slope is zero, by definition, so the first thing that changes as you step away is the curvature. Zoom in far enough and every smooth valley becomes a parabola, V(x) ≈ V(x₀) + ½k(x − x₀)², with an effective spring constant k equal to the curvature of V at the bottom, its second derivative there. Mathematicians call this cutting a Taylor series off after its quadratic term; physicists call it the reason the harmonic oscillator is everywhere.

The roster is long: a pendulum swinging through small angles, two atoms joined by a chemical bond, an atom held in a crystal by its neighbours, a circuit built from a coil and a capacitor and, most remarkably, every standing-wave pattern of the electromagnetic field in a box. Solve the quantum oscillator once and you have solved the small vibrations of nearly everything. No wonder the physicist Sidney Coleman is often quoted as saying that the career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.

To make the spring quantum, write k = mω², feed V(x) = ½·mω²x² into the time-independent Schrödinger equation of chapter 10, and ask which energies allow a well-behaved wavefunction, one that fades to zero far from the centre instead of exploding. As in the box of chapter 11, only special energies qualify. Werner Heisenberg found them first, in the 1925 paper that launched matrix mechanics, and Erwin Schrödinger rederived them in 1926. The result is a formula I would tattoo on my lab coat if the coat would only hold still: Eₙ = (n + ½)ħω, with n = 0, 1, 2, 3 and so on, where ħ = h/2π.

Each rung sits exactly ħω above the one below, a ladder of perfectly equal spacing: ½ħω, then 3/2 ħω, then 5/2 ħω, upward forever. Max Planck's desperate counting rule of 1900, which shared out the energy of an oscillator of frequency f in whole packets of hf (chapter 3), had the step right, since hf is exactly ħω, but it put the bottom rung at zero. Equal spacing has a lovely consequence. Light can move a charged oscillator only one rung at a time, a restriction called a selection rule, so every photon it absorbs or emits carries the same energy, ħω, at the classical spring's own frequency. The quantum oscillator sings the same note as its classical twin.

Do not generalize this; equal spacing is a special gift of the exact parabola. In the box of chapter 11 the levels spread apart as n². In hydrogen, chapter 5, they follow −13.6 eV/n² and crowd together toward the ionization limit. Even a real chemical bond is parabolic only near the bottom of its valley; higher up, the valley widens toward the energy at which the bond breaks, so its rungs creep closer together. Evenly spaced levels are the signature of a parabola, not a law of nature.

Set n = 0 and the energy is not zero but E₀ = ½ħω, the zero-point energy. Cool the spring to absolute zero, drain every scrap of heat, and this half-rung remains. A classical mass could rest motionless at the bottom; a quantum particle cannot, because that would mean a definite position and a definite momentum, which the uncertainty principle of chapter 8, Δx·Δp ≥ ħ/2, forbids. So the particle strikes a bargain. Squeezing the wavefunction toward the centre lowers the potential energy but raises the kinetic energy, since a narrower wave contains larger momenta; spreading it does the reverse. The cheapest compromise costs exactly ½ħω, shared equally between kinetic and potential energy, with the smallest uncertainty product nature permits, Δx·Δp = ħ/2.

Zero-point energy leaves fingerprints. In 1924, a year before Heisenberg's derivation, Robert Mulliken found that light linking the lowest vibrational levels of two electronic states of boron monoxide shifted when boron-10 replaced boron-11, impossible if those levels sat at the very bottoms of their valleys. The hydrogen molecule, H₂, holds about 0.27 eV of zero-point energy, nearly 6 percent of the depth of its valley. Its heavy twin D₂, made of deuterium, sits in essentially the same valley, since its electrons barely notice nuclear mass, but its slower vibration lets it rest about 0.08 eV lower. So breaking D₂ costs about 4.56 eV against 4.48 eV for H₂, and in reactions that must break them, bonds to deuterium typically break several times more slowly than bonds to hydrogen.

The most spectacular consequence is a liquid that refuses to freeze. Heike Kamerlingh Onnes liquefied helium in Leiden in 1908, but no further cooling produced a solid; his former student Willem Keesom finally froze it in 1926, and only by squeezing it: even near absolute zero, solid helium needs about 25 times atmospheric pressure. Helium atoms are light and attract one another feebly. Lock them into a crystal and each, caged by its neighbours, would have a zero-point jiggle comparable to the attraction holding the cage together, so the lattice shakes itself apart, as Francis Simon argued in 1934. Helium is the only element that stays liquid at ordinary pressure all the way to absolute zero. I'm mad, not wrong: nothing ever truly sits still.

What do the stationary states look like? The ground state is the most famous curve in statistics, the Gaussian or bell curve: ψ₀ is proportional to e raised to the power −x²/(2x₀²), where x₀ = √(ħ/(m·ω)) is the oscillator's natural length. Heavy masses and stiff springs make x₀ small; light masses and floppy springs make it large. For carbon monoxide, x₀ is about 4.8 picometres, or trillionths of a metre, and the ground state's spread in position is x₀ divided by √2, about 3.4 picometres, against a bond 113 picometres long: even at absolute zero, that bond's length is blurred by about 3 percent.

The excited states are the same Gaussian multiplied by a polynomial of degree n, from a family named after the French mathematician Charles Hermite. The pattern echoes the box: ψₙ has exactly n nodes, points where it crosses zero, so each rung up adds one more wiggle, more curvature and more kinetic energy. Even-n states are mirror-symmetric about the centre; odd-n states are antisymmetric and vanish at the bottom of the valley. In the ground state the most likely place to find the particle is dead centre, exactly where a classical oscillator moves fastest and spends the least time.

A classical oscillator swings between two turning points, where all its energy is potential and it momentarily stops; for level n they sit at ±√(2n + 1)·x₀. Lingering at the ends and racing through the middle, it is most likely found near the turning points. Quantum wavefunctions ignore those fences: as in chapter 12, ψ fades smoothly into the classically forbidden zone. In the ground state there is a 15.7 percent chance of finding the particle beyond the turning points, shrinking only slowly to 11.2 percent at n = 1. As n grows, the crests of the probability density crowd toward the turning points until, averaged over its wiggles, the quantum density traces the classical curve: the correspondence principle. And since the parabola rises forever, the leak never becomes an escape; the tails die off faster than any exponential.

There is a slicker way to solve the oscillator, one that never touches a differential equation. Developed by Paul Dirac, it uses two operators, mathematical machines that turn one state into another. The raising operator takes the state on rung n to rung n + 1, multiplied by √(n + 1); the lowering operator takes rung n down to rung n − 1, multiplied by √n. Each step changes the energy by exactly ħω, so together they are called ladder operators. Operators get their own chapter next session; for now, picture two robotic hands on the ladder, one lifting, one lowering.

The whole spectrum follows from two facts. Lowering the ground state gives zero, not a lower state: there is no rung beneath the bottom one. And raising then lowering differs from lowering then raising; the order of the hands matters, by exactly one unit. From these alone, the rungs must be evenly spaced, ħω apart, starting at ½ħω, and the energy can be written as ħω times (N + ½), where the number operator N reports which rung you are on.

That counting is the deep idea. The integer n is the number of quanta, each of energy ħω, that the oscillator holds above its ground state; raising adds one quantum and lowering removes one. A quantum is not a pebble rattling around inside the spring but one unit of excitation of the oscillator as a whole. In a crystal the quanta of lattice vibration are phonons; Albert Einstein's 1907 theory of solids, treating each atom as a quantized oscillator, explained why solids lose their capacity to store heat near absolute zero. Acting on fields, the same operators are renamed creation and annihilation operators, the basic grammar of quantum field theory.

A diatomic molecule is two masses on a spring. Both atoms move, so the mass in ω = √(k/μ) is the reduced mass, μ = m₁m₂/(m₁ + m₂), always smaller than the lighter atom's mass, while the bond's stiffness plays the role of k. Molecular vibrations run at roughly 10¹³ to 10¹⁴ hertz, so the photons that lift a molecule one rung are infrared. Spectroscopists quote the wavenumber, one over the wavelength, in inverse centimetres. Carbon monoxide's fundamental band is centred at 2,143 inverse centimetres, a wavelength of 4.67 micrometres, which means a rung of 0.266 eV and a bond stiffness of about 1,860 newtons per metre. Each kind of bond absorbs in its own characteristic band, so an infrared spectrum reads like a molecular fingerprint.

Two refinements enrich real spectra. First, light can only drive a vibration that changes the molecule's electric dipole moment, its separation of positive and negative charge. The symmetric molecules N₂ and O₂, 99 percent of dry air, stretch without any dipole change and barely absorb infrared at all. Carbon dioxide's bending vibration, at 667 inverse centimetres or about 15 micrometres, does change the dipole and lies near the peak of the infrared Earth radiates to space, a large part of why a trace gas can warm a planet. Second, real bonds are anharmonic, so their rungs creep together going up. Hydrogen chloride's fundamental lies at 2,886 inverse centimetres, but its first overtone, a jump from n = 0 straight to n = 2 that a perfect oscillator would forbid, lies at 5,668, about 1.8 percent short of double.

In 1926 Max Born, Werner Heisenberg and Pascual Jordan treated the electromagnetic field in a cavity as a collection of oscillators, and in 1927 Dirac built on that idea the first quantum theory of how atoms emit and absorb light. Break the field in a closed box into its modes, standing-wave patterns each with its own frequency ω. Maxwell's equations make each mode oscillate exactly like a mass on a spring, electric and magnetic energy trading places like potential and kinetic energy. Quantize each mode and its energy must be (n + ½)ħω, where n is the number of photons in it. A photon is one quantum of excitation of one field oscillator, and Planck's E = hf from chapter 3 is simply the rung spacing; emitting light raises a field oscillator by one rung.

The zero-point energy comes along too. With no photons at all, every mode still holds ½ħω, so the vacuum quivers with irreducible field fluctuations. These contribute to measurable effects, among them the tiny shift in hydrogen's energy levels that Willis Lamb and Robert Retherford measured in 1947. Yet summing ½ħω over infinitely many modes gives an infinite total. Laboratory experiments feel only differences in this energy, but gravity should feel all of it, and why the vacuum energy astronomers observe is fantastically smaller than such a sum suggests is among the deepest open problems in physics, waiting for us in chapter 25.

If every stationary state has a frozen probability density, how does a quantum spring ever swing? As in chapter 11, motion comes from superposition, and one superposition is special. In 1926 Schrödinger, seeking a bridge between his waves and Newton's mechanics, found that a ground-state Gaussian, shifted sideways and released, swings back and forth at exactly the classical frequency without spreading or changing shape, its centre tracing the path of a classical mass. This is a coherent state. It keeps the minimum uncertainty, Δx·Δp = ħ/2, at every instant, and its average position and momentum obey Newton's laws exactly.

On the ladder, a coherent state is a superposition of many rungs with a neat signature: the lowering operator returns the same state multiplied by a number. Its rung number, the photon count for light, is therefore not definite: repeated counts scatter in a Poisson distribution, the statistics of independent random events, with mean ⟨n⟩ and spread √⟨n⟩. The relative spread, 1/√⟨n⟩, shrinks as ⟨n⟩ grows, which is why billions of quanta look perfectly steady. In 1963 Roy Glauber, and independently George Sudarshan, made coherent states the foundation of the quantum theory of light; a stable laser well above threshold emits very nearly a coherent state, and Glauber shared the 2005 Nobel Prize in Physics for this work. Squeezing uncertainty from one quantity into another now even sharpens gravitational-wave detectors (chapter 24).