← quantum · contents

§ Phase v · Superposition & Time · Chapter 02

Time evolution

An eigenstate is a humming chord that never changes its loudness, only its phase. Glue two of those chords together and the wavefunction begins to slosh, on a timetable set by the difference of their energies. That single fact is why excited atoms emit light, why molecules vibrate at specific colors, and why the Schrödinger equation is, deep down, the law of a rotating arrow.

In the winter of 1925 Erwin Schrödinger took a vacation in the Swiss Alps with a notebook and (legend says) a quiet companion who was not his wife. He came back with the first six pages of an equation that would replace Newton’s force-law for the world of atoms. The version he published in early 1926 looked, at first glance, like nothing more than a fancy diffusion equation with an imaginary unit pasted on the front. But that imaginary unit changes everything. A diffusion equation forgets. The Schrödinger equation remembers. Where diffusion smears its initial condition into mush, the Schrödinger equation rotates it. Every solution carries a phase, and the phase carries the music.

The time-dependent Schrödinger equation, in its compact bra-ket form, reads

iℏ ∂_t |ψ(t)⟩ = Ĥ |ψ(t)⟩

and the whole rest of this chapter is a slow, careful reading of what it says. We have already met its star-shaped solutions, the energy eigenstates: special wavefunctions ψ_n with the property that the Hamiltonian Ĥ acting on them gives back the same shape multiplied by a number, Ĥ ψ_n = E_n ψ_n. We met the building blocks of hydrogen this way (1s, 2p, 3d, and the rest of the ladder). What we have not yet asked is how those shapes change as time ticks forward. The answer is one of the few results in physics that is simultaneously trivial to write down and deep enough to power a working laser. Plug the eigenstate ansatz into the equation, separate the time piece from the spatial piece, and you find that the spatial shape never moves. Only one thing changes: a single complex number out front, e^(-iE_n t/ℏ), winding around the unit circle at the rate E_n/ℏ.

The phase-only fact is the cleanest example of a real physical asymmetry: a wavefunction is a complex object, but the probabilities it predicts are real. The probability density at a point is |ψ(x, t)|², and the absolute value squared of e^(-iE_n t/ℏ) ψ_n(x) is just |ψ_n(x)|². The time factor has unit modulus. It cancels. The cloud you would draw if you took a million snapshots of the electron in a pure energy eigenstate is the same cloud at noon, midnight, and ten thousand years from now. This is why energy eigenstates are also called stationary states: their visible behavior is frozen, even though the invisible phase is spinning at a respectable fraction of a petahertz.

That makes the language of textbooks finally honest. When we drew the 1s orbital as a fuzzy ball and the 2p as a dumbbell, we were drawing exactly the right pictures, provided the atom is sitting in one of those states. The pictures do not move, because nothing measurable about them moves. But the moment you build a superposition, the moment you write ψ = (ψ_a + ψ_b)/√2 with two different energies in play, the cancellation breaks. The two phases tick at different rates, and a cross term appears in |ψ|² that depends on their difference. The cloud begins to throb.

The Wikipedia paragraph hides a piece of vocabulary that will pay off for the next two chapters. The set of energy eigenstates is a basis. Any wavefunction at all (any normalizable ψ) can be written as a sum of those basis states with complex coefficients A_n. And if you know what each basis state does in time (it picks up its e^(-iE_n t/ℏ) phase) then by linearity you know what every wavefunction does. Time evolution becomes a list of independent rotations, one per basis state. There is no coupling. There is no mixing. The hard problem of “how does this complicated wavefunction evolve?” reduces to “spin each piece at its own rate, then add them back up.”

This is why the Schrödinger equation is, in the bones, simple. Linear. Decoupled in the right basis. The reason quantum mechanics gets a reputation for being hard is that we are usually handed wavefunctions in the wrong basis (the position basis) and have to do work to translate. But once you do that work once, by diagonalizing the Hamiltonian, the dynamics is just a collection of phase clocks ticking at their own frequencies.

A picture helps. Imagine each basis state as a complex arrow attached to a unit circle. At time zero, both arrows point along the real axis. As time advances, each arrow rotates clockwise at angular speed E_n/ℏ. A heavier state (higher energy) rotates faster. A lighter state rotates slower. What you ever actually see (probability density) depends on the relative angle between the two arrows. They start aligned; the cloud is fattest where both basis states are large. A quarter of a beat later, the arrows are 90° apart; the cross term has vanished, and the cloud looks like a simple sum of two independent clouds. Another quarter and the arrows point opposite; the cross term flips sign, and the lobe has slid to the other side of the atom. Each full revolution returns the cloud to its starting configuration. The cloud breathes. The beat period is fixed by the energy difference.

ψ(x, 0) = (1/√2) [ ψ_a(x) + ψ_b(x) ]

ψ(x, t) = (1/√2) [ e^(-i E_a t / ℏ) ψ_a(x) + e^(-i E_b t / ℏ) ψ_b(x) ]

|ψ(x, t)|² = (1/2) [ |ψ_a|² + |ψ_b|² + 2 Re( e^(i (E_a - E_b) t / ℏ) ψ_a* ψ_b ) ]

|ψ(x, t)|² = (1/2) [ |ψ_a|² + |ψ_b|² + 2 ψ_a(x) ψ_b(x) cos(ω_ab t) ]

ω_ab = (E_a - E_b) / ℏ

A concrete example is worth a dozen formulas. Take hydrogen and build the simplest interesting superposition: |1s⟩ plus |2p_z⟩, normalized. Both are real wavefunctions (we are free to pick real basis states for a real Hamiltonian), and their energies are E_1 = −0.5 Ha and E_2 = −0.125 Ha in atomic units. The energy difference is 0.375 Ha = 3/8 Ha. Plugging into the beat formula, ω = 3/8 in atomic units, so the period is T = 2π / (3/8) ≈ 16.76 atomic time units, which is about 405 attoseconds, or 0.4 femtoseconds. Less than half a femtosecond per cycle. A single oscillation of the simplest two-orbital superposition in hydrogen takes a hundred-thousandth of the time light needs to travel one millimeter.

What does the cloud actually do during those 405 attoseconds? The 1s orbital is a round ball centered on the proton. The 2p_z orbital is a dumbbell aligned along the z-axis, positive on the +z lobe and negative on the −z lobe. Multiply 1s by 2p_z and you get a function that is positive on the top half of the atom and negative on the bottom. Multiply by cos(ω t) and the whole interference term flips sign every half-period. At t = 0 the cosine is +1, the interference adds to the top and subtracts from the bottom, and the cloud bulges upward. At t = T/4 the cosine is zero, the interference vanishes, and the cloud looks like a featureless average of the two orbitals. At t = T/2 the cosine is −1, interference subtracts on top and adds on bottom, and the cloud bulges downward. Then it climbs back up. The electron, classically speaking, is sloshing up and down the z-axis with period T.

This sloshing is the moving electric dipole that couples to the electromagnetic field. A classical accelerating charge radiates at the frequency at which it accelerates. Schrödinger’s electron in the |1s⟩ + |2p_z⟩ superposition has an oscillating expectation value of position, ⟨z(t)⟩ ∝ cos(ω t), which means an oscillating expectation value of dipole moment, which means an oscillating source for Maxwell’s equations. The atom emits light at exactly ω = (E_2 − E_1)/ℏ, which is the Lyman-alpha frequency, which is the 121.6 nm line in the ultraviolet that has been measured to staggering precision since the 1920s. This is Bohr’s frequency condition falling out of Schrödinger’s equation almost by accident. The colors of every emission spectrum in chemistry, from the orange of a sodium lamp to the green of a mercury arc to the deep red of the hydrogen-alpha line that astronomers use to map nebulae, are differences of energy levels divided by ℏ. The atom is, in this sense, a tiny radio antenna whose carrier frequency is set by quantum arithmetic.

There is a final, important way to read the same equation. Time evolution under a constant Hamiltonian is a unitary operator, which is the quantum version of a rigid rotation. Write it as Û(t) = e^(−iĤt/ℏ) and the rule for evolving any state in time is just |ψ(t)⟩ = Û(t)|ψ(0)⟩. Unitary means norm-preserving: total probability stays at 1 forever, no matter how long you wait or how complicated Ĥ is. This is the quantum version of saying that no probability ever leaks out of the universe. It is also what distinguishes Schrödinger’s equation from any thermodynamic story: a diffusion equation increases entropy, a Schrödinger equation does not. The wavefunction does not relax. It rotates.

And that rotation is what makes superposition into something dynamic instead of merely arithmetic. Adding two energy eigenstates by hand at time zero is a trivial pen-and-paper exercise. The interesting part is what happens next, automatically, while you sleep. The two pieces drift apart in phase. The interference pattern in |ψ|² begins to move. The expectation value of every operator that does not commute with Ĥ starts to oscillate. The atom that you set up in a pure 1s state is invisible to a passing photon of the wrong color; the atom you set up in (1s + 2p)/√2 is a tiny radio that broadcasts at the Bohr frequency for as long as the superposition survives. The simple act of choosing a non-stationary state is the act of bringing the atom to life.

When you turn off the math and just look at the pendulums in the hero photo, you can almost feel the same thing. Two oscillators of slightly different periods. A slow envelope. The envelope is the beat, the beat is the interference, the interference is the cross term. In the next chapter we will follow that beat into the lab, where it appears in real experiments as quantum beats: weak, slow oscillations in the fluorescence of atoms that were prepared in coherent superpositions of two excited states. The math we just did predicts those beats down to the last digit, and physicists have been using them since the 1960s to measure energy splittings with breathtaking precision.