Time and Change

Ammonia, NH₃, is a squat little pyramid: three hydrogens in a triangular base, one nitrogen perched above. Or — below. Both are perfectly good homes, mirror images, same energy. A classical marble in one of two valleys would simply stay where it was put; the wall between the valleys is far too high to climb. But this is a quantum nitrogen. It does not climb the wall. It leaks through, the way amplitude leaked through both slits in Chapter 1, and so the real molecule is never settled: put the nitrogen definitely on top, wait, and you will find the probability sloshing — top, bottom, top, bottom — about twenty-four billion round trips per second.

Nobody deduced this from a lecture; the sloshing broadcasts itself. A gas of ammonia absorbs and emits microwaves at exactly 24 GHz (a wavelength of about a centimeter), and in 1954 that oscillation was persuaded to run the first maser — the laser's microwave parent. Here is the molecule's inner life, with the knobs exposed:

Two knobs, two discoveries. The asymmetry knob tilts the two homes apart in energy — and the sloshing chokes off. Amplitude only flows freely between options of equal energy; remember that, it is the germ of every resonance. And the ground state button: press it and all motion stops dead, forever. There exist special states that do not slosh — and finding them, for any system whatsoever, is the entire technical content of quantum dynamics. Let us see why.

That upper level is not just a curiosity — it can be put to work. Sort a beam of ammonia so that only the higher-energy molecules survive, coax them past a resonant cavity tuned to 24 GHz, and each one dumps its energy into the cavity in step with the radiation already there. The result is a self-sustaining microwave oscillator of extraordinary purity — the maser (microwave amplification by stimulated emission of radiation). Figure 5.2 is the machine.

We get to derive the shape of all quantum motion from one non-negotiable demand: the probabilities must keep adding to one. An atom that is somewhere now is somewhere later. Write the evolution over time t as an operator, $|\psi (t)⟩=\hat{U}(t)|\psi (0)⟩$. Conservation of total probability says $⟨\psi (t)|\psi (t)⟩=1$ always, i.e. ${\hat{U}}^{†}\hat{U}=1$ — evolution is unitary: a rigid rotation of the state space. Nothing stretches, nothing tears; the arrow of state just turns.

Now look at an infinitesimal turn. Any unitary close to the identity has the form $\hat{U}(dt)=1-\frac{i}{\hslash }\hat{H}dt$ with $\hat{H}$ Hermitian — check it: unitarity to first order demands exactly $\hat{H}={\hat{H}}^{†}$. So a Hermitian operator generates time evolution, and stacking infinitesimal turns gives

That is the time-dependent Schrödinger equation (TDSE) — not pulled from a hat, but forced by “probability survives.” Erwin Schrödinger wrote it down in 1926, in the fourth of the six papers that founded wave mechanics; we have simply recovered it from a single demand rather than guessed it.

One caveat on that exponential, and it will matter in Section 5.5. The compact form $\hat{U}(t)=e^{-i\hat{H}t/\hslash }$ holds only when $\hat{H}$ does not itself depend on time. If it does — and a spin being shouted at by a rotating field is exactly that case — then the infinitesimal turns no longer commute with one another, the exponents refuse to add, and stacking them gives a time-ordered product instead of a single exponential. The differential equation $i\hslash {\partial }_t|\psi ⟩=\hat{H}(t)|\psi ⟩$ remains exactly true; it is only the closed-form solution that is lost. That is precisely why the lab of Section 5.6 integrates $\hat{H}(t)$ step by numerical step rather than exponentiating a matrix.

Which Hermitian operator is it? Here we take the one clue nature gives: the states that do not move. Feed an eigenstate $\hat{H}|E⟩=E|E⟩$ into (5.1):

an overall phase — invisible, as you proved in Chapter 3's hard exercise. Nothing measurable changes: these are the stationary states, the frozen bars of the widget above. And their frequency of invisible turning is $\omega =E/\hslash$ — the generator's eigenvalue is energy, and energy is nothing but the rate of the quantum phase clock. Planck's $E=\hslash \omega$ is not a law about light; it is the definition of what energy is in quantum mechanics.

Then where does visible motion come from? From superpositions. Two energy components turn at different rates, and their relative phase — which is measurable — beats at the difference frequency:

All quantum dynamics is beat notes between energy levels. Spectral lines, chemical reaction rates, the color of gold, the ammonia maser — every “how fast?” in the quantum world is answered by an energy difference. One corollary before we cash this in: using (5.1), any observable's average moves as

so whatever commutes with $\hat{H}$ is conserved — the deepest sentence in physics (symmetries make conservation laws) arriving here as a one-line consequence of Chapter 3's commutator.

Back to ammonia, now with tools. Two base states, $|L⟩$ and $|R⟩$, same energy (call it zero), and one crucial off-diagonal number: the amplitude per unit time to leak through the wall, $-A$. The Hamiltonian and its eigenstates:

The stationary states are not “left” or “right” but the symmetric and antisymmetric blends — the molecule's true energy levels, split by $2A$. Start the nitrogen at $|L⟩=(|+⟩+|-⟩)/\sqrt{2}$ and let (5.3) do the rest: the two components beat at $\omega =2A/\hslash$, and

Complete flip-flop, at the beat frequency of the doublet — and 24 GHz is nothing but $2A/h$ for ammonia's actual tunneling amplitude (about 50 µeV — a millionth of a chemical bond, which is why the wall is opaque to classical thinking and translucent to amplitudes). Tilt the two wells by ε (an electric field pulls on the molecule's dipole) and the same algebra gives a maximum transfer of $A^2/(A^2+{\epsilon }^2/4)$ — the choking you found with the slider. File that formula's shape away; in two sections it becomes the most famous curve in spectroscopy.

The same mathematics wears a more graceful costume on the Bloch sphere. Put a spin-1/2 in a magnetic field along z: $\hat{H}=\frac{\hslash {\omega }_0}{2}{\sigma }_z$, stationary states up and down, splitting $\hslash {\omega }_0$. Any other state is a superposition of the two, so by (5.3) it beats at ${\omega }_0$ — and on the sphere that beat is pure geometry: the Bloch vector sweeps a cone about the field axis. Larmor precession. The spin is a gyroscope whose rate reads the field:

Now switch on the drive and watch the second act — a tiny transverse field, rotating in step with the precession. In the spin's own rotating point of view that little field stands still, and a constant field does what constant fields do: precess the spin around itself, slowly, majestically, all the way from north to south and back. That slow roll is Rabi flopping, and the swing-pushing analogy is exact: on the beat, tiny pushes accumulate without limit; off the beat, they cancel. Nature runs on the same trick your six-year-old discovered at the playground.

Make the rotating-frame argument honest. Split the spin by $\hslash {\omega }_0$, drive at frequency ω with strength Ω, and transform to the frame rotating with the drive. The algebra (exercise 5, and exact for a circularly rotating drive) leaves a time-independent two-level Hamiltonian with diagonal splitting $\hslash \delta$ (detuning $\delta =\omega -{\omega }_0$) and off-diagonal coupling $\hslash \Omega /2$ — which is precisely the tilted ammonia molecule of Section 5.3. Same matrix, same solution, new names:

One formula, three appearances in one chapter — the tunneling doublet, the tilted molecule, the driven spin. When the same equation keeps arriving in different costumes, you have found one of nature's load-bearing walls. The Rust lab integrated the true lab-frame Schrödinger equation — the drive wiggling many times faster than the flopping it causes, with no rotating-wave shortcut anywhere — and then ran the counting experiment. The ratio it actually used is printed on the chart:

The lab Rust-QP/ch05-dynamics is our first numerical integrator — the workhorse skill for the rest of the book, where Hamiltonians stop being analytically solvable. The classic fourth-order Runge–Kutta stepper, applied to $\dot{c}=-i\hat{H}(t)c/\hslash$, is fifteen lines:

1/// i dc/dt = H(t) c  =>  dc/dt = -i H(t) c. One RK4 step of size dt.
2fn rk4_step(h: &dyn Fn(f64) -> [[C; 2]; 2], t: f64, dt: f64, y: State) -> State {
3    let f = |t: f64, y: State| -> State {
4        let m = h(t);
5        [
6            -c(0.0, 1.0) * (m[0][0] * y[0] + m[0][1] * y[1]),
7            -c(0.0, 1.0) * (m[1][0] * y[0] + m[1][1] * y[1]),
8        ]
9    };
10    let add = |a: State, b: State, s: f64| -> State { [a[0] + b[0] * s, a[1] + b[1] * s] };
11    let k1 = f(t, y);
12    let k2 = f(t + dt / 2.0, add(y, k1, dt / 2.0));
13    let k3 = f(t + dt / 2.0, add(y, k2, dt / 2.0));
14    let k4 = f(t + dt, add(y, k3, dt));
15    [
16        y[0] + (k1[0] + k2[0] * 2.0 + k3[0] * 2.0 + k4[0]) * (dt / 6.0),
17        y[1] + (k1[1] + k2[1] * 2.0 + k3[1] * 2.0 + k4[1]) * (dt / 6.0),
18    ]
19}

And a confession worth more than the code. Our first run of the driven-spin experiment failed its own assertion: the numeric flip probability disagreed with equation (5.7) by a factor of nearly one. The bug: the drive's rotation sense — one sign in one exponent — was counter-rotating against the spin's precession, pushing the swing exactly off the beat. The physics caught the physicist:

1/// Lab-frame Hamiltonian of a driven spin: splitting w0, circularly
2/// rotating transverse drive of amplitude Omega at frequency w.
3///   H(t) = (w0/2) sigma_z + (Omega/2)[cos(wt) sigma_x + sin(wt) sigma_y]
4/// Matrix form: upper-right element (Omega/2) e^{-i w t}. The sign matters:
5/// the drive must co-rotate with the spin's own precession — flip it and
6/// you drive the resonance at -w0 instead (our first buggy run proved it).
7fn rabi_h(w0: f64, omega: f64, w: f64) -> impl Fn(f64) -> [[C; 2]; 2] {
8    move |t: f64| {
9        let phase = c(0.0, -w * t).exp() * (omega / 2.0);
10        [[c(w0 / 2.0, 0.0), phase], [phase.conj(), c(-w0 / 2.0, 0.0)]]
11    }
12}

With the sense fixed, the referees pass. Each one is a race between two independent code paths — an RK4 integration on one side, a closed form or a counting experiment on the other — and each tolerance is parked at the accuracy actually achieved, with a small factor of headroom and no more. A tolerance set four orders above what your code delivers is not a referee; it is a rubber stamp that will keep stamping long after the physics has rotted. Note the order of the two assertions inside push: finite first, small second. A NaN is smaller than nothing at all.

1// --- the referees: assert FINITE before asserting small ---------------
2    let mut referees: Vec<Referee> = Vec::new();
3    let mut push = |name: &str, value: f64, tol: f64| {
4        assert!(value.is_finite(), "{name} is not finite: {value}");
5        referees.push(Referee { name: name.into(), value, tol, pass: value < tol });
6    };
7    // Tolerances sit at the achieved accuracy (~1.3x headroom), never above it.
8    push("Norm drift, time-independent H (ammonia)", norm_err, 2e-14);
9    push("Norm drift, time-dependent H(t) (driven spin)", rabi_norm_err, 6e-14);
10    push("Ammonia flip: RK4 vs sin²(A t)", flip_err, 1e-12);
11    push("Larmor: RK4 ⟨σx⟩ vs cos(ω₀ t)", prec_err, 2e-12);
12    push("Rabi traces: lab frame vs rotating frame", rabi_err, 5e-11);
13    push("Resonance peak: RK4 P(t*) vs Ω²/(Ω²+δ²)", peak_err, 1e-12);
14    push("Counting experiment: |measured − theory| in σ", count_sigma, 3.5);

And here is that scoreboard, read live out of the JSON the lab just wrote — not transcribed into this page by hand, which is how published numbers go stale:

Two of them are worth a second look. The norm drift is measured twice, because conserving probability under a frozen Hamiltonian is the easy half; the driven spin's $\hat{H}(t)$ turns over many times per Rabi period, and it is that harder number the scoreboard reports. And the last referee is a statistical one: the whole detuning sweep of synthetic atoms, graded in standard deviations rather than absolute error, because a counting experiment that agreed to 10⁻¹² would mean the noise had been faked. When you write your own labs, keep the assertions merciless — they are the difference between a simulation and a cartoon.

1. (F) Show that $\hat{U}=e^{-i\hat{H}t/\hslash }$ is unitary if and only if $\hat{H}$ is Hermitian, and derive equation (5.4) from the Schrödinger equation and its conjugate.
2. (F) For the ammonia Hamiltonian (5.5), carry out the two-line diagonalization and verify (5.6). Then add the tilt ε and derive the choked maximum $A^2/(A^2+{\epsilon }^2/4)$.
3. (C) In the driven-spin widget, set δ/Ω = 0, switch the drive on (which relaunches the spin from the north pole with the clock at zero), and read the elapsed-t counter when $⟨{\sigma }_z⟩$ returns to $+1$. Simulated time runs 1:1 with the wall clock, so that number should be 2π/Ω for the displayed Ω. Then find the detuning at which the spin never drops below the equator, and verify it is δ = Ω.
4. (F, hard) Energy–time uncertainty, done honestly: there is no time operator, so derive the Mandelstam–Tamm relation instead. Combine Robertson (4.5) for $\hat{H},\hat{A}$ with equation (5.4) to show $\Delta E\cdot {\tau }_A\ge \hslash /2,{\tau }_A\equiv \frac{\Delta A}{|d⟨\hat{A}⟩/dt|}$ — the time for any observable to change by one standard deviation. Evaluate $\tau$ for the ammonia flip and check it against (5.6).
5. (F, hard) Do the rotating-frame transformation without shortcuts: with $\hat{R}(t)=e^{+i\omega t{\sigma }_z/2}$ and the lab Hamiltonian of the Rust lab, compute ${\hat{H}}^{\prime }=\hat{R}\hat{H}{\hat{R}}^{†}+i\hslash \dot{\hat{R}}{\hat{R}}^{†}$ and show it is exactly time-independent for a circularly rotating drive — then show it is not for a linearly polarized one, and identify the leftover term (the seed of the Bloch–Siegert shift in exercise 6).
6. (P) Change the lab's drive to linear polarization, $\Omega \cos (\omega t){\sigma }_x$. The resonance survives (half the amplitude works with you, half against), but the peak moves. Sweep carefully and measure the shift for several Ω; compare with the Bloch–Siegert prediction ${\delta }_{\text{peak}}\approx {\Omega }^2/4{\omega }_0$.
7. (P, hard) Spin echo. Simulate 1,000 spins with random detunings drawn from a Gaussian (an inhomogeneous sample). Apply a π/2 pulse, watch the net transverse signal $⟨{\sigma }_x⟩$ decay as the spins fan out (T₂* dephasing) — then hit a π pulse at time τ and watch the fan close into an echo at 2τ. Plot signal vs time. You have reproduced Hahn's 1950 discovery, the foundation of MRI contrast — and proved this “decay” was never true information loss.