The quarter cycle a turning point costs

The action that knows where every path ends closes on the second life of the action. In quantum mechanics every path contributes a phase equal to its action divided by Planck’s constant, the classical paths are the ones whose phases are stationary, and where two of them merge the simple approximation needs a repair: a quarter-cycle phase lost at every focus a path passes through, which reappears as a measurable shift in energy levels. This essay is about that shift — where it comes from, how to count it, and what it looks like when a neutron is used to measure it.

The oldest form of quantum mechanics had no such shift. Bohr’s rule for the hydrogen atom, generalised by Sommerfeld in 1916, said that the allowed orbits of a periodic system are those whose action — the area enclosed by the orbit in the plane of position and momentum, $\oint p\,dx$ — is a whole number of Planck constants. It was a triumph for hydrogen and a failure almost everywhere else, and the failures had a pattern that took decades to state. The rule was always wrong by a fixed fraction of $h$, and the fraction depended on nothing but how the orbit turns round.

The area each state is allowed #

For a harmonic oscillator of angular frequency $\omega$, the orbit at energy $E$ is an ellipse in phase space of area $2\pi E/\omega$. Setting that to $nh$ gives $E = n\hbar\omega$ — including a state of zero energy, which the motion that cannot be stopped shows cannot exist, because a particle at rest at the bottom of a well would have a definite position and a definite momentum at once. The true levels are $(n + \frac12)\hbar\omega$.

For a particle in a box with hard walls, the orbit is a rectangle — constant speed one way, instantaneous reversal, constant speed back. The box that allows only some energies fits a whole number of half-wavelengths between the walls and gets levels proportional to $(n+1)^2$, counting from zero. The area of the rectangle at those energies is $(n+1)h$: one whole unit more than Bohr’s rule.

Neither set of energies is put into the drawing. Each is found as an eigenvalue of the Schrödinger equation written on a grid of six thousand points, and the orbit is then drawn at that energy and its area measured. The oscillator’s areas come out as 0.500, 1.500, 2.500, 3.500 Planck constants to three decimal places; the box’s as 1.000, 2.000, 3.000, 4.000. Both are whole numbers plus a fixed offset, and the offsets are a half and one. The obvious guess is that each is a property of its potential. It is not. Each is a count of the orbit’s turning points, weighted by how sharply the orbit turns.

A quarter cycle at a smooth turn #

The quantum wave that corresponds to an orbit oscillates wherever the particle could classically be, with a local wavelength set by the local momentum, $\lambda = h/p$. The semiclassical approximation writes the wave as a cosine of the accumulated phase, $\int k\,dx$ with $k = p/\hbar$, and a closed orbit gives a consistent wave only if the phase accumulated round it is a whole number of cycles. That is Bohr’s rule, and it would be exact if the phase were the whole story.

At a turning point the momentum falls to zero, the local wavelength becomes infinite, and the approximation fails; something happens there that the phase integral does not count. What happens depends on how the particle turns round.

The drawing asks the question directly. The exact wave near a smooth turning point — a uniform force, the simplest case — is computed by starting deep in the forbidden region, where it must decay, and integrating towards the allowed side. Far from the turning point it oscillates like a semiclassical wave, and fitting it there finds the phase the semiclassical wave must carry to match: 0.7881 radians, against $\pi/4 = 0.7854$. A wave that turns round smoothly leaves as though a quarter of a cycle had been spent at the turning point. The phases 0 and $\pi/2$ visibly miss. The residual difference of three thousandths of a radian is the finite distance over which the fit was made and shrinks further out.

The number on the drawing is the offset of the standing wave, and it is half the cost of the turn. The standing wave near a turning point is the sum of the wave arriving and the wave leaving, and its offset $\phi$ is the average of their phases; the outgoing wave is shifted from the incoming one by $2\phi$. An offset of $\pi/4$ is a reflection phase of $\pi/2$ — a quarter of a full cycle of $2\pi$ — and that is the cost the rule counts.

A hard wall does something different. The wave must vanish at the wall, so the incident and reflected waves cancel there and the reflection costs exactly half a cycle. A smooth turning point lets the wave leak a little way into the forbidden region before it turns — the exact curve in the drawing reaches past $x = 0$ and decays over a distance of order one — and the leak is what makes the loss a quarter rather than a half.

With those two costs the rule becomes $\oint p\,dx = (n + \mu/4)h$, where $\mu$ counts a one for every smooth turning point and a two for every hard wall. The harmonic oscillator has two smooth turning points: $\mu = 2$, $n + \frac12$. The box has two walls: $\mu = 4$, $n + 1$. The integer $\mu$ is called the Maslov index, after Viktor Maslov, who put it on a general footing in 1965; Joseph Keller had stated the same rule for waves in 1958, and Einstein had seen in 1917 that the classical rule needed rewriting in terms of invariant tori before any of it could be generalised.

A bounce that only one offset gets right #

The test of a counting rule is a case with one of each.

A particle resting on a hard floor in a uniform gravitational field turns round at the floor abruptly and at the top of its bounce smoothly. The rule says $\mu = 2 + 1 = 3$, so the levels should satisfy $\oint p\,dx = (n + \frac34)h$. The exact levels are zeros of the Airy function, and here they are computed, as before, from the Schrödinger equation on a grid.

The distinction is stark. With $\mu = 3$ the rule gets the ground state within three quarters of a per cent and the higher levels within hundredths of a per cent, the error falling steadily as the levels rise. With $\mu = 2$, the offset right for a smooth well, it is a quarter out at the bottom and still nearly two per cent out at the tenth level; $\mu = 4$, right for a box, errs by as much the other way. Only the count that matches the orbit’s actual turning points converges. The rule is not a fit with an adjustable half; it is a count, and the count has to be done.

Why the error falls with $n$ is part of the same picture. The quarter cycle is exact only far from the turning point compared with the local wavelength there. For higher levels the wavelength near the turning point is a smaller fraction of the orbit, the region where the approximation fails is proportionally smaller, and the rule becomes more nearly exact. Where the quantum picture hands back the old one argues that large quantum numbers bring back classical physics only in a particular, averaged sense; the corrected rule is one of the places where that return is exact in the limit and quantitatively measurable on the way.

The same quarter in a string and a rainbow #

The counting is a property of waves, not of quantum mechanics, and it can be checked on waves nobody would call quantum. A guitar string fixed at both ends is the box: two hard reflections, half a cycle each, and only some notes fit finds the allowed wavelengths are those that put a whole number of half-wavelengths between the ends, exactly the $n + 1$ of the rectangle in phase space. Sound in a pipe closed at one end and open at the other has one reflection of each kind and a series of odd harmonics, the acoustic cousin of the bouncer’s $n + \frac34$ with the open end’s reflection playing the smooth turn.

Light supplies the fold in its purest form. The angle the rainbow has to be finds the rainbow at the angle where the rays leaving a raindrop pile up and turn back — a caustic, where neighbouring rays cross, which is exactly what a turning point is in one dimension. On the bright side the ray picture predicts light and on the dark side none, and the wave that fills in the boundary is an Airy function, the same function the bouncer’s states are made of. The fringes below the rainbow are its oscillations, and George Airy’s 1838 theory placed them using the same quarter-cycle shift at the caustic. A beam of light passing through a focus picks up a similar extra phase, a quarter cycle for a line focus and a half for a point, which is the Gouy phase that decides where the modes of a laser cavity resonate. In each case the wave has passed through a place where the rays converge, and in each case the geometry of the convergence sets a fixed fraction of a cycle.

When the rule is exact, and when it is close #

For three of the five potentials the corrected rule is exact at every level, to the precision of the computation: the box, the harmonic well, and the half well — a harmonic well with a hard wall at its centre, whose levels are the odd states of the full oscillator and whose rule is $n + \frac34$. Those three are special: their exactness follows from symmetries of the underlying equation that the approximation happens to respect. For the bouncer and for a quartic well, $x^4$, the rule is an approximation, poorest for the ground state — eighteen per cent low for the quartic — and closing on the exact values as the quantum number rises.

This is the honest shape of the result. The quarter-cycle correction does not make semiclassical quantisation exact; it makes it right in the limit and wrong by a controlled amount elsewhere. What it removes is the error that does not shrink — Bohr’s rule is out by half a unit of action at every level of the oscillator, no matter how high — and what is left is an error that vanishes as the quantum number grows.

A neutron’s heights above a mirror #

The bouncer is not a textbook invention. In 2002 a group at the Institut Laue-Langevin in Grenoble let ultracold neutrons — neutrons moving at a few metres per second — slide horizontally over a flat mirror under a second absorbing surface, and measured how many got through as the gap between the two was narrowed. Below about fifteen micrometres almost none did. The neutrons were in the quantum states of the bouncer.

The natural length for a neutron in the Earth’s gravity is 5.87 micrometres, and the lowest state’s classical turning height is 13.7 micrometres, with the probability peaking about eight micrometres above the mirror. A neutron cannot be held lower than about ten micrometres above a mirror, however slowly it moves — the zero-point energy of the bouncer, 1.41 pico-electronvolts, keeps it up. The next states are at 24.0, 32.4 and 39.8 micrometres. Later experiments drove transitions between these states with a vibrating mirror and measured their energy differences to parts in a thousand, which makes them a laboratory test of gravity at micrometre distances; a new force of short range would move the levels.

The experiment works because the neutron’s motion separates. Horizontally it slides at a few metres per second over a mirror some centimetres long, and that motion is entirely classical; vertically it has an energy of a pico-electronvolt, and that motion is entirely quantum. The vertical part is exactly the bouncer: a hard floor, since the mirror’s surface reflects slow neutrons as a wall; a smooth turning point in the Earth’s field above it; and nothing else, since a neutron has no charge for stray electric fields to push. Its natural length, $(\hbar^2/2m^2g)^{1/3}$, contains the neutron’s mass twice and the acceleration of gravity once, so the measured heights are at the same time a measurement of $g$ at micrometre scale and of Planck’s constant over the neutron’s mass.

The first level’s height, computed with $\mu = 3$ instead of the exact Airy zero, would be 0.8 per cent low — a tenth of a micrometre. With Bohr’s original rule the ground state would sit on the mirror.

Where the counting rule stops #

One dimension, or motion that separates into one-dimensional pieces. The rule as drawn applies to an orbit on a line. In more dimensions it applies to motion that lies on tori in phase space — systems with as many conserved quantities as dimensions — and each independent loop round the torus gets its own action and its own Maslov index. Einstein’s point of 1917 was that for chaotic motion there are no tori at all.

Isolated turning points. The quarter cycle assumes the turning point is far, in wavelengths, from any other feature of the potential. Two turning points close together, as at the top of a barrier or the bottom of a shallow well, interfere, and the phase lost is neither a quarter nor a half but something in between that depends on the energy — the regime in which the wall that is not quite a wall lets a particle tunnel.

Grid-computed exact levels. The comparison energies here come from a finite-difference solution of the Schrödinger equation on six thousand points. Its own error is a few parts in a million for the lowest levels, far below the differences being measured.

A state is not an orbit #

The orbit figure draws a closed curve for each state and shades its area, which is the rule’s own picture, and it misleads in one way worth naming. A quantum state is not an orbit. Its probability is spread over the whole ellipse and leaks outside it, into the forbidden region the classical orbit never reaches — the exact wave in the second figure reaches past the turning point before it decays. The area rule describes which energies occur, not where the particle is. And none of the figures shows the geometric origin of the quarter cycle in the general theory, where the Maslov index counts the times a family of classical paths folds over itself — the caustics at which neighbouring trajectories focus, the conjugate points of the longest way round is the shortest clock — and a turning point is simply the one-dimensional case of a fold.

Still open: quantising what has no tori #

The corrected rule needs closed orbits on tori. A chaotic system has none; its classical paths wander over the whole energy surface. Martin Gutzwiller showed in 1971 that the density of a chaotic system’s quantum levels can still be written as a sum over its periodic orbits, each contributing a phase from its action and its own Maslov index. But the number of periodic orbits grows exponentially with their length, and the sum does not converge in general. Techniques for resumming it have reproduced the levels of particular systems — the helium atom’s doubly excited states, billiards of certain shapes — and whether the sum can be made to work in general, and which properties of a quantum system its classical orbits actually fix, remains open.

The habit worth carrying away is to ask what an approximation does at the place where it breaks. A formula that fails at isolated points can still be exact everywhere else, provided the failure is replaced by the right constant, and the constant usually counts something simple. The quarter cycle at a smooth turn and the half cycle at a wall are that constant for waves; Bohr’s rule was wrong by exactly their sum at every level, and correcting it needed no new physics, only a count of how many times the orbit turns round and how.