The angle the uncertainty principle cannot be written for

Sharpness has to be paid for found the uncertainty principle inside Fourier analysis: a wave that is short must be made of many wavelengths, and the widths of a function and of its transform trade against each other exactly. The questions that can be asked together rederived it from matrices, showing that two quantities can be definite together only when their operators commute. The fastest a state can stop being itself found that energy and time do not form such a pair, because time has no operator, and replaced the naive relation with a sharper theorem about how fast a state can change.

Angle and angular momentum are the next pair that looks like position and momentum and is not. The angular momentum about an axis is the quantity that produces rotations, exactly as momentum produces translations, and its operator is $-i\hbar\,\partial/\partial\varphi$, exactly as momentum’s is $-i\hbar\,\partial/\partial x$. It is tempting to write $\Delta\varphi\,\Delta L \ge \hbar/2$ and move on. The relation is false, and a textbook state violates it as badly as possible. The reason is that an angle lives on a circle and a position does not, and the repair shows what an uncertainty relation really measures.

A sharp angle, and a spread of turning #

A particle on a ring, or a rigid rotor turning about one axis, has a wavefunction $\psi(\varphi)$ that must return to itself after one turn. It can therefore be written as a Fourier series in $e^{im\varphi}$, and the angular momentum can only take the values $m\hbar$ with $m$ a whole number.

The figure draws three states, each as its distribution of angle on the left and of angular momentum on the right. The first is tightly bunched around one angle, and its angular momentum is spread over several values, with a spread of 2.21ħ. The second is loosely bunched and its angular momentum is concentrated on the three values nearest zero. The third has no preferred angle at all. Its probability is the same everywhere on the circle, and its angular momentum is exactly zero: a single bar.

The first two behave as the Fourier rule predicts. A narrow angular distribution needs many values of $m$ to build it, a wide one needs few. The third is where the naive relation breaks. Its angular momentum has zero spread. Its angle, measured on the interval from $-\pi$ to $\pi$, has a spread of $\pi/\sqrt{3}$, which is finite, because an angle cannot be spread further than all the way round. The product is zero.

The rule that is broken on a circle #

The failure is not confined to that one state.

The figure computes the product $\Delta\varphi\,\Delta L$ for a whole family of states, called von Mises states after the statistician whose distribution on a circle they follow: the probability of each angle is proportional to $e^{\kappa\cos\varphi}$, and $\kappa$ sets how tightly the state is bunched. For large $\kappa$ the state is a narrow bump on the circle, the circle is irrelevant, and the rotor behaves like a particle on a line. The product sits just above $\hbar/2$, as Heisenberg requires. As $\kappa$ falls the product falls with it. It crosses $\hbar/2$ at $\kappa = 1.58$ and goes on falling to zero as the state spreads round the circle.

The derivation of the naive relation uses the commutator $[\varphi, L] = i\hbar$, obtained by the same algebra that gives $[x, p] = i\hbar$. For a line the algebra is sound. For a circle it is not, and the reason is concrete. Multiplying a periodic wavefunction by $\varphi$ produces a function that jumps by $2\pi\psi$ where $\varphi$ wraps from $\pi$ back to $-\pi$. That new function is not periodic, so it is not a wavefunction on the circle, and the commutator has been evaluated on something outside the space it was meant to act on. The contradiction the third state produces is the price of that step. The angle, as a number between $-\pi$ and $\pi$, is simply not a good quantum observable on a circle, for the same reason that it has a discontinuity where the interval’s ends meet.

The relation that holds #

What is a good observable on a circle is any periodic function of the angle. The natural ones are $\cos\varphi$ and $\sin\varphi$, which together are the position of a point on the unit circle, and their commutators with the angular momentum are clean:

$$[L, \cos\varphi] = i\hbar\sin\varphi, \qquad [L, \sin\varphi] = -i\hbar\cos\varphi.$$

Each gives a Robertson inequality: $\Delta L\,\Delta\cos\varphi \ge \tfrac{\hbar}{2}|\langle\sin\varphi\rangle|$ and $\Delta L\,\Delta\sin\varphi \ge \tfrac{\hbar}{2}|\langle\cos\varphi\rangle|$. Squaring and adding them gives a single relation. Write $R = |\langle e^{i\varphi}\rangle|$ for the length of the average position on the unit circle — one for a state sharply at one angle, zero for a state spread evenly. The spreads of cosine and sine add to $1 - R^2$, the circular spread, and the relation becomes

$$\Delta L^2 \ge \frac{\hbar^2}{4}\,\frac{R^2}{1 - R^2}.$$

The figure plots the angular-momentum spread against the circular spread for the von Mises family and for a few wrapped Gaussians — Gaussian distributions wound round the circle — together with the bound. Every state lies above it. The von Mises states lie close: at $\kappa = 2$ the spread is $0.591\hbar$ against a bound of $0.487\hbar$. Nothing approaches it from below.

The relation behaves correctly at both ends. When the state is sharply bunched, $R$ is close to one and $1 - R^2$ is approximately the square of the angle’s ordinary spread, so the relation reduces to Heisenberg’s: sharp angles still cost spread-out angular momentum. When the state is spread round the circle, $R$ goes to zero and the bound goes to zero, allowing a state of definite angular momentum, as there must be. The third state of the first figure is not a violation of this relation; it is its extreme case.

The lesson is about what an uncertainty relation measures. The spread of a quantity is a meaningful measure of its indefiniteness only when the quantity lives on a line. On a circle the natural measure of spread is how far the average of the point on the circle falls short of the rim, and with that measure the Fourier trade-off is restored.

A slice of a beam #

The relation has been tested with light. A light beam can carry orbital angular momentum, a twist of its wavefronts around its axis, in whole-number units of $\hbar$ per photon, and the angle around the axis is its conjugate.

The figure computes the orbital-angular-momentum spectrum of a beam passed through an aperture shaped like a slice of pie. The spectrum is the Fourier series of the aperture, a sinc-squared function, and its width is inversely proportional to the slice’s angle: a 90-degree slice has its first zeros at $\pm4$, a 30-degree slice at $\pm12$. Cutting the beam’s angle down spreads its twist out, exactly as the Fourier rule says.

The spectrum shows one more subtlety. A slice with hard edges has, strictly, an infinite spread of angular momentum, because the sinc’s tails fall only as $1/m^2$ and the variance sums them weighted by $m^2$. Real apertures have edges that are not perfectly sharp, and real detectors see a finite range of $m$. Franke-Arnold, Barnett, Padgett and colleagues measured such spectra in 2004 and tested a form of the relation that Barnett and Pegg had derived for an angle measured within a chosen window of $2\pi$, in which the bound is reduced by the probability that the state has at the window’s edge. The measurements agreed with it. The window-edge term is another face of the same problem: an angle measured on an interval is fine until the state has weight near where the interval wraps round.

The phase of a few photons #

The same structure appears wherever a quantity comes in whole numbers and its conjugate is periodic, and the most important example is the phase of a light wave. The number of photons in a single mode of light is a whole number, and the phase of the wave is an angle. They form a circle-and-integers pair, exactly like angle and angular momentum, with one extra constraint: the number cannot be negative.

The figure computes the phase distribution of a coherent state — the state of an ideal laser, whose photon number has a Poisson distribution — for three mean photon numbers. With ten photons the phase is sharply peaked, with a circular spread of 0.026, and the pair behaves like position and momentum: a coherent state is a minimum-uncertainty state, and its number spread of $\sqrt{10}$ is matched by a phase spread of about $1/(2\sqrt{10})$. With two photons the phase is broader, with a circular spread of 0.195. With half a photon on average it is broad indeed, 0.620, and the distribution has spread well round the circle. The vacuum, with no photons, has its phase spread evenly: an empty mode has no phase to measure.

That is not a technical limitation. A measurement of a light wave’s phase needs a reference and a detector, and in principle both can be perfect. The phase is indefinite because the state does not have one, in exactly the way that a rotor in a state of definite angular momentum does not have an angle. The noise that the noise pushed below the floor described in the amplitude and phase of laser light is this indefiniteness at large photon numbers, where the circle has been unrolled into a line and Heisenberg’s form applies. At small photon numbers the circle shows, and it is one reason that defining a quantum phase operator was a problem that took until the 1980s to settle, when Pegg and Barnett constructed one on a truncated space and took the limit at the end.

Molecules lined up for a moment #

A gas of molecules is a gas of rotors, and the trade between orientation and angular momentum can be watched in it. In thermal equilibrium the molecules point in every direction, each in a mixture of rotational states of definite angular momentum. Strike the gas with a laser pulse much shorter than a rotation period, and the pulse’s electric field gives every molecule a brief torque towards the field’s direction. That kick puts each molecule into a superposition of many rotational states — a wide spread of angular momentum — and by the relation above that is what a sharp orientation needs.

The molecules do not line up during the pulse. They line up shortly afterwards, when the components of their superposition, each turning at its own rate, momentarily come into phase at the angle the kick favoured. Then the alignment disperses, as the components drift apart. The energies of a rigid rotor are proportional to $J(J+1)$, whole-number multiples of a single rotational constant, so after a definite time every component has turned through a whole number of cycles, and the alignment reappears, sharply. This is the same revival that the return a classical cloud never makes found for a quantum packet in a well whose levels are spaced by integers, and it has been used to align molecules in gases at a known moment for imaging and chemistry experiments. For nitrogen the revival period is about 8.4 picoseconds.

The sharpness of the momentary alignment is bounded by the spread of angular momentum the pulse gave. A stronger kick spreads the angular momentum more and lines the molecules up more tightly, and the circular form of the relation says how tightly: the alignment, measured by how close the average orientation comes to the rim of the circle, is limited by $\Delta L$. Experiments measure alignment by the average of $\cos^2\theta$, which is a periodic function of the angle and a good observable on the sphere, exactly the kind of quantity the naive relation could not use and the correct one does.

One kind of angular momentum is not a rotor at all. The angular momentum that is not a rotation describes spin, which has no angle conjugate to it, because there is nothing that turns. A spin-half particle’s state returns to itself only after two full turns, the turn that has to be made twice, which is impossible for a wavefunction of an angle on a circle, since that must return after one. The whole-number values of orbital angular momentum are a direct consequence of the circle; spin’s half-integers are a sign that spin is not the angular momentum of anything going round.

The same pair in a superfluid ring #

A superfluid in a ring is the macroscopic version of the rotor. Its whole condensate shares one wavefunction whose phase must return to itself round the ring, so the phase can wind by a whole number of turns, and each winding corresponds to a quantum of circulation, the quantisation the whirlpool that comes in one size described for a vortex. The winding number and the phase at a point are conjugate, like angular momentum and angle.

The trade between them is what makes the superfluid’s flow stable. A state with a definite winding number — a persistent current round the ring — has a phase that is completely indefinite at any one point of the ring but perfectly correlated from point to point. The current can persist for years because changing the winding number would require the phase to slip by a whole turn somewhere, which costs energy. A Josephson junction is the opposite case: two condensates coupled weakly, with their phase difference nearly definite and their number difference correspondingly spread.

What the circle does not change #

Three things about the rotor are exactly as for a particle on a line, and they are worth stating so that the difference is not overstated.

Fourier’s trade-off is intact. The angle distribution and the angular-momentum distribution are still a Fourier pair, and narrowing one still broadens the other. Only the measure of width had to change.

The operator for angular momentum is unproblematic. $-i\hbar\,\partial/\partial\varphi$ acts on periodic functions and gives periodic functions, has whole-number eigenvalues, and generates rotations. The trouble is entirely on the angle’s side.

Heisenberg’s form is recovered where it should be. For states tightly bunched in angle, which are the ones that behave classically, the circle is invisible and the familiar relation holds to high accuracy, as the second figure showed. The breakdown is confined to states spread over a large part of the circle, which is where the circle’s topology matters.

Still open: the best measure of spread on a circle #

The relation derived here is one of several valid inequalities for angle and angular momentum, each built on a different way of measuring the spread of an angle: the circular spread used here, the Barnett–Pegg spread measured in a chosen window, and entropic relations that use the information content of the two distributions rather than any variance. The entropic relations have the advantage of treating both sides alike and not needing a window. They have been tested with light carrying orbital angular momentum, and they are tighter than the variance-based relations for some states and looser for others. Which measure is the most useful depends on the application — imaging with twisted light, quantum communication using many values of angular momentum, and rotation sensing all use different ones. Whether there is a single natural choice, or whether the circle simply admits several equally good ways of saying how spread a state is, is not settled.

The habit worth carrying away is to ask where a variable lives before writing an uncertainty relation for it. A commutator derived on a line is valid only on a line; an angle lives on a circle, where multiplying by the angle is not an operation on periodic functions, and the spread of an angle must be measured by how far its average falls short of the rim rather than by a variance. With that one change the trade-off between a sharp angle and a spread of angular momentum is restored, and a state of definite angular momentum is no longer a paradox but the limit where the angle has nowhere left to be.