Illustrated Physics
physics, drawn
Series

Spin — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

    The angular momentum that is not a rotation

    An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed.

    part 1 · quantum
  2. Four product states, and the four combinations that have a total spin. The four ways two spin-halves can be arranged, on the left, and the four combinations of them that are eigenstates of the total spin, on the right. Two of the products — both up and both down — are already eigenstates. The other two are not: one spin up and the other down does not specify a total spin, because it does not say which spin is which, and the states that do are the sum and the difference. The eigenvalues printed beside them are computed by applying S² as a matrix in the product basis and reading the result off, then solving s(s + 1) for s: |↑↑⟩ gives 2ħ², so s = 1; (|↑↓⟩ + |↓↑⟩)/√2 gives 2ħ², so s = 1; |↓↓⟩ gives 2ħ², so s = 1; (|↑↓⟩ − |↓↑⟩)/√2 gives 0ħ², so s = 0. The column on the far right is the eigenvalue of the operator that swaps the two particles: the three states with s = 1 come back unchanged and the one with s = 0 comes back with a minus sign. That sign is the whole of the difference. It is why the three are called a triplet and the one a singlet, why they behave differently in a magnetic field, and — through the requirement that the total state of two electrons be antisymmetric — why the two families occupy space differently before any force between them has been mentioned.

    Four states, and one of them is odd

    Two spin-halves make four states, and they split three and one rather than into four of a kind. Three come back unchanged when the two particles are swapped and one comes back with a minus sign — and that single sign decides how far apart two electrons sit before any force between them has been mentioned, and why hydrogen gas is two gases that do not interconvert.

    part 2 · quantum
  3. What comes back after one turn, and what needs two. A spin-½ pointing along z and rotated about the x axis through 720°, with the rotation integrated step by step rather than evaluated from a formula. The direction of the spin — the quantity a Stern–Gerlach magnet, a compass or any other instrument reports — is back where it started after 360°, exactly as the orientation of any other object would be. The state is not: its overlap with the state it began in has reached −1 there, and returns to +1 only after 720°. At 360° the overlap is -1.000 and ⟨σz⟩ is 1.000; At 720° the overlap is 1.000 and ⟨σz⟩ is 1.000. Both curves come off one integration of dψ/dθ = −(i/2)σx ψ whose norm is checked before anything is drawn, so the factor of two between their rates is a property of the propagation rather than of two separate formulae that were chosen to differ.

    The turn that has to be made twice

    Turn a spin-½ through a full circle and it does not come back. The direction it points in does, and every measurement on it does, but the state itself has changed sign — and a second full turn is needed before anything is where it started. The sign is invisible on one spin and measurable the moment a superposition has one branch turned and the other not.

    part 3 · quantum
  4. The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all.

    Two states where the counting says three

    A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

    part 4 · quantum
  5. No beam is narrow enough and wide enough at once. Three lengths at the far end of a Stern-Gerlach magnet 10 centimetres long with a gradient of 1000 tesla a metre, against the width of the beam entering it, for an electron at 100 electronvolts. The spin splitting is a horizontal line at 2.9e-6 metres: it does not depend on the beam's width. The Lorentz blurring rises in proportion to the width, because the field a particle sees depends on where in the beam it is, and the divergence-free condition ties a gradient in one component to a gradient in another. It overtakes the splitting at 1.0e-9 metres. Narrower than that and diffraction has already spread the beam by 3.9e-3 metres, which is larger still. There is no width at which the splitting is the largest of the three, and the magnet's length and gradient cancel out of the comparison entirely — so no magnet helps.

    The experiment that defines spin and cannot be done on it

    A Stern–Gerlach magnet separates magnetic moments and is how spin was discovered. It cannot be made to work on a free electron, and the obstruction is not the apparatus: the field gradient that splits the beam also deflects the charge by an amount that varies across it, and the ratio of the splitting to that blurring comes out as the de Broglie wavelength over the beam width — with the magnet's length and gradient cancelling exactly.

    part 5 · quantum

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