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Revolvix

Spinor Rotation / The 720° Return Physics

\[\begin{aligned} R(2\pi)\,\psi &= e^{-i\,2\pi\,\hat{n}\cdot\boldsymbol{\sigma}/2}\,\psi \\ &= -\psi \end{aligned}\]
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Revolvix is a Pixelated Physics law character, one of our quantum designs. A spin-½ particle must turn 720°, not 360°, to come back to itself: one full turn flips the sign of its wavefunction. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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The physics

Rotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.

Revolvix waits tables, and his trick is the tray.

Hold a tray of drinks flat on your palm and turn it one full circle under your arm without letting go. The tray stays level, but your elbow ends up twisted. Keep turning the same way for a second full circle and the arm untwists, everything back as it started. This is the plate trick, also called the Balinese cup trick, a cousin of Dirac’s string or belt trick. Paul Dirac liked to show with strings that a rotation by 360° is not always the same as no rotation at all; Max Newman published the mathematics of Dirac’s string problem in 1942.

For a spin-½ particle that is literally true. Its state is a two-component spinor, and the operator for a rotation by angle θ contains θ/2. Turn it by 2π and the spinor comes back as minus itself, the first line on this page. Turn it by 4π and it is home again. The rotations of space contain loops that one full turn cannot undo but two can.

A minus sign on a whole state cannot be seen on its own. It shows up only relative to something else. In 1975 two groups made that comparison. Helmut Rauch’s team in Vienna and Samuel Werner’s team in Missouri split neutrons into two paths through a perfect silicon crystal, let a magnetic field precess the spins along one path, and recombined them. The interference pattern repeated not every 360° of precession but every 720°, exactly as the minus sign demands.

So Revolvix brings your drinks back the long way round. After one lap the tray looks fine to you; only he, comparing it with the path not taken, knows the sign is wrong.

One turn isn’t enough for me.

Equations

\[\begin{aligned} R(2\pi)\,\psi &= e^{-i\,2\pi\,\hat{n}\cdot\boldsymbol{\sigma}/2}\,\psi \\ &= -\psi \end{aligned}\]
\[R(4\pi)\,\psi = +\psi\]
SymbolMeaningUnit
\(R(\theta)\) operator that rotates a spin-1/2 state by angle θ about the axis n \(\mathrm{1}\)
\(\psi\) two-component spinor state of a spin-1/2 particle \(\mathrm{1}\)
\(\hat{n}\) unit vector along the rotation axis \(\mathrm{1}\)
\(\boldsymbol{\sigma}\) the three Pauli matrices (σx, σy, σz) \(\mathrm{1}\)
\(2\pi, 4\pi\) one full turn (360°) and two full turns (720°) \(\mathrm{rad}\)

Sources

  1. Rauch et al. (1975) Verification of coherent spinor rotation of fermions, Phys. Lett. A 54, 425
  2. Werner, Colella, Overhauser & Eagen (1975) Observation of the Phase Shift of a Neutron Due to Precession in a Magnetic Field, Phys. Rev. Lett. 35, 1053
  3. Newman (1942) On a String Problem of Dirac, J. London Math. Soc. s1-17, 173

Source Rauch, H. et al. (1975). Verification of coherent spinor rotation of fermions. Physics Letters A 54, 425–427.doi.org/10.1016/0375-9601(75)90798-7

Field notes

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