The Double Helix
Screw symmetry — Q1, Part 7 (helical solenoid)
Pitch d
k = 2π/d
z travelled
rotation α
Pure translation is not a symmetry. As you slide along z without
rotating, the helix visibly drifts — the geometry you see keeps changing.
An infinite helix is unchanged only by a combined move: a shift
\(\epsilon\hat z\) along the axis together with a rotation
\(\alpha=k\epsilon\) about it. Infinitesimally
\(\vec r\,'=\vec r+\epsilon(\hat z+k\,\hat z\times\vec r)\), so the
symmetry generator is \(\vec a=\hat z+k\,\hat z\times\vec r\) (the yellow
arrow — it lies tangent to the strand). With pitch \(d\),
\(k=2\pi/d\). By Noether's theorem the conserved quantity is
\(Q=\vec p\cdot\vec a = p_z + k\,J_z\).
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