Private study companion. Please enter the access password.
The Future of Now · UChicago QSNC companion
🧭 The Three Spin Arrows
MODULE 1 · EXERCISE 3
The Pauli operators X, Y, Z — and the direction arrow r·σ
🎯 The big idea (for everyone)
A qubit has three basic spin moves, named X, Y, and Z.
Think of them as "flip," "flip-with-a-twist," and "stand-up/down."
Part (a) — they clash! Doing one move then a different one gives the
exact opposite of doing them in the other order. We say they anti-clash (anticommute).
Parts (b) & (c) — point an arrow. Pick a direction arrow r in 3D and mix the
moves: r·σ = rx·X + ry·Y + rz·Z. Square it and you just get the length of r, squared, times "do nothing."
So its only stretch numbers are +length and −length.
Executive / UChicago layer: (a) {σi, σj} = σiσj + σjσi = 0 for i ≠ j
(and σi² = I). (b) (r·σ)² = |r|² I via
σiσj = δijI + i·εijkσk. (c) Since (r·σ)² = |r|²I and tr(r·σ)=0,
the eigenvalues are ±|r|.
🎮 Point the arrow r
Move the three sliders to aim the arrow. Watch the two stretch
numbers always come out as +length and −length — perfectly mirror-image, no matter where you point.
rx0.30
ry-0.50
rz0.80
|r| (length)0.99
eigenvalue ++0.99
eigenvalue −-0.99
(r·σ)² = |r|²I?✓
💥 See the anti-clash (part a)
Press a pair. We compute both orders and add them. If they
anti-clash, the sum is all zeros — a perfect cancel!
🧠 Quick quiz — earn your stars
1) Two different spin moves (like X and Y)…
2) If you point arrow r and square r·σ, you get…
3) The two stretch numbers of r·σ are always…
🏆 Brilliant! You proved Pauli anticommutation and found the eigenvalues ±|r| of r·σ.