Spin-1/2 Operators

Equivalent forms

\sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix},\;\sigma_y = \begin{pmatrix}0&-i\\i&0\end{pmatrix},\;\sigma_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}

[\hat{S}_i, \hat{S}_j] = i\hbar\,\epsilon_{ijk}\hat{S}_k

Three 2×2 matrices generate every qubit gate, every electron's magnetic behavior, and every fermion's symmetry.

Symbol	Name	SI unit	Dimension	Range
$S_z$	Spin-z eigenvalueoutput Measured z-component of spin	J*s	ML^2T^-1	-5.272859e-35 – 5.272859e-35
$hbar$	Reduced Planck constant h/(2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$theta$	Polar angle Polar angle of Bloch-sphere state	rad	1	0 – 3.14159265359
$phi$	Azimuthal angle Azimuth of Bloch-sphere state	rad	1	0 – 6.28318530718

Unit systems

Symbol	SI	CGS	Imperial
$S_z$	J*s	erg*s	ftlbfs
$hbar$	J*s	erg*s	ftlbfs
$theta$	rad	rad	rad
$phi$	rad	rad	rad

Where it holds

Exact for spin-1/2 particles (electrons, protons, neutrons, quarks). Higher spins use (2s+1)-dimensional matrices generalizing the algebra.

Dimensional analysis

S_i \to [M*L^2*T^-1]

hbar *

\sigma _i / 2 \to [M*L^2*T^-1]

* dimensionless

Both sides are angular momentum.

\checkmark

Discovery

Wolfgang Pauli · 1927

Pauli introduced the matrices to incorporate the recently discovered electron spin (Uhlenbeck & Goudsmit, 1925) into Schrödinger's wave mechanics. They became the generators of SU(2), the structure of all spin-1/2 systems.

Try this

How do you describe something that isn't a vector but isn't a number either?

Electron spin can't be visualized as a tiny arrow — it lives in a 2D complex space. What are the operators that act on it, and why do they obey angular-momentum algebra?

Research status: stable

Real-world applications

Qubits and quantum computing (every qubit is spin-1/2 algebraically)
NMR and MRI imaging
Spintronics (spin valves, MRAM)
Quantum cryptography (BB84 uses Pauli measurements)

Common misconceptions

Spin is not classical rotation — a 'spinning' electron would exceed light speed at its surface
$\sigma _i^2 = I$ , so squared measurement always gives +1 (in units of hbar/2)
Spin needs $720^{\circ}$ (not $360^{\circ})$ to return to itself — half-integer spin geometry

Experimental verification

Stern–Gerlach (1922) showed two discrete deflections — direct measurement of

S_z = \pm

hbar/2. Electron g-factor

\approx 2.0023 (g = 2

from Dirac, 0.1% from QED). NMR experiments measure Larmor precession of S in magnetic fields. Single-spin readout in NV centers and quantum dots.

Derivation

Demand operators S_i obey [S_i,

S_j] = i

*hbar*eps_ijk*S_k (angular-momentum algebra) on a 2D Hilbert space.

The Pauli matrices sigma_i are the unique (up to unitary) traceless Hermitian

2\times 2

matrices satisfying

\sigma _i*\sigma _j = \delta _ij*I + i*eps_ijk*\sigma _k

.

Setting

S_i =

(hbar/2)*sigma_i recovers the commutator with the correct hbar.

The eigenvalues

\pm

hbar/2 are the only measurable spin-z outcomes.

Limiting cases

|psi> = |↑>⟶ <S_z> = +hbar/2, <S_x> = <S_y>

= 0

Spin-up eigenstate of sigma_z.

|psi> = (|↑> + |↓>)/sqrt(2)⟶ <S_x> = +hbar/2, <S_z>

= 0

Spin-x eigenstate — completely uncertain in z.

[S_x, S_y]⟶

= i

*hbar*S_zNon-commutation — basis of spin uncertainty relations.

What if…

What if [S_x,

S_y] = 0

?

Spin components could be measured simultaneously — no quantum measurement weirdness, but also no qubits.

What if hbar/2 were replaced by hbar?

We'd be describing spin-1 (3D representation, photons) instead of spin-1/2 — three possible outcomes per measurement.

What if you rotate a spin state by

360^{\circ}

?

It picks up a -1 phase — measurable in neutron interferometry (Werner 1975).

1

Eigenvalues of sigma_x

Given ·

sigma x:: [[0,1],[1,0]]

Find · eigenvalues

Steps

Characteristic equation: $\lambda ^2 - 1 = 0$
$\lambda = +1$ → eigenvector (1, $1)/\sqrt{2} =$ |+>
$\lambda = -1$ → eigenvector (1,- $1)/\sqrt{2} =$ |->
Physical S_x eigenvalues: $\pm$ hbar/ $2 \approx \pm 5.27e-35 J\cdot s$

Result ·

det(\sigma _x - \lambda *I) = \lambda ^2 - 1 = 0 \to \lambda = \pm 1

2

Expectation <S_z> for general state

Given ·

psi:: (cos(theta/2)|↑> + sin(theta/2)*exp(i*phi)|↓>)

Find · <S_z>

Steps

<S_z> = (hbar/2) * <psi|sigma_z|psi>
= (hbar/2) * (|c↑| $^{2} - |c$ ↓| $^{2})$
= (hbar/2) * cos(theta) — projection on z-axis of Bloch vector

Result · <S_z> = (hbar/

2)*(\cos ^{2}(\theta /2) - \sin ^{2}(\theta /2))

= (hbar/2)*cos(theta)

Heisenberg Uncertainty Principle→Pauli Exclusion Principle→Stern–Gerlach Deflection→Dirac Equation→