Quantum Mechanics · Chapter 10

Spin

The angular momentum a particle carries with nothing to rotate. The Stern–Gerlach experiment, the two-dimensional state space of spin one-half and its picture on a sphere, the Pauli matrices, measurement along any axis, spinors that change sign under a full turn, precession in a magnetic field, and the discovery that every two-level system is a spin in disguise.

S^_{z} = \frac{ℏ}{2} σ_{z}

∣ ↑ ⟩

∣ ↓ ⟩

σ_{x} σ_{y} = i σ_{z}

Lessons

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05Measuring spin along any axisA Stern–Gerlach magnet can be turned to any direction $\mathbf{n}$, and it then measures the spin component $\mathbf{n}\cdot\hat{\mathbf{S}}$. Its outcomes are again $\pm\hbar/2$, because no direction is special, and its eigenstates are up and down along $\mathbf{n}$. A spin prepared up along one axis and measured along another at angle $\alpha$ comes out up with probability $\cos^{2}(\alpha/2)$: certainly at 0, half and half at $90^\circ$, never at $180^\circ$. The average of the measured component is $\cos\alpha$ — exactly the projection a classical arrow of unit length would have.Read lesson →

06Spinors and rotationsRotations act on spin states through the exponentials of the Pauli matrices, and those exponentials have a surprise: because $(\mathbf{n}\cdot\vec{\sigma})^{2}=1$, the exponential of a rotation by $\theta$ is $\cos(\theta/2)-i\mathbf{n}\cdot\vec{\sigma}\sin(\theta/2)$, with half angles. A full turn gives $-1$. Objects that change sign under a full turn and need two turns to come back are spinors. The sign cannot be seen on one spin alone, since it is an overall phase, but it shows in interference, and neutrons sent round a magnetic field showed it in 1975.Read lesson →

07Spin precessionPut a spin in a magnetic field along $z$. Its energy depends only on $\hat{S}_{z}$, so the probabilities of up and down along $z$ never change — but the transverse components do: the average spin vector turns about the field at a steady rate, the Larmor frequency, like a gyroscope under gravity. For an electron in one tesla that is about 28 GHz; for a proton about 43 MHz. Magnetic resonance, from chemistry labs to hospital scanners, listens to this precession.Read lesson →

08Two-level systemsAny system with two relevant states — the ammonia molecule of the dynamics chapter, an atom with two levels, a photon’s polarisation, a superconducting qubit — has a Hamiltonian that is a Hermitian two-by-two matrix. Every such matrix is a number plus $\mathbf{a}\cdot\vec{\sigma}$. So every two-level system is a spin one-half in an effective magnetic field $\mathbf{a}$: its energies are the number plus and minus $|\mathbf{a}|$, and its states precess about $\mathbf{a}$. What was learned about spin applies to them all.Read lesson →

09Chapter practiceBring the chapter together: spin as intrinsic angular momentum, the Stern–Gerlach experiment, the Bloch sphere, the Pauli matrices, measurement along any axis, spinors and their sign, precession in a field and the two-level systems that are all spins in disguise. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Spin

Build the foundation