Spin and Spinors

Spin is an internal quantum degree of freedom that carries angular momentum under rotations. It obeys the same Lie algebra as orbital angular momentum, but it is not constructed from a position and momentum through $\mathbf R\times\mathbf P$. Spinors are the Hilbert-space objects on which half-integer-spin rotations act, and their transformation law reveals structure that ordinary three-dimensional vectors cannot represent.

Intrinsic angular momentum
gives finite spin multiplets
represented by spinors and spin matrices
measured along a chosen axis
rotated by $SU(2)$
and coupled to magnetic fields through a magnetic moment.

This chapter develops that chain from the conceptual meaning of spin to practical calculations. Spin-$1/2$ receives special attention because every pure state has a Bloch-sphere representation and every Hermitian operator is built from the identity and Pauli matrices. The same angular-momentum structure then extends to arbitrary spin.

Canonical Boundaries

This page owns the chapter map and the relations among its main formulas. Detailed derivations, experimental histories, and applications remain at their canonical homes.

Topic	Canonical home	Role here
conceptual meaning and misconceptions	What Spin Is and Is Not	separates intrinsic spin from mechanical rotation
spin as an angular-momentum representation	Spin as Intrinsic Angular Momentum	owns the internal Hilbert-space and multiplet structure
two-dimensional spin space	Spin-1/2 Hilbert Space	fixes the basis and spinor convention
spin-$1/2$ matrix calculus	Pauli Matrices	owns the Pauli algebra and operator expansion
geometry of pure spin-$1/2$ states	Bloch Sphere	connects spinors, rays, and expectation values
arbitrary-axis projective measurements	Spin Measurements	owns projectors, probabilities, and state update
modern analyzer formalism	Stern–Gerlach Revisited	connects internal states to separated paths
finite spin rotations	Spin Rotations	owns the half-angle rotation operator
sign under a full turn	Spinors and 2π Rotations	distinguishes a ray from relative phase
static magnetic coupling	Spin in Magnetic Fields	owns Zeeman energies and sign conventions
precession dynamics	Larmor Precession	derives operator and Bloch-vector motion
classical-looking spin states	Spin Coherent States	introduces the rotated highest-weight family
representations beyond spin one-half	Higher Spin Systems	explains larger multiplets and state geometry
magnetic-moment conventions	Magnetic Moments and g-Factors	separates charge signs, magnetons, and $g$ factors

The general measurement postulates belong to Projective Measurement. The historical 1922 experiment belongs to Stern–Gerlach Experiment. Qubits as information carriers belong to Quantum Information, while the spin interpretation of the Bloch sphere belongs here. Relativistic particle spin and Lorentz representations begin in From Spin to Relativistic Representations.

Spin Without Mechanical Rotation

Spin is angular momentum because its components generate rotations and satisfy

$$[S_i,S_j] = i\hbar \sum_k\epsilon_{ijk}S_k.$$

It is intrinsic because it acts on an internal factor of the Hilbert space rather than arising from spatial motion. In a simple nonrelativistic one-particle model,

$$\mathcal H = L^2(\mathbb R^3) \otimes \mathcal H_s.$$

Orbital operators act on the first factor and spin operators on the second. Consequently,

$$\begin{gathered} [S_i,R_j]=0, \\ [S_i,P_j]=0, \\ [S_i,L_j]=0. \end{gathered}$$

in this elementary product model. When both kinds of angular momentum are present, the generator of simultaneous rotations is

$$\mathbf J = \mathbf L+\mathbf S.$$

Orbital angular momentum	Spin angular momentum
$\mathbf L=\mathbf R\times\mathbf P$	no position-space cross-product definition
acts on spatial wavefunctions	acts on an internal spin space
integer $\ell$ for ordinary scalar wavefunctions	integer or half-integer $s$
spatial probability distribution can carry orbital structure	spin state need not have a classical spatial shape

The shared commutator algebra permits $\mathbf L$ and $\mathbf S$ to be added. It does not make their physical origins identical.

Spin Multiplets

For fixed spin $s$, choose simultaneous eigenstates of $S^2$ and $S_z$:

$$S^2|s,m\rangle = \hbar^2s(s+1)|s,m\rangle,$$ $$S_z|s,m\rangle = \hbar m|s,m\rangle.$$

The allowed labels are

$$\begin{gathered} s=0,\frac12,1,\frac32,\ldots, \\ m=-s,-s+1,\ldots,s. \end{gathered}$$

Thus one irreducible spin-$s$ multiplet has dimension

$$\dim\mathcal H_s=2s+1.$$

The ladder operators $S_\pm=S_x\pm iS_y$ move between adjacent $m$ values without changing $s$. Their normalized action is the general angular-momentum result developed in Ladder Operators.

The local commutator algebra is compatible with both integer and half-integer $s$. Globally, integer-spin representations descend to ordinary representations of $SO(3)$, whereas half-integer-spin states require the double cover $SU(2)$ or, equivalently, a projective action of spatial rotations on rays.

Spin One-Half State Space

For spin $1/2$, the Hilbert space is $\mathbb C^2$. In the standard $S_z$ basis,

$$|\uparrow\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \qquad |\downarrow\rangle = \begin{pmatrix}0\\1\end{pmatrix}.$$

They obey

$$S_z|\uparrow\rangle = \frac{\hbar}{2}|\uparrow\rangle, \qquad S_z|\downarrow\rangle = -\frac{\hbar}{2}|\downarrow\rangle.$$

A normalized pure state is

$$|\psi\rangle = \alpha|\uparrow\rangle + \beta|\downarrow\rangle,$$

with

$$|\alpha|^2+|\beta|^2=1.$$

The two complex amplitudes contain four real parameters. Normalization removes one and global phase removes another, leaving two physical parameters. This is why pure spin-$1/2$ rays form a two-dimensional sphere. The relative phase between $\alpha$ and $\beta$ remains observable through measurements along axes other than $z$.

The basis labels $\uparrow$ and $\downarrow$ do not describe permanent little arrows. They denote eigenstates of one specified component, here $S_z$. Changing the quantization axis changes the basis.

Pauli Calculus

In the $S_z$ basis, the Pauli matrices are

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

Their compact product identity is

$$\sigma_i\sigma_j = \delta_{ij}I + i\sum_k\epsilon_{ijk}\sigma_k.$$

It implies both

$$[\sigma_i,\sigma_j] = 2i\sum_k\epsilon_{ijk}\sigma_k$$

and

$$\{\sigma_i,\sigma_j\} = 2\delta_{ij}I.$$

Physical spin components are

$$S_i=\frac{\hbar}{2}\sigma_i.$$

The distinction matters dimensionally: $\sigma_i$ is dimensionless, whereas $S_i$ has units of angular momentum. Every Hermitian operator on a two-dimensional Hilbert space has a unique expansion

$$A=a_0I+\mathbf a\cdot\boldsymbol\sigma,$$

with real $a_0$ and $\mathbf a$. For Hamiltonians, $a_0$ shifts both energies equally and $\mathbf a$ selects a preferred axis in spin space.

Bloch-Sphere Geometry

After removing global phase, every pure spin-$1/2$ state can be written

$$|\hat{\mathbf n}\rangle = \cos\frac{\theta}{2}|\uparrow\rangle + e^{i\phi} \sin\frac{\theta}{2}|\downarrow\rangle,$$

where

$$\hat{\mathbf n} = \begin{pmatrix} \sin\theta\cos\phi\\ \sin\theta\sin\phi\\ \cos\theta \end{pmatrix}.$$

This direction is recovered from expectation values:

$$\langle\boldsymbol\sigma\rangle = \hat{\mathbf n}, \qquad \langle\mathbf S\rangle = \frac{\hbar}{2}\hat{\mathbf n}.$$

A normalized spin-$1/2$ ray is represented by a point $\hat{\mathbf n}$ on the sphere. The spinor uses half-angles, while the expectation-value direction transforms as an ordinary vector.

The same state has density operator

$$\rho = \frac12 \left( I+\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

Mixed spin-$1/2$ states replace $\hat{\mathbf n}$ by a vector $\mathbf r$ with $|\mathbf r|<1$, filling the Bloch ball. That density-operator treatment belongs to Bloch Sphere for Density Operators.

For $s>1/2$, a direction on a sphere no longer specifies every pure state. Spin coherent states form an important sphere inside the larger projective state space, but generic higher-spin states contain additional multipole information.

Measuring a Spin Component

A spin measurement must specify an axis. Along a unit vector $\hat{\mathbf a}$, the measured observable is

$$S_{\hat a} = \hat{\mathbf a}\cdot\mathbf S = \frac{\hbar}{2} \hat{\mathbf a}\cdot\boldsymbol\sigma.$$

Because

$$(\hat{\mathbf a}\cdot\boldsymbol\sigma)^2=I,$$

the outcomes are $\pm\hbar/2$, with projectors

$$P_\pm^{(\hat a)} = \frac12 \left( I\pm \hat{\mathbf a}\cdot\boldsymbol\sigma \right).$$

For a state with Bloch vector $\mathbf r$,

$$p_\pm = \operatorname{Tr} \left( \rho P_\pm^{(\hat a)} \right) = \frac12 \left( 1\pm\mathbf r\cdot\hat{\mathbf a} \right).$$

As a useful geometric example, prepare the $+\hat{\mathbf n}$ state and measure along $\hat{\mathbf a}$. If the angle between the axes is $\gamma$, then

$$p_+ = \cos^2\frac{\gamma}{2}, \qquad p_- = \sin^2\frac{\gamma}{2}.$$

The half-angle probabilities are a spinor effect. Orthogonal laboratory axes have $\gamma=\pi/2$ and therefore give equal probabilities, not deterministic opposite outcomes.

Different components generally do not commute:

$$[S_{\hat a},S_{\hat b}] = i\hbar (\hat{\mathbf a}\times\hat{\mathbf b}) \cdot\mathbf S.$$

Thus preparing an eigenstate along one axis usually destroys sharpness along another. The general rules for Born probabilities, selective update, and nonselective measurement are developed in Core Formalism; this chapter supplies their spin-$1/2$ realization.

Stern–Gerlach as a Spin Analyzer

An ideal Stern–Gerlach device uses an inhomogeneous magnetic field to correlate magnetic-moment projection with spatial path. In modern spin-$1/2$ notation, an analyzer oriented along $\hat{\mathbf a}$ implements the two projectors $P_\pm^{(\hat a)}$.

The device therefore plays three related roles:

Laboratory use	Quantum description
separate an incoming beam	correlate spin eigenstates with distinct paths
block one output	prepare a selected spin eigenstate
detect both outputs	measure a spin component

A sequence exposes noncommutativity. Preparing $+z$, measuring $x$, selecting $+x$, and then measuring $z$ gives $+z$ and $-z$ with equal probabilities. The intermediate $x$ selection does not merely reveal a pre-existing $z$ value; it prepares a new state.

The original Stern–Gerlach experiment used neutral silver atoms, whose beam splitting cannot be identified naively with a bare free-electron trajectory. Its historical interpretation and atomic details are kept separate from the ideal analyzer model.

Rotating Spinors

The unitary operator for an active rotation by angle $\theta$ about $\hat{\mathbf n}$ is

$$U(\hat{\mathbf n},\theta) = \exp\!\left( -\frac{i\theta}{\hbar} \hat{\mathbf n}\cdot\mathbf S \right).$$

For spin $1/2$,

$$U(\hat{\mathbf n},\theta) = \exp\!\left( -\frac{i\theta}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

Using $(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=I$ gives

$$\begin{aligned} U(\hat{\mathbf n},\theta) &= \cos\frac{\theta}{2}\,I \\ &\quad -i\sin\frac{\theta}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma. \end{aligned}$$

The spinor carries the half-angle, but its Bloch vector rotates through the physical angle $\theta$. If $\mathcal R\in SO(3)$ is the corresponding vector rotation, then

$$U (\mathbf a\cdot\boldsymbol\sigma) U^\dagger = (\mathcal R\mathbf a) \cdot\boldsymbol\sigma.$$

This relation is a practical bridge between $SU(2)$ matrix calculations and ordinary three-dimensional geometry. It also makes the two-to-one map explicit: $U$ and $-U$ induce the same $SO(3)$ rotation of Bloch vectors.

The 2π Spinor Sign

For spin $1/2$, one full physical turn gives

$$U(\hat{\mathbf n},2\pi)=-I,$$

whereas two full turns give

$$U(\hat{\mathbf n},4\pi)=I.$$

More generally, the central element associated with a $2\pi$ rotation acts on a spin-$s$ irreducible multiplet as

$$U_s(2\pi) = (-1)^{2s}I.$$

Integer-spin state vectors return after $2\pi$; half-integer-spin vectors acquire a minus sign. This is a global representation-theoretic distinction, not a consequence visible from the infinitesimal commutators alone.

The sign does not change an isolated ray because

$$|\psi\rangle \sim e^{i\alpha}|\psi\rangle.$$

In particular, $|\psi\rangle$ and $-|\psi\rangle$ represent the same pure state. The sign becomes observable when it is relative to another coherent amplitude. If only one arm of an interferometer applies a $2\pi$ spin rotation, then schematically

$$\begin{aligned} \frac{|a\rangle+|b\rangle}{\sqrt2} \otimes|\chi\rangle &\longmapsto \\ \frac{|a\rangle-|b\rangle}{\sqrt2} \otimes|\chi\rangle. \end{aligned}$$

Recombining the paths detects the changed relative phase. The Bloch sphere cannot display this sign because $U$ and $-U$ act identically on the Bloch vector.

Spin in a Magnetic Field

A magnetic field couples to a magnetic moment:

$$H = -\boldsymbol\mu\cdot\mathbf B.$$

If the effective magnetic moment is proportional to spin,

$$\boldsymbol\mu=\gamma\mathbf S,$$

then

$$H = -\gamma\mathbf S\cdot\mathbf B.$$

For spin $1/2$ this becomes

$$H = -\frac{\gamma\hbar}{2} \mathbf B\cdot\boldsymbol\sigma.$$

Take $\mathbf B=B_0\hat{\mathbf z}$. The $S_z$ eigenstates are energy eigenstates with

$$E_{+z} = -\frac{\gamma\hbar B_0}{2},$$ $$E_{-z} = +\frac{\gamma\hbar B_0}{2}.$$

The positive level spacing is

$$\Delta E = \hbar|\gamma|B_0.$$

The absolute value gives the spacing, but it does not tell which state is lower. That ordering depends on the sign of $\gamma B_0$. For an electron, the magnetic moment points opposite to the spin in the standard convention, so a verbal claim such as “aligned is lower” is ambiguous unless it says whether spin or magnetic moment is aligned with the field.

A fixed external field breaks full rotational symmetry down to rotations about the field axis. Accordingly, the spin component parallel to the field is conserved, while transverse components evolve.

Larmor Precession

For the Zeeman Hamiltonian, the Heisenberg equation gives

$$\frac{d\mathbf S}{dt} = \gamma\, \mathbf S\times\mathbf B.$$

The expectation value and the spin-$1/2$ Bloch vector obey the same rotation equation. In a constant field of magnitude $B_0$, the precession-frequency magnitude is

$$\omega_L=|\gamma|B_0.$$

The sign of $\gamma$ fixes the sense of precession. For $\mathbf B=B_0\hat{\mathbf z}$ and the signed frequency $\omega=\gamma B_0$,

$$\begin{aligned} S_x(t) &= S_x(0)\cos\omega t +S_y(0)\sin\omega t, \\ S_y(t) &= S_y(0)\cos\omega t -S_x(0)\sin\omega t. \end{aligned}$$

An energy eigenstate aligned with the field axis acquires only a global phase and has a stationary Bloch vector. Precession requires transverse coherence: the two Zeeman eigencomponents acquire a changing relative phase. Time-dependent transverse fields and Rabi oscillations are a further dynamical problem, introduced in Rabi Oscillations: First Encounter.

Spin Coherent States

For general spin $s$, a spin coherent state is obtained by rotating the highest-weight state $|s,s\rangle$:

$$|\hat{\mathbf n};s\rangle = R(\hat{\mathbf n})|s,s\rangle.$$

It satisfies

$$\hat{\mathbf n}\cdot\mathbf S |\hat{\mathbf n};s\rangle = \hbar s |\hat{\mathbf n};s\rangle,$$

and

$$\langle\hat{\mathbf n};s| \mathbf S |\hat{\mathbf n};s\rangle = \hbar s\hat{\mathbf n}.$$

These states are the most classical-looking members of a spin-$s$ multiplet. Their relative transverse fluctuations scale down as $s$ grows, which makes them useful in semiclassical spin dynamics, many-body magnetism, and spin path integrals.

For $s=1/2$, every pure state is spin coherent. For $s>1/2$, coherent states form only a two-parameter subset of all pure states. A generic higher-spin state cannot be reconstructed from the direction $\langle\mathbf S\rangle$ alone.

Higher Spin Systems

The same algebra produces a hierarchy of finite-dimensional representations:

Spin $s$	Magnetic labels $m$	Dimension	Common multiplet name
$0$	$0$	$1$	singlet or scalar
$1/2$	$-1/2,1/2$	$2$	doublet
$1$	$-1,0,1$	$3$	triplet
$3/2$	$-3/2,-1/2,1/2,3/2$	$4$	quartet
$2$	$-2,-1,0,1,2$	$5$	quintet

In the standard ordered basis, $S_z$ is diagonal and $S_\pm$ supplies the off-diagonal matrix elements. This constructs all three spin matrices for any $s$ without guessing them.

Higher spin also permits observables beyond the vector polarization $\langle\mathbf S\rangle$. Spin $1$ and above can carry quadrupole alignment, and progressively higher ranks appear as $s$ increases. This is why a three-component Bloch vector is complete for a two-level density matrix but incomplete for a generic higher-spin density operator.

The labels integer and half-integer describe representation class, not particle count. A composite system of several spin-$1/2$ constituents can have an integer total spin, which is developed in Addition of Angular Momentum, beginning with Tensor Product Representations.

Magnetic Moments and g Factors

For angular momentum $\mathbf J$, a magnetic moment is often written

$$\boldsymbol\mu = \gamma\mathbf J = g\frac{q}{2m}\mathbf J.$$

Here $q$ is signed charge, $m$ is a mass scale appropriate to the model, and $g$ is dimensionless. The formula is useful only when those conventions are stated.

With $e>0$, the Bohr magneton is

$$\mu_B = \frac{e\hbar}{2m_e}.$$

For an electron,

$$\boldsymbol\mu_L = -\mu_B\frac{\mathbf L}{\hbar},$$

where the leading orbital factor is $g_L=1$, while

$$\boldsymbol\mu_S = -g_s\mu_B\frac{\mathbf S}{\hbar},$$

with $g_s$ close to $2$. The factor near two is not obtained by modeling spin as a classical rotating charged shell. Its relativistic origin belongs to the Dirac theory, and the small anomalous correction belongs to quantum electrodynamics.

For atoms with coupled $\mathbf L$ and $\mathbf S$, an effective Landé factor describes the magnetic moment projected onto total $\mathbf J$. The coupling derivation belongs to angular-momentum addition and atomic spectroscopy rather than this chapter overview.

A Reliable Workflow

For a spin problem, proceed in this order:

1. Identify the spin representation. Record $s$, the Hilbert-space dimension $2s+1$, and whether spatial degrees of freedom are also present.
2. Choose and state a basis. For spin $1/2$, name the axis defining $|\uparrow\rangle$ and $|\downarrow\rangle$.
3. Separate dimensionless matrices from observables. Use $S_i=(\hbar/2)\sigma_i$ rather than interchanging $S_i$ and $\sigma_i$.
4. Specify the physical operation. Distinguish state rotation, apparatus rotation, Hamiltonian evolution, and measurement.
5. Use projectors for probabilities. For axis $\hat{\mathbf a}$, compute with $P_\pm^{(\hat a)}$ instead of relying on a sketch alone.
6. Track signed magnetic conventions. Keep $q$, $\gamma$, $g$, and the direction of $\mathbf B$ explicit until the energy ordering is known.
7. Check invariant quantities. Probabilities must sum to one, rotations preserve norm, and unitary evolution preserves Bloch-vector length for a pure isolated spin-$1/2$ state.

Chapter Map

Read this page	When the question is
What Spin Is and Is Not	Why is spin not literal mechanical rotation?
Spin as Intrinsic Angular Momentum	How does spin realize the angular-momentum algebra?
Spin-1/2 Hilbert Space	What is a two-component spinor state?
Pauli Matrices	Which matrix identities make spin-$1/2$ calculations efficient?
Bloch Sphere	How do pure spin-$1/2$ rays become directions on a sphere?
Spin Measurements	How are arbitrary-axis probabilities and updates computed?
Stern–Gerlach Revisited	How does a magnetic analyzer implement a spin measurement?
Spin Rotations	How does $SU(2)$ rotate a spinor?
Spinors and 2π Rotations	Why does a full turn produce a sign, and when can it matter?
Spin in Magnetic Fields	How are Zeeman energies and signs organized?
Larmor Precession	How does a spin direction evolve in a static field?
Spin Coherent States	Which higher-spin states look most classical?
Higher Spin Systems	What changes beyond the two-state case?
Magnetic Moments and g-Factors	How do charge, magnetons, and $g$ factors enter?

Reading Paths

Core spin-$1/2$ formalism

1. What Spin Is and Is Not
2. Spin-1/2 Hilbert Space
3. Pauli Matrices
4. Bloch Sphere
5. Spin Rotations

Measurement and experiment

1. Spin Measurements
2. Stern–Gerlach Revisited
3. Sequential Measurements
4. Spin Problems

Magnetic dynamics

1. Magnetic Moments and g-Factors
2. Spin in Magnetic Fields
3. Larmor Precession
4. Rabi Oscillations: First Encounter

Representation and geometry bridge

1. Spin as Intrinsic Angular Momentum
2. Spinors and 2π Rotations
3. Higher Spin Systems
4. Spin Coherent States
5. From SU(2) Spinors to Lorentz Spinors

Distinctions Worth Keeping

Do not conflate	Why
spin vector and spinor	$\langle\mathbf S\rangle$ transforms as a vector; the state transforms in a spin representation
Pauli matrix and spin observable	$\sigma_i$ is dimensionless; $S_i=(\hbar/2)\sigma_i$ carries angular momentum
global phase and relative phase	only relative phase can alter interference or other observable statistics
$SU(2)$ element and $SO(3)$ rotation	$U$ and $-U$ produce the same ordinary vector rotation
basis label and physical direction	“up” is always up along a specified quantization axis
spin and magnetic moment	they are proportional only after a signed, system-dependent gyromagnetic ratio is specified
Zeeman splitting and precession	splitting is spectral; precession requires a superposition with transverse coherence
Bloch sphere and all higher-spin states	the sphere is complete for pure spin $1/2$, but only labels coherent states for larger $s$

Common Mistakes

• Modeling spin as a tiny extended body literally rotating in space.
• Forgetting the factor $\hbar/2$ between $\boldsymbol\sigma$ and $\mathbf S$.
• Asking for “the spin” without naming the measured component.
• Treating a Bloch vector as the two-component spinor itself.
• Using the physical angle instead of the half-angle in a spin-$1/2$ rotation operator.
• Claiming that a $2\pi$ sign is either directly observable as an absolute phase or completely meaningless; it is meaningful as a relative phase.
• Inferring the lower Zeeman state from the word “aligned” without tracking the sign of the magnetic moment.
• Calling every two-level system a physical spin-$1/2$ without identifying how rotations act on it.
• Extending the pure-state Bloch sphere unchanged to arbitrary higher-dimensional spin spaces.

Cross-Links

• Rotations and Orbital Angular Momentum
• Projective Representations
• Rays and Global Phase
• Two-Level System
• Time Reversal for Spin-1/2 Particles
• Addition of Angular Momentum
• Pauli Matrix Identity Index
• Spin Problems
• From Spin to Relativistic Representations
• From SU(2) Spinors to Lorentz Spinors

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• H. Rauch and S. A. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics, 2nd ed., Oxford University Press, 2015.
• A. M. Perelomov, Generalized Coherent States and Their Applications, Springer, 1986.

Exercises

1. Show that the arbitrary-axis operators

$$P_\pm = \frac12 \left( I\pm\hat{\mathbf n}\cdot\boldsymbol\sigma \right)$$

are orthogonal projectors that sum to the identity.

Solution

For a unit vector $\hat{\mathbf n}$, the Pauli product identity gives

$$(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=I.$$

Therefore

$$\begin{aligned} P_\pm^2 &= \frac14 \left( I \pm2\hat{\mathbf n}\cdot\boldsymbol\sigma +I \right) \\ &= P_\pm. \end{aligned}$$

The sum is $P_++P_-=I$, and

$$P_+P_- = \frac14 \left[ I-(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2 \right] = 0.$$

Thus the two ranges are orthogonal and complete.

2. Consider

$$|\psi\rangle = \frac{\sqrt3}{2}|\uparrow\rangle + \frac{i}{2}|\downarrow\rangle.$$

Find its Bloch vector and the probability of obtaining $+\hbar/2$ in an $S_x$ measurement.

Solution

For a normalized spinor $(\alpha,\beta)^T$,

$$\mathbf r = \begin{pmatrix} 2\operatorname{Re}(\alpha^*\beta)\\ 2\operatorname{Im}(\alpha^*\beta)\\ |\alpha|^2-|\beta|^2 \end{pmatrix}.$$

Here $\alpha=\sqrt3/2$ and $\beta=i/2$, so

$$\mathbf r = \begin{pmatrix} 0\\ \sqrt3/2\\ 1/2 \end{pmatrix}.$$

For an $x$-axis measurement,

$$p(+x) = \frac12(1+r_x) = \frac12.$$

3. Begin in $|+x\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt2$ and apply an active rotation by $\pi$ about $z$. Find the final ray and its Bloch vector.

Solution

The rotation matrix is

$$U(\hat z,\pi) = \begin{pmatrix} e^{-i\pi/2}&0\\ 0&e^{i\pi/2} \end{pmatrix}.$$

Hence

$$\begin{aligned} U(\hat z,\pi)|+x\rangle &= \frac{-i|\uparrow\rangle+i|\downarrow\rangle}{\sqrt2} \\ &= -i|-x\rangle. \end{aligned}$$

The global factor $-i$ does not change the ray. The Bloch vector is therefore $(-1,0,0)$, as expected from rotating $+x$ by $\pi$ about $z$.

4. A spin-$1/2$ state initially points along $+x$ and evolves under

$$H=-\gamma B_0S_z.$$

For $\gamma B_0>0$, find its Bloch vector as a function of time and state its initial direction of motion.

Solution

Let $\omega=\gamma B_0>0$. The precession equations give

$$\mathbf r(t) = \begin{pmatrix} \cos\omega t\\ -\sin\omega t\\ 0 \end{pmatrix}.$$

At $t=0$, $d\mathbf r/dt=(0,-\omega,0)$, so the Bloch vector initially moves from $+x$ toward $-y$. Replacing $\gamma$ by a negative value reverses the sense of precession without changing the positive frequency magnitude $|\gamma|B_0$.

5. Use a $z$-axis rotation to show that a $2\pi$ rotation acts as $(-1)^{2s}I$ throughout a spin-$s$ multiplet.

Solution

On a basis state,

$$e^{-i2\pi S_z/\hbar}|s,m\rangle = e^{-i2\pi m}|s,m\rangle.$$

Every allowed $m$ differs from $s$ by an integer, so

$$e^{-i2\pi m} = e^{-i2\pi s} = (-1)^{2s}.$$

The factor is therefore the same on every state in the irreducible multiplet. It is $+1$ for integer $s$ and $-1$ for half-integer $s$.