4.4.1 Classical Spin#

Prompts

A classical spin can be represented by a unit vector \(\boldsymbol n\) on the sphere of possible spin-axis directions. What is moving when the spin precesses?
Starting from torque balance, why does the equation for \(\boldsymbol n(t)\) contain a transverse gyroscopic term \(-\dot{\boldsymbol n}\times S\boldsymbol n\)?
Compare the spin-axis equation with the Lorentz force for a charged particle moving on a sphere. Which term plays the role of the monopole magnetic field?
Why does the vector potential for this effective monopole have a string singularity, and how does requiring a single-valued quantum wavefunction quantize \(S/\hbar\)?

Lecture Notes#

Overview#

For the purposes of §4.4, the useful classical picture is simple and deep: a spin is an oriented axis. Its direction is a point \(\boldsymbol n\) on the unit sphere, and precession is motion of that point on the sphere. The surprising fact is that the gyroscopic term in the equation of motion has exactly the form of a Lorentz force from a magnetic monopole sitting at the center of that sphere. This turns the classical spin problem into the classical limit of a charged particle on a sphere threaded by monopole flux. Quantizing that flux gives half-integer spin and prepares the Dirac monopole and monopole harmonics of §4.4.2–§4.4.3.

Spin Axis and Angular Momentum#

Assume the spinning object is spherically symmetric, so a single moment of inertia \(I\) describes rotations. The configuration variable is the direction of the spin axis,

(162)#\[\begin{split} \begin{split} \boldsymbol n &= \boldsymbol n(t), \\ \boldsymbol n^2 &= 1. \end{split} \end{split}\]

The main object is the angular momentum carried by this moving axis:

(163)#\[ \boldsymbol L = \boldsymbol S+I\boldsymbol\Omega = S\boldsymbol n+I\boldsymbol n\times\dot{\boldsymbol n}. \]

This formula sets up the whole problem.

\(\boldsymbol S=S\boldsymbol n\) is the spin angular momentum along the body axis. The magnitude \(S\) is fixed; the direction \(\boldsymbol n(t)\) can move.
\(I\boldsymbol\Omega\) is the transverse angular momentum of spinning body as its axis moves on the unit sphere, where \(I\) is the moment of inertia and \(\boldsymbol\Omega=\boldsymbol n\times\dot{\boldsymbol n}\) is the angular velocity associated with changing the axis direction. It is transverse to \(\boldsymbol n\).

So the dynamics is not about a particle moving through ordinary space. It is about a point \(\boldsymbol n\) moving on the sphere of possible spin directions, while carrying both the intrinsic spin \(S\boldsymbol n\) and the transverse angular momentum \(I\boldsymbol n\times\dot{\boldsymbol n}\).

Derivation: kinematics of the spin axis

If an angular velocity \(\boldsymbol\Omega\) rotates the axis, then

(164)#\[ \dot{\boldsymbol n}=\boldsymbol\Omega\times\boldsymbol n. \]

Because \(\boldsymbol n^2=1\), differentiating gives \(\boldsymbol n\cdot\dot{\boldsymbol n}=0\): the velocity of \(\boldsymbol n\) is tangent to the sphere. We choose \(\boldsymbol\Omega\) to be the transverse angular velocity that changes the axis direction; any component parallel to \(\boldsymbol n\) would describe additional rotation about the axis and is already contained in \(S\boldsymbol n\). With this transverse choice, cross (164) with \(\boldsymbol n\) from the left and apply the BAC-CAB identity \(\boldsymbol a\times(\boldsymbol b\times\boldsymbol c)=\boldsymbol b\,(\boldsymbol a\cdot\boldsymbol c)-\boldsymbol c\,(\boldsymbol a\cdot\boldsymbol b)\):

\[\begin{split} \begin{split} \boldsymbol n\times\dot{\boldsymbol n} &= \boldsymbol n\times(\boldsymbol\Omega\times\boldsymbol n) \\ &= \boldsymbol\Omega\,(\boldsymbol n\cdot\boldsymbol n) - \boldsymbol n\,(\boldsymbol n\cdot\boldsymbol\Omega) \\ &= \boldsymbol\Omega, \end{split} \end{split}\]

using \(\boldsymbol n\cdot\boldsymbol n=1\) and \(\boldsymbol n\cdot\boldsymbol\Omega=0\). Hence

(165)#\[ \boldsymbol\Omega=\boldsymbol n\times\dot{\boldsymbol n}. \]

Equation of Motion for the Spin Axis#

The equation of motion comes from torque balance,

\[ \dot{\boldsymbol L}=\boldsymbol\tau. \]

The transverse spin-axis dynamics is

(166)#\[ \boxed{ I\left.\ddot{\boldsymbol n}\right|_\perp = \boldsymbol\tau\times\boldsymbol n - \dot{\boldsymbol n}\times\boldsymbol S } \]

Here the vertical bar means “take the tangent component on the spin sphere”:

(167)#\[ \left.\ddot{\boldsymbol n}\right|_\perp \equiv \ddot{\boldsymbol n}-\boldsymbol n\left(\boldsymbol n\cdot\ddot{\boldsymbol n}\right). \]

Read the right-hand side term by term:

Applied torque: \(\boldsymbol\tau\times\boldsymbol n\) is the force that the external torque produces on the spin axis.
Gyroscopic term: \(-\dot{\boldsymbol n}\times\boldsymbol S\) is the sideways response caused by the pre-existing spin angular momentum \(\boldsymbol S=S\boldsymbol n\).

The second term is the key structure: it has the form of a monopole Lorentz force on the sphere of spin directions.

Derivation: spin-axis equation

Starting from (163),

\[ \boldsymbol L=S\boldsymbol n+I\boldsymbol n\times\dot{\boldsymbol n}. \]

Taking a time derivative gives

\[ \boldsymbol\tau =\dot{\boldsymbol L} =S\dot{\boldsymbol n}+I\boldsymbol n\times\ddot{\boldsymbol n}, \]

where the term \(\dot{\boldsymbol n}\times\dot{\boldsymbol n}\) vanishes. Cross both sides with \(\boldsymbol n\) on the right:

\[ \boldsymbol\tau\times\boldsymbol n =S\dot{\boldsymbol n}\times\boldsymbol n +I(\boldsymbol n\times\ddot{\boldsymbol n})\times\boldsymbol n. \]

The last factor is the tangent acceleration,

\[ (\boldsymbol n\times\ddot{\boldsymbol n})\times\boldsymbol n =\ddot{\boldsymbol n}-\boldsymbol n(\boldsymbol n\cdot\ddot{\boldsymbol n}) =\left.\ddot{\boldsymbol n}\right|_\perp. \]

Using \(\boldsymbol S=S\boldsymbol n\) then gives (166).

Spin as a Charged Particle on a Monopole Sphere#

Now compare (166) with the Lorentz-force equation for a charged particle constrained to a sphere. Using the same vertical-bar notation for the tangent component,

\[ \boxed{ m\left.\ddot{\boldsymbol x}\right|_\perp = \boldsymbol E_\perp +q\dot{\boldsymbol x}\times\boldsymbol B } \]

The dictionary is

Spin-monopole dictionary

\[\begin{split} \begin{array}{ccl} \text{charged particle} & & \text{classical spin axis}\\[2mm] m & \leftrightarrow & I\\ \boldsymbol x & \leftrightarrow & \boldsymbol n\\ \left.\ddot{\boldsymbol x}\right|_\perp & \leftrightarrow & \left.\ddot{\boldsymbol n}\right|_\perp\\ \boldsymbol E_\perp & \leftrightarrow & \boldsymbol\tau\times\boldsymbol n\\ q\boldsymbol B(\boldsymbol n) & \leftrightarrow & -S\boldsymbol n \end{array} \end{split}\]

Thus the gyroscopic force is the Lorentz force of a radial monopole field on the spin sphere.

If we write the monopole field as

\[ \boldsymbol B(\boldsymbol n)=\frac{g}{4\pi}\boldsymbol n, \]

then the spin system corresponds to

(168)#\[ \frac{qg}{4\pi}=-S. \]

The sign depends on orientation conventions. The important point is the magnitude: the spin angular momentum \(S\) is proportional to the electric-magnetic coupling \(qg/(4\pi)\). In dimensionless form,

(169)#\[\begin{split} \begin{split} s_{\mathrm{spin}} &= \frac{S}{\hbar}, \\ |s_{\mathrm{mono}}| &= \left|\frac{qg}{4\pi\hbar}\right|. \end{split} \end{split}\]

This is the bridge to the monopole quantum number used in §4.4.3.

Monopole Vector Potential on the Spin Sphere#

A radial monopole field on the unit sphere can be generated locally by a vector potential. One northern-patch choice for the spin sphere is

(170)#\[ \boldsymbol A(\boldsymbol n)=S\frac{\cos\theta-1}{\sin\theta}\,\boldsymbol e_\varphi. \]

It satisfies

(171)#\[ \nabla_\perp\times\boldsymbol A=-S\boldsymbol n, \]

which is exactly the effective monopole field in (166).

Why a string appears

The potential (170) diverges at the south pole. This is not a physical force singularity; the field strength is smooth on the sphere away from the monopole at the center. The singularity says that one gauge patch cannot cover a sphere threaded by nonzero monopole flux.

This is the same obstruction that becomes the Dirac string in §4.4.2.

Check: curl of the spin-sphere potential

For a vector potential with only a \(\varphi\) component on the unit sphere,

\[ (\nabla_\perp\times\boldsymbol A)\cdot\boldsymbol n =\frac{1}{\sin\theta}\partial_\theta\left(A_\varphi\sin\theta\right). \]

Substitute \(A_\varphi=S(\cos\theta-1)/\sin\theta\):

\[\begin{split} \begin{split} A_\varphi\sin\theta &= S(\cos\theta-1), \\ \partial_\theta\left[S(\cos\theta-1)\right] &= -S\sin\theta, \\ \frac{1}{\sin\theta}\partial_\theta\left(A_\varphi\sin\theta\right) &= -S. \end{split} \end{split}\]

Therefore \(\nabla_\perp\times\boldsymbol A=-S\boldsymbol n\).

Quantization from Berry Phase#

The monopole flux through the spin sphere appears as a Berry phase. For a closed path \(\mathcal C\) of the spin axis,

(172)#\[ \Phi_{\mathrm{Berry}}[\mathcal C]=\frac{1}{\hbar}\oint_\mathcal C \boldsymbol A\cdot\mathrm d\boldsymbol l. \]

For a loop enclosing solid angle \(\Omega\),

(173)#\[ \Phi_{\mathrm{Berry}}=-\frac{S}{\hbar}\Omega. \]

A full sphere has \(\Omega=4\pi\). Single-valuedness requires the phase around the full flux to be physically trivial:

\[\begin{split} \begin{split} \frac{4\pi S}{\hbar} &= 2\pi n, \\ n &\in \mathbb Z. \end{split} \end{split}\]

Therefore

Spin quantization as monopole flux quantization

(174)#\[\begin{split} \begin{split} S &= \hbar\frac{n}{2}, \\ s &\equiv \frac{S}{\hbar} = \frac{n}{2}. \end{split} \end{split}\]

Half-integer spin is the statement that monopole flux through the spin sphere is quantized.

Summary#

A classical spin direction is a point \(\boldsymbol n\) on the sphere of possible axes.
For a spherical top, the spin-axis dynamics is governed by \(I\left.\ddot{\boldsymbol n}\right|_\perp=\boldsymbol\tau\times\boldsymbol n-\dot{\boldsymbol n}\times S\boldsymbol n\).
The gyroscopic term has the same form as a Lorentz force from a radial monopole field on the spin sphere.
The effective monopole vector potential has a string singularity, so it must be described patchwise.
Quantizing the Berry phase of the monopole flux gives \(S=\hbar n/2\), preparing the Dirac monopole and monopole harmonics in §4.4.2–§4.4.3.

Homework#

1. Spin-axis kinematics. Let \(\boldsymbol n(t)\) be a unit vector and suppose its motion is generated by an angular velocity \(\boldsymbol\Omega\) through \(\dot{\boldsymbol n}=\boldsymbol\Omega\times\boldsymbol n\).

(a) Show that \(\boldsymbol n\cdot\dot{\boldsymbol n}=0\).

(b) Assuming \(\boldsymbol\Omega\) is tangent to the sphere, show that \(\boldsymbol\Omega=\boldsymbol n\times\dot{\boldsymbol n}\).

2. Steady precession. A spherical top with spin angular momentum \(S\) in a torque field \(\boldsymbol{\tau}=-\boldsymbol{n}\times\boldsymbol{B}\) (gyromagnetic coupling) has the spin-axis equation of motion

\[ I\,\left.\ddot{\boldsymbol{n}}\right|_\perp=\boldsymbol{\tau}\times\boldsymbol{n}-\dot{\boldsymbol{n}}\times S\boldsymbol{n}. \]

(a) Look for a steady-precession solution \(\boldsymbol{n}(t)=(\sin\theta\cos\Omega t,\sin\theta\sin\Omega t,\cos\theta)\) at fixed angle \(\theta\) around \(\hat{z}\). Substitute and find the precession frequency \(\Omega(B,S,I,\theta)\).

(b) In the gyroscopic limit \(I\Omega\ll S\), recover the Larmor-like result \(\Omega=B/S\) — independent of \(I\) and \(\theta\).

(c) Explain physically why the spin \(S\) acts like a “rotational mass” for the spin-axis orientation, distinct from the moment of inertia \(I\).

3. Spin-monopole dictionary. Compare

\[ I\left.\ddot{\boldsymbol n}\right|_\perp=\boldsymbol\tau\times\boldsymbol n-\dot{\boldsymbol n}\times S\boldsymbol n \]

with the Lorentz-force equation

\[ m\left.\ddot{\boldsymbol x}\right|_\perp=\boldsymbol E_\perp+q\dot{\boldsymbol x}\times\boldsymbol B. \]

(a) Give the dictionary between \(m,\boldsymbol x,\left.\ddot{\boldsymbol x}\right|_\perp,\boldsymbol E_\perp,\boldsymbol B\) and \(I,\boldsymbol n,\left.\ddot{\boldsymbol n}\right|_\perp,\boldsymbol\tau\).

(b) If \(\boldsymbol B=g\boldsymbol n\), show that the analogy identifies \(qg/(4\pi)=-S\) up to orientation convention.

4. Curl of the monopole potential. On the unit sphere, take

\[\begin{split} \begin{split} A_\theta &= 0, \\ A_\varphi &= S\frac{\cos\theta-1}{\sin\theta}. \end{split} \end{split}\]

(a) Use

\[ (\nabla_\perp\times\boldsymbol A)\cdot\boldsymbol n =\frac{1}{\sin\theta}\partial_\theta(A_\varphi\sin\theta) \]

to show \(\nabla_\perp\times\boldsymbol A=-S\boldsymbol n\).

(b) Where is this vector potential singular?

5. Latitude Berry phase. Consider a loop at fixed polar angle \(\theta\) on the spin sphere.

(a) Compute

\[ \oint \boldsymbol A\cdot\mathrm d\boldsymbol l \]

using \(A_\varphi=S(\cos\theta-1)/\sin\theta\).

(b) Show that the result is \(-S\Omega(\theta)\), where \(\Omega(\theta)=2\pi(1-\cos\theta)\) is the solid angle of the cap.

6. Single-valuedness across spin values. The Berry-phase factor accumulated for a loop at polar angle \(\theta\) is \(\mathrm{e}^{-\mathrm{i}S\Omega(\theta)/\hbar}\), where \(\Omega(\theta)=2\pi(1-\cos\theta)\). The full sphere encloses \(\Omega=4\pi\), and single-valuedness of the wavefunction forces \(S=\hbar n/2\) for integer \(n\).

(a) For \(n=2\) (spin-1), verify single-valuedness by computing the Berry-phase factor over the full sphere. Show the result is \(\mathrm{e}^{-4\pi\mathrm{i}}=1\).

(b) For \(n=1\) (spin-1/2), compute the phase over the full sphere (\(\mathrm{e}^{-2\pi\mathrm{i}}=1\)) and over a single equatorial \(2\pi\) rotation (\(\mathrm{e}^{-\mathrm{i}\pi}=-1\)). Explain why this gives the famous “spin-1/2 needs \(4\pi\) to return to itself.”

(c) For \(n=0\) (zero spin), explain why no quantization condition arises and discuss what this says about the “spinless” limit.

7. Connecting to Dirac monopoles. In §4.4.2 a charged particle near a magnetic monopole has dimensionless monopole quantum number

\[ s=\frac{qg}{4\pi\hbar}. \]

(a) Compare this with the spin quantum number \(s=S/\hbar\).

(b) Explain why the identification \(S\leftrightarrow |qg|/(4\pi)\) makes spin quantization and Dirac quantization the same mathematical condition.

(c) Preview §4.4.3: why should wavefunctions on the spin sphere be related to monopole harmonics rather than ordinary spherical harmonics?