The goal of physics is to understand the what, how, and why of the universe. The twist is that sometimes one sometimes has to introduce auxiliary physical quantities as stepping stones towards such an understanding. An exemplar of this in classical mechanics is the introduction of orbital angular momentum \(\textbf L:=\textbf x\times\textbf p\) as an auxiliary physical quantity in understanding orbital mechanics and more generally dynamics in isotropic potentials \(V(r)\) thanks to the conservation law \(\dot{\textbf L}=0\) in such situations. In quantum mechanics, a similar phenomenon occurs. Always, the goal is to understand how states \(|\psi(t)\rangle\in\mathcal H\) of quantum systems (e.g. electrons, qubits) evolve with time \(t\) in the presence of a particular potential energy environment \(V(\textbf x,t)\). Therefore, in principle all that has to be done is to solve the Schrodinger equation \(i\hbar\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle\) with a suitable initial condition \(|\psi(0)\rangle\). For \(\frac{\partial H}{\partial t}=0\), the solution is immediate \(|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle\) (otherwise see interaction picture and Dyson series) and to then further compute the unitary time evolution operator \(e^{-iHt/\hbar}\) one diagonalizes it via a resolution of the identity \(e^{-iHt/\hbar}=\sum_{E\in\Lambda_H}e^{-iEt/\hbar}|\rangle E\langle E|\). However, this is still not particularly useful unless one can actually find the \(H\)-eigenstates \(|E\rangle\) and the spectrum \(\Lambda_H\). As with linear algebra on finite-dimensional vector spaces where one typically chooses some convenient basis for the vector space to perform explicit calculations in, with quantum mechanics one also typically selects some convenient basis for state space (e.g. the \(\textbf X\)-eigenbasis) which allows explicit computation of \(\langle \textbf x|E\rangle\) and also the energies \(E\). A striking difference however is that whereas in the finite-dimensional case this is typically done by finding the zeroes of the characteristic polynomial \(\det(H-E1)=0\) which would be like extracting the energies \(E\) first before subsequently extracting the energy eigenstates \(|E\rangle\), in quantum mechanics it is often the case that it may happen the other way around as a byproduct of the fact that PDEs and polynomials have different techniques for solving them (although exceptions to this exist such as Dirac’s auxiliary construction of the creation/raising and annihilation/lowering ladder operators to extract the spectrum \(\Lambda_{H_{\text{QHM}}=\{\hbar\omega_0(n+1/2):n=1,2,3,…\}\) of \(H_{\text{QHM}}\) in a completely \(\mathcal H\)-basis free manner without knowing the \(H_{\text{QHM}}\)-eigenstates \(|\hbar\omega_0(n+1/2)\rangle\)).
All that being said then, as with the classical Kepler problem it is often useful to define auxiliary observables to aid in solving the relevant PDEs. Specifically, one can promote the classical orbital angular momentum vector \(\textbf L:=\textbf x\times\textbf p\) to a quantum vector observable \(\textbf L:=\textbf X\times\textbf P\) (quantum mechanics also has the spin angular momentum observable \(\textbf S\) as a novel surprise).
The philosophy for classifying all (finite-dimensional, complex, continuous/smooth) \(SU(2)\) representations will be to proceed retrospectively by a sequence of reductions:
- To classify all \(SU(2)\) representations, it suffices to classify all \(SU(2)\) irreducible representations since \(SU(2)\) has a compact topology (see Heine-Borel theorem), so \(SU(2)\) irreducible representations will then span (via direct sums) all \(SU(2)\) representations (see Peter-Weyl theorem).
- To classify all \(SU(2)\) irreducible representations, it suffices to classify all \(\frak{su}\)\((2)\) irreducible representations since \(SU(2)\) has a simply connected topology, so all its irreducible Lie algebra (\(\frak{su}\)\((2)\)) representations will integrate uniquely up to \(SU(2)\) irreducible representations.
- To classify all \(\frak{su}\)\((2)\) irreducible representations, it suffices to classify all \(\frak{sl}\)\(_2(\textbf C)\) irreducible representations since \(\frak{sl}\)\(_2(\textbf C)\cong\frak{su}\)\((2)\otimes_{\textbf R}\textbf C\) is the complexification of \(\frak{su}\)\((2)\), and we’re only interested in complex representations.
When one sees the Pauli matrices for the first time:
\[\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}\]
\[\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix}\]
\[\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\]
they can be confusing since it is not clear where they come from. The answer is simply: any spin \(s=1/2\) quantum particle has a qubit state space \(\mathcal H\cong\textbf C^2\) which is spanned by the simultaneous \(\textbf S^2\) and \(S_3\)-eigenstates \(|3\hbar^2/4,\hbar/2\rangle\) and \(|3\hbar^2/4,-\hbar/2\rangle\). For brevity, henceforth denote these angular momentum eigenstates respectively as \(|\hbar/2\rangle\) and \(|-\hbar/2\rangle\). Then clearly, one has:
\[[S_3]_{|\hbar/2\rangle,|-\hbar/2\rangle}^{|\hbar/2\rangle,|-\hbar/2\rangle}=\begin{pmatrix}\hbar/2&0//0&-\hbar/2\end{pmatrix}=\frac{\hbar}{2}\sigma_3\]
Thus, this explains why the \(\sigma_3\) Pauli matrix is so simple/diagonal! One can perform a similar computation using \(S_1=(S_++S_-)/2\) and \(S_2=(S_+-S_-)/2i\) (cf. \(\cos(\theta)=(e^{i\theta}+e^{-i\theta})/2\) and \(\sin(\theta)=(e^{i\theta}+e^{-i\theta})/2i\)) and standard Condon-Shortley normalizations to obtain:
\[S_1|\hbar/2\rangle = \hbar/2|-\hbar/2\rangle\]
\[S_1|-\hbar/2\rangle=\hbar/2|\hbar/2\rangle\]
\[S_2|\hbar/2\rangle=i\hbar/2|-\hbar/2\rangle\]
\[S_2|-\hbar/2\rangle=-i\hbar/2|\hbar/2\rangle\]
All this is to say that the raising and lowering operators are UNDERRATED! Anyways, I propose to call Pauli’s identity for spin-1/2 quantum particles:
\[[\textbf S]_{|\hbar/2\rangle,|-\hbar/2\rangle}^{|\hbar/2\rangle,|-\hbar/2\rangle}=\frac{\hbar}{2}\boldsymbol{\sigma}\]
One can proceed in an entirely analogous fashion for spin \(s=1\) quantum particles (e.g. \(W\) and \(Z\) bosons), deriving analogs \(\Sigma_1,\Sigma_2,\Sigma_3\) of the Pauli matrices for them in the \(S_3\)-eigenbasis (except that these matrices no longer have any special name).
\[\Sigma_1=\frac{1}{\sqrt{2}}\begin{pmatrix} 0&1&0\\1&0&1\\0&1&0\end{pmatrix}\]
\[\Sigma_2=\frac{i}{\sqrt{2}}\begin{pmatrix} 0&-1&0\\1&0&-1\\0&1&0\end{pmatrix}\]
\[\Sigma_3=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&-1\end{pmatrix}\]
where now \([\textbf S]_{|\hbar\rangle,|0\rangle,|-\hbar\rangle}^{|\hbar\rangle,|0\rangle,|-\hbar\rangle}=\hbar\boldsymbol{\Sigma}\) and so forth for \(s=3/2\), \(s=2\), etc.
A simple classical model of a compass needle is to treat it as a magnetic dipole \(\boldsymbol{\mu}\) in the presence of an external magnetic field \(\textbf B_{\text{ext}}\) such as \(\textbf B_{\text{Earth}}\). If the compass needle has rotational inertia \(I\), then an external couple of \(\boldsymbol{\tau}_{\text{ext}}=\boldsymbol{\mu}\times\textbf B_{\text{ext}}\) acts on it, so denoting \(\theta:=\angle\boldsymbol{\mu},\textbf B_{\text{ext}}\), one has the equation of motion:
\[I\ddot{\theta}=\muB_{\text{ext}}\sin(\theta)\]
Thus the compass needle behaves precisely like a pendulum, which looks like simple harmonic oscillation in the small-angle approximation \(\theta\to 0\) (of course in practice there is also damping effects which cause the compass needle to align precisely \(\theta=0\) in the steady state, arguably an example of where friction is actually useful). One more simplification in the model is that for paramagnetic compasses I suspect that \(\boldsymbol{\mu}\) itself is a function of \(\theta\), with greater alignment \(\theta\to 0\) inducing a stronger magnetic dipole moment \(\boldsymbol{\mu}(\theta)\).
However, a more interesting case arises when the magnetic dipole moment \(\boldsymbol{\mu}\) arises from the intrinsic spinning of the object. Then the magnetic dipole moment is \(\boldsymbol{\mu}=\gamma\textbf L\) where \(\gamma\) is the gyromagnetic ratio. Analogous to how \(\dot{\textbf v}=(q/m)\textbf v\times\textbf B_{\text{ext}}\) gives rise to a precession in \(\textbf v(t)\) (corresponding to helical spiralling along \(\textbf B_{\text{ext}}\) in \(\textbf x(t)\) with angular frequency \(\omega_0=qB_{\text{ext}}/m), such a gyroscopic magnetic dipole moment leads to a mathematically analogous equation of motion \(\dot{\textbf L}=\gamma\textbf L\times\textbf B_{\text{ext}}\) which indicates that \(\textbf L(t)\) precesses around \(\textbf B_{\text{ext}}\) with angular frequency \(\omega_0=\gammaB_{\text{ext}}\) called the Larmor angular frequency. Since it is the case that \(\boldsymbol{\mu}\propto\textbf S\) for quantum particles, one might anticipate on the basis of classical physics that the spin angular momentum of quantum particles will also precess about an external magnetic field \(\textbf B_{\text{ext}}\) and indeed, understood in a suitable sense, this is what quantum mechanics predicts will happen too and underlies the technology of magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy.