In this article I will apply the ideas from part 1 to the theory of rotations in three dimensions. (The theory of rotations in an arbitrary number of dimensions is similar, but for reasons of familiarity and simplicity I will stick to 3 dimensions).

Consider the rotation of a function of space in the x-y plane (anticlockwise about the z-axis): \(\psi(x,y,z) \to \psi(x \cos \phi+ y \sin \phi, -x \sin \phi+ y \cos \phi, z)\) . For an infinitesimal rotation we get \(\psi(x+y \mathrm{d}\phi,-x\mathrm{d}\phi+y,z)=\psi(x,y,z)+ \left(y \frac{\partial \psi}{\partial x} -x \frac{\partial \psi}{\partial y}\right)\mathrm{d}\phi\) , so infinitesimally the rotations are \(\text{Id} + i (r_x p_y - r_y p_x) \mathrm{d}\phi\) where \(p_i = -i \frac{\mathrm{d}}{\mathrm{d}x_i}\) is the ith component of momentum. The commutation relations are \([x_i,x_j]=0, [p_i,p_j]=0, [x_i,p_j]=i \delta_{ij} \text{Id}\) . Note that in spherical coordinates \(x=r \cos \phi \sin \theta, y=r \sin \theta \sin \phi, z = r \cos \theta\) , \(r_x p_y - r_y p_x = -i \frac{\partial}{\partial \phi}\) . With this choice of coordinates it is easy to see finite rotations can be achieved by exponentiation: \(e^{t \frac{\partial}{\partial \phi}} f(r,\theta,\phi) = \sum_{n=0}^{\infty} t^n \frac{\partial^n f}{{\partial \phi}^n}(r,\theta,\phi) = f(r,\theta,\phi+t)\) , where the final equality is the Taylor expansion.

There are similar equations for rotations about the other coordinates, we define the ith component of angular momentum by \(L_i = \epsilon_{ijk} r_j p_k\) (Einstein summation convention will be used throughout this article). The three components of angular momentum generate rotation about an arbitrary axis; rotations about the axis through the unit vector (a,b,c) are generated by \(a L_x + bL_y + c L_z\) . Rotations don’t commute, infinitesimally this corresponds to the components of angular momentum not commuting \([L_i,L_j] = i \epsilon_{ijk} L_k\) .

There is another, quadratic, invariant associated to these operators \(L^2 = L_x^2 + L_y^2 +L_z^2\) which commutes with each component of angular momentum \([L,L^2]=0\) . These sorts of conserved quantities that are polynomials in the generators are called Casimir invariants; they are not a property of a Lie algebra but rather of a representation of a Lie algebra (or the universal enveloping algebra).

A maximally commuting set is given by \(L_z \text{ and } L^2\) (the choice of z is arbitrary, but conventional). These operators are invariant under rotations in the \(L_x-L_y\) plane:

\(\begin{bmatrix} L_x' \\ L_y' \end{bmatrix} = \begin{bmatrix} \cos \phi& \sin \phi \\ \sin \phi & \cos \phi\end{bmatrix} \begin{bmatrix} L_x \\ L_y \end{bmatrix}\) ,

and in fact the rotations are infinitesimally generated by \(L_z\) . Just as in the case of the simple harmonic oscillator we take the eigenstates of this transformation \(L_{\pm} = L_x \pm i L_y\) (notice that these are Hermitian conjugates of each other).

From algebra the commutation relations follow \([L_{\pm},L_{\mp}]=\pm 2 i L_z\) , \([L_z,L_{\pm}] = \pm L_{\pm}\) , and \(L_{\pm}L_{\mp} = L^2 - L_z^2 \pm L_z\) , \([L^2,L_{\pm}]=0\) . Again these commutation relations allow us to find the eigenvalues of the operator \(L^2\) as well as of the operator \(L_z\) (since they commute they are simultaneously diagonalizable).

Suppose \(\psi\) is a simultaneous, normalized, eigenvector of \(L^2\) and \(L_z\) ; \(L^2 \psi = \lambda \psi, L_z \psi = m \psi, \psi^{\dagger}\psi=1\) . Taking inner products over the Hilbert space gives \(0 \leq (L_{\pm} \psi)^{\dagger} L_{\pm} \psi = \psi^{\dagger} L^2 \psi - \psi^{\dagger} L_z^2 \psi \mp \psi^{\dagger} L_z \psi = \lambda - m^2 \mp m\) , which implies that \(m(m \pm 1) \leq \lambda\) . This implies that we can’t keep raising or lowering the z-component of angular momentum forever, there must be states for which e.g. \(L_{-} \psi_{\mbox{min}} = 0\) and \(L_{+} \psi_{\mbox{max}} = 0\) . Then \(L^2 \psi_{\mbox{min}} = (L_{+}L_{-} + L_z^2 - L_z) \psi_{\mbox{min}} = m_{\min}(m_{\min}-1) \psi_{\mbox{min}}\) . Similarly \(L^2 \psi_{\mbox{max}} = (L_{-}L_{+} + L_z^2 + L_z) \psi_{\mbox{max}} = m_{\max}(m_{\max}+1) \psi_{\mbox{max}}\) .

Suppose the maximum state is reached from the minimum state by raising n times, that is; substituting into the equations for \(\lambda\) above implies \(m_{\text{min}} = -\frac{n}{2}\) , \(m_{\text{max}}=\frac{n}{2}\) , \(\lambda =\frac{n}{2}(\frac{n}{2}+1)\) .

It is in fact possible to show that it must be an integer for \(L_i = i \epsilon_{ijk} r_j p_k\) (following an argument from Ballentine’s quantum mechanics textbook page 127): set \(q_{\pm} = \frac{1}{\sqrt{2}}(x \pm p_y)\) , \(p_{\pm} = p_x mp y\) . Then \(0=[q_{\pm},q_{\mp}] = [p_{\pm},p_{\mp}] = [q_{\pm},p_{\mp}]\) , and \([q_{\pm},p_{\pm}]=i=[q_{\mp},p_{\mp}]\) . These are the commutation relations for independent components of momentum and position. Then \(L_z = \frac{1}{2} (q_{+}^2 + p_{+}^2) - \frac{1}{2} (q_{-}^2 + p_{-}^2)\) which is the difference between two simple harmonic oscillators. Thus \(L_z\) must have eigenvalues of the form \((n+1/2) - (m+1/2) = (n-m)\) for suitable integers \(n,m\) ; consequently the eigenvalues of \(L_z\) must be integers. Similar to the simply harmonic oscillator by solving a first order differential equation in terms of the lowering operator we can find solutions to the second order differential operator \(L^2\) ; the solutions are the spherical harmonics (which are related to complex exponentials and Legendre polynomials), and as before we get Rodrigues’ type formulae.

Just as for the Simple Harmonic Oscillator the algebra of the angular momentum operators allowed us to find all the eigenvalues of a quadratic operator, and determine all its eigenfunctions in a particular basis. Incidentally the half-integer solutions can be realised, but not as rotations in 3 dimensions, they correspond to rotaions in spinor space – a rotation through 360 degrees in spinor space is only a rotation of 180 degrees in three dimensional space.