Commutation Relations

The three components of angular momentum cannot be measured simultaneously with arbitrary precision. This single fact is encoded in their commutation relations, and from those relations the entire spectrum of angular momentum follows.

The fundamental relation

For the orbital operators of the previous lesson, a direct calculation with the differential forms gives

and by the cyclic $x\to y\to z$ symmetry,

These three relations can be packaged compactly using the Levi-Civita symbol $\varepsilon_{ijk}$:

A sample derivation

Consider $[\hat{L}_x, \hat{L}_y]$. Using $\hat{L}_x = y\hat{p}_z - z\hat{p}_y$ and $\hat{L}_y = z\hat{p}_x - x\hat{p}_z$, expand the commutator. Most terms cancel because operators acting on different coordinates commute (for example $y$ commutes with $\hat{p}_x$). The surviving contributions come from the single non-commuting pair $[z, \hat{p}_z] = i\hbar$, which leaves

The abstract Lie algebra

Any set of three Hermitian operators obeying $[\hat{J}_i, \hat{J}_j] = i\hbar\,\varepsilon_{ijk}\hat{J}_k$ is called an angular momentum. This is the Lie algebra of $\mathrm{SU}(2)$, and it does not require an underlying $\mathbf{r}\times \mathbf{p}$. Spin is the prime example: it has no spatial wavefunction, yet its operators obey exactly these relations. We will exploit this generality throughout the module.

A concrete check with spin-1/2

The smallest nontrivial representation uses the Pauli matrices. Setting $\hbar = 1$ and $S_i = \tfrac{1}{2}\sigma_i$, you can verify the algebra by direct matrix multiplication:

In the exercise you will compute this commutator explicitly and read off the proportionality constant, confirming the fundamental relation in a finite-dimensional setting.

Try it

Build the spin-1/2 operators from the Pauli matrices and compute $[S_x, S_y]$. Compare it to $i\,S_z$ and return the proportionality constant (it should be $1$).

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