Checkpoint: Angular Momentum Algebra

This checkpoint pulls together the whole module: the algebra, the spectrum, the ladder operators, and their matrix elements. Work through the reasoning before running your answer.

What you have built

Over the last eleven lessons you constructed the complete quantum theory of angular momentum from a single seed — the commutator $[\hat{L}_i, \hat{L}_j] = i\hbar\,\varepsilon_{ijk}\hat{L}_k$:

• Operators. $\hat{L}_x, \hat{L}_y, \hat{L}_z$ are Hermitian observables; $\hat{L}^2$ is the Casimir that commutes with all of them.
• Spectrum. Joint eigenstates $|\ell, m\rangle$ satisfy $\hat{L}^2|\ell,m\rangle = \hbar^2\ell(\ell+1)|\ell,m\rangle$ and $\hat{L}_z|\ell,m\rangle = \hbar m|\ell,m\rangle$, with $m = -\ell, \dots, +\ell$.
• Ladders. $\hat{L}_\pm = \hat{L}_x \pm i\hat{L}_y$ step $m$ by $\pm 1$ with the coefficient $\hbar\sqrt{\ell(\ell+1) - m(m\pm 1)}$, terminating at the edges of the multiplet.

The challenge problem

Take a spin-$\tfrac{3}{2}$ system, $\ell = \tfrac{3}{2}$, whose four states are labeled $m \in \{-\tfrac32, -\tfrac12, +\tfrac12, +\tfrac32\}$. Apply the lowering operator to the state $m = +\tfrac12$:

Substituting $\ell = \tfrac32$ and $m = +\tfrac12$ (with $\hbar = 1$):

Try it

Compute the lowering coefficient for $\hat{L}_-$ acting on $\big|\tfrac32, +\tfrac12\big\rangle$ and return it. The answer is $2$.

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