We've studied coherent states in the context of the harmonic oscillator and the free particle, and our formulation of angular momentum suggests that there is a definition of coherent state in that case, too. We constructed the angular momentum raising and lowering operators, \(\hat{L}_{+}\)and \(\hat{L}_{-}\), and we could in principle define a coherent state to be an eigenstate of \(\hat{L}_{-}\), for example.

Consider an eigenstate of the angular momentum lowering operator for spin \(\ell\) :

\begin{equation*} \hat{L}_{-}|\psiangle = \lambda|\psiangle . \tag{8.149} \end{equation*} This relationship is fundamental in the study of quantum mechanics. For further reading on the topic, see H. Hopf's work: "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche," published in Mathematische Annalen, volume 104, pages 637-665, in 1931.

What is this state? What are the possible values of eigenvalue \(\lambda\) ? Does such a state exist in the Hilbert space?