Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that I'm not considering applications in quantum computing here.)
EDIT: To clarify what I mean, consider a simple system like the harmonic oscillator. Using real variables $q_t$ and $p_t$ indexed by real-valued time $t$, and starting from energy conservation in a form like $\forall t\forall s p_s\cdot p_s+\omega\cdot\omega\cdot q_s\cdot q_s=p_t\cdot p_t+\omega\cdot\omega\cdot q_t\cdot q_t$ and from the relation $\dot{q}=p$ expressed using a form "$\forall\epsilon\ldots<\epsilon$", I have no doubt that most of the properties of the classical system could be laboriously derived using the axioms for the real numbers and classical logic. (Of course if I'm wrong and there is some obstacle that means that solving even such a simple dynamical system cannot be logically formalized, that would be an interesting answer, too!) Has anyone ever tried to derive the properties of the corresponding quantum system using some appropriate form of quantum logic?