Intrinsic resonance representation of quantum mechanics

The choice of basis states in quantum calculations can be influenced by several requirements, and sometimes a very natural basis suggests itself. However often one retreats to a “merely complete” basis, whose coefficients in the eigenstates carry little physical insight. We suggest here an optimal representation, based purely on classical mechanics. “Hidden” constants of the motion and good actions already known to the classical mechanics are thus incorporated into the basis, leaving the quantum effects to be isolated and included by small matrix diagonalizations. This simplifies the hierarchical structure of couplings between “zero-order” states. We present a (non-perturbative) method to obtain such a basis-state as solutions to a certain resonant Hamilton–Jacobi equation.