This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudodifferential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised nondegenerate potential well, yielding uniform estimates in the energy $E$ . This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\hstrok)$ and gives improvements and new proofs for many of the results in the field. In the completely resonant case, we show that the pseudodifferential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained