On the basis of analytical transfer matrix theory, we fine a generalized quantization condition. By introducing a new type of modified momentum, our quantization condition has the same form as the Bohr–Sommerfeld formula. Numerical and analytical comparisons show that the present method is exact.
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.