Exact quantization rule and the invariant

We present and prove the exact quantization rules both for the one-dimensional Schrodinger equation and for the three-dimensional Schrodinger equation with a spherically symmetric potential. In the exact quantization rule, in addition to the usual term N pi, there is an integral term, called the correction term. For the quantum systems with a so-called shape invariant potential in the supersymmetric quantum mechanics, we find that the correction term is an invariant I independent of the number of nodes in the wave functions. In those systems, the invariant can be determined with the help of the energy and the wave function of the ground state, and then, the energy levels of all the bound states can be easily calculated from the exact quantization rule. Conversely, the validity of the calculated energy levels shows that the correction term is an invariant in those quantum system with a shape invariant potential. The systems with a shape invariant potential we calculated in this paper are the one-dimensional systems with a finite square well, with the harmonic oscillator potential, with the Morse potential and its generalizations, with the Rosen-Morse potentials, with the Poschl-Teller potentials, with the Eckart potential, and with the Hulthen potential, and the three-dimensional systems of harmonic oscillators and the hydrogen atom.