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When the scalar potential is larger than the vector potential there are very few exactly solvable Klein–Gordon equations. Based on a general transformation between the unequal scalar and vector potential, in this paper, we employ two semiclassical methods to determine the bound state energy spectrum of the Klein–Gordon equation. To illustrate this procedure, the scalar potentials are chosen as the linear, exponential and linear plus Coulomb potentials and the corresponding energy spectra are analytically obtained. It is shown that the energy spectrum can be obtained by a simple algebraic method and our proposal methods can be extended to discuss the quasi-exactly solvable cases.

In this paper, a quantum action-angle variable in two dimensions is defined in the context of the quantum Hamilton-Jacobi theory, and then a quantization rule, which can give the exact bound state energy spectrum without solving the motion equation for the wavefunction, is derived from the singularities of the quantum momentum function. As illustrative examples, the Coulomb potential and the harmonic oscillator are considered.

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