A recent method called asymptotic Taylor expansion (ATEM) is applied to determine the analytical expression for eigenfunctions and numerical results for eigenvalues of the Schr？dinger equation for the bistable potentials. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical results for eigenvalues. It is shown that the results are obtained by a simple algorithm constructed for a computer system using symbolic or numerical calculation. It is observed that ATEM produces excellent results consistent with the existing literature. 1. Introduction There is no doubt that an interesting problem in fundamental quantum mechanics for lecturers and advanced undergraduate and graduate students in physics and applied mathematics is to obtain the exact solutions of the Schr？dinger equation for any type of potential. It is well known that the Schr？dinger equation, proposed by Erwin Schr？dinger in 1926, is a second-order differential equation that describes how the quantum state of a physical system changes with time [1]. It is as central to quantum mechanics as Newton's laws are to classical mechanics. It is also well known that the quantum mechanics is established on some certain postulates, and in any introductory quantum physics textbook these postulates can be found with the application of the time-independent Schr？dinger equation for a single particle in one dimension as a simple and basic example. Unfortunately, there are not so many potentials that can be solved exactly such as the Coulomb, harmonic oscillator, and P？schl-Teller potentials. Since one of the source of progress of the science depends on the study of the same problem from different point of view, various methods have been suggested such as numerical calculation [2, 3], the variational [2, 4], the perturbation [5], the WKB [6, 7], the shifted expansion [8, 9], the Nikiforov-Uvarov (NU) [10, 11], the supersymmetry (SUSY) [12, 13], the generalized pseudospectral (GPS) [14], the asymptotic iteration method (AIM) [15], and other methods [16–21] to find the approximate solutions of the potentials that are not exactly solvable. In this study we will apply a new formalism based on the Taylor series expansion method, namely, asymptotic Taylor expansion method (ATEM) [22], to bistable potentials. These type of potentials have been used in the quantum theory of molecules as a crude model to describe the motion of a particle in the presence of two centers of force [23–29]. It is mentioned in [22] that the taylor series Method [30, 31] is an old
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