Double-well potential

The so-called double-well potential is one of a number of quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical properties since it permits in many cases explicit calculation without over-simplification. Thus the "symmetric double-well potential" served for many years as a model to illustrate the concept of instantons as a pseudo-classical configuration in a Euclideanised field theory. In the simpler quantum mechanical context this potential served as a model for the evaluation of Feynman path integrals. or the solution of the Schrödinger equation by various methods for the purpose of obtaining explicitly the energy eigenvalues. The "inverted symmetric double-well potential", on the other hand, served as a nontrivial potential in the Schrödinger equation for the calculation of decay rates and the exploration of the large order behavior of asymptotic expansions. The third form of the quartic potential is that of a "perturbed simple harmonic oscillator" or ′′pure anharmonic oscillator′′ having a purely discrete energy spectrum. The fourth type of possible quartic potential is that of "asymmetric shape" of one of the first two named above. The double-well and other quartic potentials can be treated by a variety of methods—the main methods being (a) a perturbation method (that of B. Dingle and H.J.W. Müller-Kirsten) which requires the imposition of boundary conditions, (b) the WKB method and (c) the path integral method.. All cases are treated in detail in the book of H.J.W. Müller-Kirsten. The large order behavior of asymptotic expansions of Mathieu functions and their eigenvalues (also called characteristic numbers) has been derived in a further paper of R.B. Dingle and H.J.W. Müller. The main interest in the literature has (for reasons related to field theory) focused on the symmetric double-well (potential), and there on the quantum mechanical ground state.