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.
[1] In the simpler quantum mechanical context this potential served as a model for the evaluation of Feynman path integrals.
[2][3] 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[4] and the exploration of the large order behavior of asymptotic expansions.
[5][6][7] 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.
Müller-Kirsten[8]) 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.
[9] 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.
[10] 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.
The case of the ground state is mediated by pseudoclassical configurations known as instanton and anti-instanton.
As pseudoclassical configurations these naturally appear in semiclassical considerations—the summation of (widely separated) instanton-anti-instanton pairs being known as the dilute gas approximation.
The ground state eigenenergy finally obtained is an expression containing the exponential of the Euclidean action of the instanton.
The stability of the instanton configuration in the path integral theory of a scalar field theory with symmetric double-well self-interaction is investigated using the equation of small oscillations about the instanton.
[11] As stated above, the instanton is the pseudoparticle configuration defined on an infinite line of Euclidean time that communicates between the two wells of the potential and is responsible for the ground state of the system.
The configurations correspondingly responsible for higher, i.e. excited, states are periodic instantons defined on a circle of Euclidean time which in explicit form are expressed in terms of Jacobian elliptic functions (the generalization of trigonometric functions).
In addition this method requires matching of different branches of solutions in domains of overlap.
The application of boundary conditions finally yields (as in the case of the periodic potential) the nonperturbative effect.
In terms of parameters as in the Schrödinger equation for the symmetric double-well potential in the following form the eigenvalues for
are found to be (see book of Müller-Kirsten, formula (18.175b), p. 425) Clearly these eigenvalues are asymptotically (
Observe that terms of the perturbative part of the result are alternately even or odd in
In field theory contexts the above symmetric double-well potential is often written (
becomes the vacuum instanton solution, Perturbation theory along with matching of solutions in domains of overlap and imposition of boundary conditions (different from those for the double-well) can again be used to obtain the eigenvalues of the Schrödinger equation for this potential.
In terms of parameters as in the Schrödinger equation for the inverted double-well potential in the following form the eigenvalues for
are found to be (see book of Müller-Kirsten, formula (18.86), p. 503) The imaginary part of this expression agrees with the result of C.M.
[12] This result plays an important role in the discussion and investigation of the large order behavior of perturbation theory.
In terms of parameters as in the Schrödinger equation for the pure anharmonic oscillator in the following form the eigenvalues for
This is an important aspect of the solutions of the differential equation for quartic potentials.
instantons), and the WKB method, though with the use of elliptic integrals and the Stirling approximation of the gamma function, all of which make the calculation more difficult.
Moreover in each of these cases (double-well, inverted double-well, cosine potential) the equation of small fluctuations about the classical configuration is a Lamé equation.