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.
Contents
- The symmetric double well
- The inverted double well potential
- The pure anharmonic oscillator
- General comments
- References
Thus the "symmetric double-well potential" served for many years as a model to illustrate the concept of instanton 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 Dingle and Müller) 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 Müller-Kirsten.
The symmetric double-well
The main interest in the literature has (for reasons related to field theory) focussed on the symmetric double-well (potential), and there on the quantum mechanical ground state. Since tunneling through the central hump of the potential is involved, the calculation of the eigenenergies of the Schrödinger equation for this potential is nontrivial. The case of the ground state is mediated by pseudoclassical configurations known as instanton and anti-instanton. In explicit form these are hyperbolic functions. As pseudoclassical configurations these naturally appear in semi-classical 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. This is an expression containing the factor
The stability of the instanton configuration in the path integral theory of a scalar field theory with symmetric double-well self-interaction is investigated by investigating the equation of small oscillations about he instanton. One finds that this equation is a Pöschl-Teller equation (i.e. a second order differential equation like the Schrödinger equation with Pöschl-Teller potential) with nonnegative eigenvalues. The nonnegativity of the eigenvalues is indicative of the stability of the instanton.
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). The evaluation of the path integral in these cases involves correspondingly elliptic integrals. The equation of small fluctuations about these periodic instantons is a Lamé equation whose solutions are Lamé functions. In cases of instability (as for the inverted double-well potential) this equation possesses negative eigenvalues indicative of this instability, i.e. decay (as shown in the paper of Liang, Müller-Kirsten and Tchrakian referred to above).
Application of the perturbation method of Dingle and Müller (applied originally to the Mathieu equation, i.e. a Schrödinger equation with cosine potential) requires exploitation of parameter symmetries of the Schrödinger equation for the quartic potential. One expands around one of the two minima of the potential. 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
Clearly these eigenvalues are asymptotically (
In field theory contexts the above symmetric double-well potential is often written (
and the instanton is the solution
(
The equation of small fluctuations
with
Since all eigenvalues
In the more general case of
where
The inverted double-well potential
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 this case, however, one expands around the central trough of the potential. The results are therefore different from those above.
In terms of parameters as in the Schrödinger equation for the inverted double-well potential in the following form
the eigenvalues for
The imaginary part of this expression agrees with the result of Bender and Wu (see their formula (3.36) and set
The pure anharmonic oscillator
In terms of parameters as in the Schrödinger equation for the pure anharmonic oscillator in the following form
the eigenvalues for
More terms can easily be calculated. Observe the coefficients of the expansion are alternately even or odd in
General comments
The above results for the double-well and the inverted double-well can also be obtained by the path integral method (there via periodic instantons, cf. 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.