Eigenvalues for a Pure Quartic Oscillator

The oscillator with a quartic anharmonicity, with Hamiltonian
has been extensively treated in the literature. Consider the pure quartic oscillator, in which the quadratic term is missing: . For simplicity, take . The Schrödinger equation thus reduces to
.
No analytic solution has been found, but accurate numerical computations have been carried out [1]. This Demonstration applies the operator method [2], a generalization of the canonical operator formulation for the harmonic oscillator, for potential energies that are functions of even powers of . For larger values of the matrix dimension, the results obtained here are comparable with the published results.
The computation results in a secular equation for the eigenvalues, which are plotted as red lines on the graph superposed on the potential energy curve. The number of eigenvalues shown is equal to the selected matrix dimension. For comparison, the first eight eigenvalues according to the WKB method are shown as thin gray lines.

SNAPSHOTS

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DETAILS

The ladder operators can be defined by
,
,
with an adjustable parameter introduced. The nonvanishing matrix elements of the Hamiltonian can then be computed, giving
,
,
.
The eigenvalues are then determined using the built-in Wolfram Language function Eigenvalues for selected dimensions 1 to 8.
The WKB method determines the eigenvalues using the integral . The resulting energies are given by .
A classical realization of a quartic oscillator can be approximated by a particle attached to two Hooke's law springs [3].
References
[1] P. M. Mathews, M. Seetharaman, S. Raghavan and V. T. A. Bhargava, "A Simple Accurate Formula for the Energy Levels of Oscillators with a Quartic Potential," Physics Letters A, 83(3), 1981 pp. 118–120. doi:10.1016/0375-9601(81)90511-9.
[2] S. M. Blinder, "Ammonia Inversion Energy Levels Using Operator Algebra." arxiv.org/abs/1809.08178.
[3] M. Alexander. "The Quartic Oscillator in Classical Physics." (Feb 21, 2019) www2.chem.umd.edu/groups/alexander/chem691/Quartic_oscillator.pdf.
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