# Eigenvalues for a Pure Quartic Oscillator

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The oscillator with a quartic anharmonicity, with Hamiltonian

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Contributed by: S. M. Blinder (March 2019)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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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