# Ammonia Inversion: Classical and Quantum Models

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The ammonia molecule has a trigonal pyramidal configuration, with the nitrogen atom connected to three hydrogen atoms. The molecule readily undergoes inversion at room temperature, like an umbrella turning itself inside out in a strong wind. The energy barrier to this inversion is 24.2 kJ/mol. A resonance frequency of 24.87 GHz, in the microwave region, corresponds to the energy splitting of the two lowest vibrational levels. This transition is exploited in the ammonia maser.

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

Open content licensed under CC BY-NC-SA

## 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 Eigenvalue routine for selected dimensions 2 to 10.

Reference

[1] S. M. Blinder, "Ammonia Inversion Energy Levels Using Operator Algebra." arxiv.org/abs/1809.08178.

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