Quaternion, Harmonic Oscillator, and High-Dimensional Topological States

Sergio Giardino

Kengatharam Thirulogasanthar, Adam Krzyżak, Qutaibeh Deeb Katatbeh

B. Muraleetharan, Kengatharam Thirulogasanthar

Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols, and related quantities are analyzed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are also obtained.

John P. Lowe

Sergio Giardino

The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.

John P. Lowe, Kirk A. Peterson