In quantum mechanics, the Schrödinger equation is a partial differential equation describing the timely modification of quantum systems. The equation was postulated by Schrödinger about a century ago and has stayed at the helm of quantum mechanics being an equivalence of Newton’s second law in classical mechanics. With a broad range of problems arising from the Schrodinger representation, researchers have adopted solutions based on exponential time-dependent perturbation theories. In particular, Floquet-Magnus expansion and Fer expansion have attracted significant research attention.
Herein, Dr. Eugene Stephane Mananga from the City University of New York and New York University, recently explored the two potential expansion schemes developing in the NMR field. The Floquet-Magnus and Fer expansion were specifically used to investigate the spin dynamics process in a spin-locking radiation experiment. The aim was to use the expansion schemes in the calculation of the effective Hamiltonians and propagators to improve the existing understanding of the spin-locking radiation for the benefit of solid-state NMR and magnetic resonance in general. The work is currently published in the journal, Chemical Physical Letters.
The proposed criterion for the Floquet-Magnus expansion is presented through the iterative based approach. Additionally, theoretical analysis was vital in controlling the systems during spin radiation. In the past analysis, it was concluded that Fer expansion was advantageous in calculating correct higher-order as compared to its counterparts. To ascertain this result, the author examined a magic-echo sequence using the Fer expansion and compared the results to the existing literature. Finally, the applications and relationship between the two approaches was established.
Comparing and contrasting the two, the two orders of both expansions were observed to be identical for time-dependent Hamiltonian. At higher orders, however, the discrepancy between the two orders was evident. For instance, the third order of Floquet-Magnus expansion looked more complicated than Fer expansion. The results were in good agreement with the existing resulting regarding the Floquet-Magnus expansion and Fer expansion expansions and more so the simplicity and efficiency of Fer expansion in the calculation of higher-order corrections. However, for solving spin dynamics problems in solids, the Floquet-Magnus expansion was more suitable and less complex to use. Since the two approaches are developed generally for different purposes, it is not guaranteed that they may be equal at a particular instance.
In summary, the study presented the applications and a comparison of the Floquet-Magnus and Fer expansion approaches of the Schrodinger equation in spin-locking radiation of solid-state NMR. The results could be extended to the construction and implementation of numerical integrators. The work is identified by the Advances in Engineering committee as a useful contributor to the general field of spin dynamics. In a statement to Advances in Engineering, Dr. Eugene stated that his theoretical and numerical investigations will particularly be of great interest in advancing not only the solid-state NMR but the general magnetic resonance community at large.
Mananga, E. (2019). Application of Floquet-Magnus and Fer expansion approaches during spin-locking radiation in solid-state NMR. Chemical Physics Letters, 730, 153-164.