Abstract
This short review article presents theories used in solid-state nuclear magnetic resonance spectroscopy. Main theories used in NMR include the average Hamiltonian theory, the Floquet theory and the developing theories are the Fer expansion or the Floquet-Magnus expansion. These approaches provide solutions to the time-dependent Schrodinger equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance in particular. Methods of these expansion schemes used as numerical integrators for solving the time dependent Schrodinger equation are presented. The action of their propagator operators is also presented. We highlight potential future theoretical and numerical directions such as the time propagation calculated by Chebychev expansion of the time evolution operators and an interesting transformation called the Cayley method.
Highlights
The Schrodinger equation is the fundamental equation of physics for describing quantum mechanical behavior
The overall goals of this review article is to support theories in nuclear magnetic resonance (NMR) in order to continue to a) apply the average Hamiltonian theory to problems including: a class of symmetrical radio-frequency pulse sequences in the NMR of rotating solids, the symmetry principles in the design of NMR multiple-pulse sequences, the composite pulses, and the problems still unsolved such as the AHT for 3 spins [58]-[68]; b) use the Floquet theory in the study of several magic-angle spinning (MAS) NMR experiments on spin systems with a periodically time-dependent Hamiltonian such as the multiple-multimode Floquet-theory in NMR [69]; c) enhance the performance of the Floquet-Magnus expansion by considering fundamental questions that arise when dealing with this approach [34]
We have thoroughly reviewed the abiding applications of average Hamiltonian theory, Floquet theory, and Floquet-Magnus expansion from very different perpectives in spin quantum physics of nuclear magnetic resonance
Summary
The Schrodinger equation is the fundamental equation of physics for describing quantum mechanical behavior. For instance in field such as nuclear magnetic resonance (NMR), much effort still needs to be done to explore several problems using the time-dependent Schrodinger equation These problems include but are not limited to medical imaging, crystallography, ultra short strong laser pulses, biological systems, chemical structures and composition, spin dynamics of superconductors and semiconductors [7]-[26]. The overall goals of this review article is to support theories in NMR in order to continue to a) apply the average Hamiltonian theory to problems including (but not limited to): a class of symmetrical radio-frequency pulse sequences in the NMR of rotating solids, the symmetry principles in the design of NMR multiple-pulse sequences, the composite pulses, and the problems still unsolved such as the AHT for 3 spins [58]-[68]; b) use the Floquet theory in the study of several magic-angle spinning (MAS) NMR experiments on spin systems with a periodically time-dependent Hamiltonian such as the multiple-multimode Floquet-theory in NMR [69]; c) enhance the performance of the Floquet-Magnus expansion by considering fundamental questions that arise when dealing with this approach [34]. It is noteworthy that unifying or combining two and more theories known in NMR will continue to provide a framework for treating time-dependent Hamiltonian in quantum physics and NMR in a more efficient way that can be extended to all types of modulations
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