Formulation of Resonance Conditions

A resonance condition is a condition on the variables of a system (external field, orientation, etc.) such that two energy levels differ by a fixed energy δ. This paper discusses the formulation of exact theoretical resonance conditions by methods which do not require knowledge of the energy levels themselves. An exact resonance condition for an n‐level system can be written as an algebraic equation of degree ½n(n−1) in δ². The determination of this equation from the characteristic equation for the energies is an old algebraic problem discussed by Lagrange and by Cayley, the formulation of the equation of differences. Relevant properties are reviewed, and polynomial and determinantal formulas are given. Application is made to a particle of spin $\mathrm{S\leqq }\frac{5}{2}$ , with a second‐order tensor zero‐field term, in a uniform external magnetic field. A second approach is to construct an operator $\mathcal{F}\mathrm{=\{(}{\mathcal{K}}_1{\mathbf{\times 1}}_2-{\mathbf{1}}_1\mathbf{\times }{\mathcal{K}}_2{)}^2\}$ , in a ½n(n−1)‐dimensional space of antisymmetric two‐particle states, whose eigenvalues are the squares of the differences of the eigenvalues of the Hamiltonian $\mathcal{K}$ . The resonance condition is then det $|\mathcal{F}{\mathrm{-\delta }}^2\text{I|=0}$ . A feature of this approach is that transition intensities can be calculated directly from the appropriate eigenvector of $\mathcal{F}$ , without knowledge of the individual eigenvectors of $\mathcal{K}$ . For a system of many levels the exact resonance condition may be very complicated, and knowledge of the resonance condition is by no means equivalent to knowledge of the resonance fields.