The Quantum Mechanics Solver 21 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 20 Magnetic Excitons Quantum field theory deals with systems possessing a large number of degrees of freedom. This chapter presents a simple model where we study the magnetic excitations of a long chain of coupled spins. We show that one can associate the excited states of the system with quasi-particles that propagate along the chain. We recall that for any integer k N e2lnkn N N if k pN with p integer n 1 0 otherwise. The Molecule CsFeBr3 Consider a system with angular momentum equal to 1 . j 1 in the basis j m common to J2 and Jz. . What are the eigenvalues of J2 and Jz . For simplicity we shall write j m a where a m 1 0 1. Write the action of the operators J Jx iJy on the states a . . In the molecule CsFeBr3 the ion Fe2 has an intrinsic angular momentum or spin equal to 1. We write the corresponding observable J and we note a the eigenstates of Jz. The molecule has a plane of symmetry and the magnetic interaction Hamiltonian of the ion Fe2 with the rest of the molecule is Hr DJ D 0 . r h2 z What are the eigenstates of H and the corresponding energy values Are there degeneracies 204 20 Magnetic Excitons Spin-Spin Interactions in a Chain of Molecules We consider a one-dimensional closed chain made up with an even number N of Cs Fe Br3 molecules. We are only interested in the magnetic energy states of the chain due to the magnetic interactions of the N Fe2 ions each with spin 1. We take ct1 ct2 n W 1 0 _1 to be the orthonormal basis of the states of the system it is an eigenbasis of the operators J where J is the spin operator of the n-th ion n 1 N . The magnetic Hamiltonian of the system is the sum of two terms H Ho Hi where u - D n 2 Ho -2 v Jz h ----------1 n 1 has been introduced in and H1 is a nearest-neighbor spin-spin interaction term N Hi A Jn Jn 1 A 0 . n 1 To simplify the notation of H1 we define J J . We assume that H1 is a small perturbation compared to H0 A C D and we shall treat it in first order perturbation .
