Quantum theory of spin waves in finite samples

We present the formalism for the quantum theory of spin waves in finite samples of arbitrary shape. The sample shape is assumed such that the magnetization per unit volume and the internal demagnetizing field are constant in direction and magnitude everywhere, though this restriction may be lifted. We proceed within the framework of continuum theory, with both dipolar interactions and exchange interactions between the spins included. We derive a prescription for normalizing the spin wave eigenfunctions, and also provide a representation of the operators associated with the transverse components of the magnetization density. Completeness relations, which form the basis of expansion of arbitrary functions in terms of spin wave eigenfunctions, are derived as well. The theory may be employed to describe the interaction of spin wave quanta with external probes and other phenomena where the quantum nature of spin waves enters. We use the formalism to obtain an expression for the spatial and temperature dependence of the magnetization within a ferromagnetic nanosphere, at low temperatures where spin wave theory is applicable. We explore this issue with an explicit calculation.