Quantum Theory of Spin Waves

Abstract

When the spacing between spins is small enough that the wave functions of the particles overlap, spins are coupled to each other through a quantum mechanical mechanism called the exchange interaction. In the simplest treatment of this interaction, spins are coupled only to their nearest neighbors, giving rise to what is known as the Heisenberg Hamiltonian. The ground and excited states of molecular hydrogen are explored in Problem 2.1 as a way of gaining some insight into the origin of the exchange interaction. The quantum mechanical harmonic oscillator and its associated raising and lowering operators in the occupation number (or second quantization) formalism are important to many phenomena in physics, including spin waves in particular. The harmonic oscillator is considered in Problems 2.2 and 2.3. Spin raising and lowering operators discussed in Problem 2.4, and spin wave excitations on a linear chain are considered in Problems 2.5 and 2.6.