Quantum Theory in a Nutshell

Abstract

In principle, the two-state quantum system can be discussed in an elegant and short manner by employing various subtle operator techniques. We do not follow this route. Instead, we go one by one through all essential steps. We first establish the spin matrices for spin one half. It turns out that the most general 2×2 Hamiltonian matrix can be expressed in terms of the Pauli spin matrices. Therefore, every two-state quantum system, whatever the underlying interaction, can be treated as an $s = \frac{1}{2}$ effective-spin system. We derive the expectation values of energy- and spin-operators and their uncertainties, and we solve the two-state Schrödinger equation for two simple cases.