4. The Principles of Quantum Mechanics

Since quantum computing is based on concepts and techniques developed in quantum mechanics, some understanding of the latter is necessary, starting with a history of its development. At the microscopic level, we discuss Louis de Broglie’s proposal that waves behave like particles and particles behave like waves. We then discuss Heisenberg’s uncertainty principle—using a geometrical approach—and explore the Schrödinger equation that underpins the entire framework of quantum mechanics, obtaining several results. Schrödinger’s equation is difficult to interpret, however. A reasonable interpretation of quantum mechanics in probabilistic terms was provided by Max Born. We progress to formulation in quantum mechanics, basing our approach on the axiomatic point of view proposed by Heisenberg and Dirac. To provide a framework for quantum mechanics we list seven postulates, which we define, in turn, in terms of operators. quantum mechanics works with complex numbers, vectors and other entities that are defined in what is called the Hilbert space. We do not explicitly define this space, but introduce the bracket notation of Paul Dirac and describe certain properties—such as the inner product—as groundwork for dealing with concepts in the chapters ahead.