The mystery of square root of minus one in quantum mechanics, and its demystification

Abstract

To most physicists, quantum mechanics must embrace the imaginary number i = square root of minus one is at least a common belief if not a mystery. We use the famous example pq -qp = h/(2 pi i) to demonstrate the possible elimination of i when constructing this noncommutative relationship. We then discuss the role of i in the formulation of Schroedinger's wave equation. Common to the original development of these two quantum theories was the use of complex exponential to represent the fundamental variables (i.e., p, q, and the wave function). Understanding this complex function from the right perspective, as we suggest in this essay, removes the mysteries surrounding the complex nature of quantum mechanics.