Abstract:

Physical states are represented by kets in complex vector space, while physical observables are represented by Hermitian operators. The eigenkets of these operators span the space of state kets. The probabilistic interpretation then allows these abstract descriptions to be connected with the outcomes of physical measurements. Last, the Schrödinger equation determines the time-evolution of a ket in such a way that preserves probability. All these constitute the basic ideas that any introductory course in quantum mechanics would aim to deliver. Traditionally, instructors would begin by presenting these fundamental postulates of quantum mechanics and then gradually show how they enable one to have a coherent understanding of different quantum phenomena. To most instructors this may seem to be the most satisfying way of teaching the subject. However, as recent literature has indicated, most students encounter difficulties while learning quantum mechanics. Given that we do not experience quantum phenomena directly in our macroscopic world, it is no wonder that instructors would find it very challenging to accomplish the latter. We believe the other main problem faced by many students is the mathematics that is required. This is the problem that we will address in this work. It is usual to present the necessary mathematics together with the postulates, like in many good textbooks. Our experience with teaching introductory quantum mechanics has suggested that it may be more appropriate to introduce the necessary mathematics separately, like in introductory classical mechanics and electromagnetism textbooks. In this work, we present a coherent way of introducing all the important mathematical ideas required in order to understand the significance of the mathematical formulations of the fundamental postulates. We will show how this can be achieved by exploring the relevant details of rotations in Euclidean-three space – something most students would find familiar. Our students have used the materials that we have developed, and their feedback has been encouraging.

Keywords:

Introductory Quantum Mechanics, Course Development.