Bridging a quantum-mechanical barrier [engineering education]

Most engineers need a basic understanding of quantum mechanics, and for this the typical college introduction is enough. Such an introduction tends to be axiomatic but physically unenlightening. The rules of quantum mechanics (QM) are a fait accompli; justified because they work. This is a good beginning, but those needing to learn more face a special quantum barrier: more advanced texts often continue in the axiomatic tradition. Infinite-dimensional spaces, amplitude vectors, matrices, and operators in lieu of momentum and energy—these and other concepts may be introduced with little rationalization or intuition. To an inquiring student, QM can quickly become bizarre; something to be manipulated but not understood. This need not be so. Here we present one teaching solution. We show that many of QM’s key ideas can be reasoned out from just a few physical assumptions with mathematics that is low in dimension, linear, and accessible to many. Some results include a simple reason for believing probability amplitudes are more natural to use than probabilities, the duality of functions as vectors and vice versa, an example of how a matrix, an operator, and a physical value might be related, the Heisenberg uncertainty principle, and Schrodinger’s equation.