Understanding the Schrodinger equation as a kinematic statement: A probability-first approach to quantum

Quantum technology is seeing a remarkable explosion in interest due to a wave of successful commercial technology. As a wider array of engineers and scientists are needed, it is time we rethink quantum educational paradigms. Current approaches often start from classical physics, linear algebra, or differential equations. This chapter advocates for beginning with probability theory. In the approach outlined in this chapter, there is less in the way of explicit axioms of quantum mechanics. Instead the historically problematic measurement axiom is inherited from probability theory where many philosophical debates remain. Although not a typical route in introductory material, this route is nonetheless a standard vantage on quantum mechanics. This chapter outlines an elementary route to arrive at the Schr\"odinger equation by considering allowable transformations of quantum probability functions (density matrices). The central tenet of this chapter is that probability theory provides the best conceptual and mathematical foundations for introducing the quantum sciences.