ABSTRACT
We assume that some readers have not had previous exposure to quantum mechanics, other than the cursory introductory undergraduate level discussion. As such, we develop the subject from scratch, and we follow, at least for the first few sections of the chapter, a typical introductory course. We give a quick overview of the problems that forced physicists in the early part of the last century to propose quantum mechanics, and we formally present its fundamental postulates. However, the rest of the chapter takes a different course compared to the typical introductory treatment. The harmonic oscillator is introduced, but not from the analytical point of view, rather, the machinery of matrix quantum mechanics is unveiled and brought to bear on the problem. The rest of the chapter develops three powerful deterministic tools to solve complex problems in quantum mechanics, the Discrete variable representation (DVR), the Lanczos algorithm for the tridiagonalization of sparse Hamiltonians, and the theory of angular momentum. In the typical introductory curriculum of quantum mechanics, the harmonic oscillator is solved by the method of power series. In advanced courses, the same problem is solved with creation and annihilation operators, making a later transition to quantum field theory easier for the advanced student of quantum mechanics. Our approach is somewhere in between. It is important to realize that the power series solution is no different from other approaches, in that a vector space is used to transform the differential equation into an algebraic problem. The power series method is important because, for the harmonic oscillator, the finiteness of the solution at both asymptotes is the condition that ultimately yields a quantized energy, and this is mathematically more transparent in the power series approach.
