Usually, introductory quantum mechanics starts with Schrodinger’s equation and using partial differential equations as the mathematical tools. For example, the hydrogen atom problem is resolved with spherical harmonics and Laguerre polynomials. A possible shortcoming with this approach is that the readers become submerged in pages and pages of mathematical formulas and lose the conceptual understanding of the physics. Historically, before Erwin Schrodinger discovered the partial differential equation format, Heisenberg and Pauli developed the algebraic approach of quantum mechanics, and resolved several basic problems in quantum mechanics, including harmonic oscillator, angular momentum, and the hydrogen atom. From a pedagogic point of view, the succinct notation of the algebraic approach, especially the Dirac notation, could be conceptually more directly related to the underlying physics. From a practical point of view, to handle the problems with the utilization of solar energy, analytic approach for partial differential equations is not useful because numerical calculations and perturbation methods are the norm. Furthermore, the more advanced methods of quantum mechanics, such as quantum electrodynamics, rely on the algebraic method rather than the partial differential equation method.
This Appendix is a brief summary of the algebraic approach to quantum mechanics, exemplified by the problems of the harmonic oscillator, angular momentum, and hydrogen atom. For clarity, we use the Dirac notation, and adding a hat on an operator to distinguish it from a (in general complex) number.