This course seeks to outline the mathematical underpinnings of the present theory of the microscopic world. Beginning with a brief review of the dilemmas faced by physics at the turn of the 20th century, we will examine solutions to the Schrödinger equation and their interpretations as wave functions of probability. Various model systems will be studied: bound states of the "particle-in-a-box," the harmonic oscillator, the hydrogen atom, as well as unbound examples of quantum mechanical tunneling and reflection of free electrons. If time permits, we will examine the description and consequences of the electron's "spin" on the structure and stability of matter. A final project will involve creating a spreadsheet/computer program to solve quantum mechanical problems numerically.