The study of any discipline requires some grounding in fundamentals. Without this common experience, there is little hope of communicating any complex concepts. For example, in order to make use of a textbook, one must be comfortable with reading. In a mathematically intensive discipline such as physical chemistry, ones comfort level must extend to following discussions that incorporate mathematics and mathematical equations and relationships. As an example, consider the proof of conservation of energy as a means to frame a discussion of this concept. 1.1: Some Newtonian Physics Consider the definition of acceleration (a) as the first time-derivative of velocity (v) and the second time-derivative of position (x). 1.2: Some Vectors and Dot Products The concepts of linear combinations and orthogonality show up repeatedly in quantum chemistry. But these are generally not new concepts to students at this level, as the same concepts are used to describe forces and motions in a standard physics course in classical mechanics. 1.3: Classical Description of a Wave on a String The mathematics used in solving quantum mechanical problems follow be the same basic process for each of the different problems we will examine. In this section, those mathematics will be developed in order to describe a (hopefully) familiar problem in classical physics. 1.4: Failures of Classical Physics Imagine being a scientist in the year 1900. At the time, there was significant debate in society as to whether or not science was a valuable discipline for study. 1.5: On Superposition and the Weirdness of Quantum Mechanics In order to better appreciate the fascinating (and sometimes shocking!) results of the quantum world, let’s consider some measurable properties of electrons. Consider in particular two specific properties they exhibit. 1.6: References 1.7: Vocabulary and Concepts 1.8: Problems Thumbnail: The Photoelectric effect require quantum mechanics to describe accurately (CC BY-SA-NC 3.0; anonymous via LibreTexts).