Physical Conception of Mathematical Functions and Differentials

Science consists of interrogating nature by experimental means and expressing the underlying patterns and relationships between measured properties by theoretical means. Thermodynamics is the science of heat, work, and other energy-related phenomena. 

An experiment may generally be represented by a set of stipulated control conditions, denoted x1 x2. x , that lead to a unique and reproducible experimental result, denoted z. Symbolically, the experiment may be represented as an input-output relationship, 

Mathematically, such relationships between independent (x1 x2, x ) and dependent (z) variables are represented by Junctions 

We first wish to review some general mathematical aspects of functional relationships, prior to their specific application to experimental thermodynamic phenomena. 

Two important aspects of any experimentally based functional relationship are (1) its differential dz, i.e., the smallest sensible increment of change that can arise from corresponding differential changes (dx, dx2. dxn) in the independent variables and (2) its degrees of freedom n, i.e., the number of control variables needed to determine z uniquely. How small is the magnitude of dz (or any of the dxf) is related to specifics of the experimental protocol, particularly the inherent experimental uncertainty that accompanies each variable in question. 

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Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

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