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02 April 2004

Secrets of a Theorist, Part 1: How to Understand an Equation

George E. Hrabovsky, President, MAST

Note: George asked me to convey his regrets at not getting a Mathematics Corner or Mind of a Theorist Column in this week, however he promises to make up for it next week, as only he can. -ed.

 

Introduction

I have decided to write a new series of features, as next year marks the centennial of what has become known as modern physics. 

There are many definitions of modern physics. Some consider it to be the development of relativity, while others consider it to be the development of quantum mechanics.  I consider it to be the realization that it is impossible to separate the observer from what is being observed. In any case, what I want to present is a series of short essays and tutorials about how theoretical physics is done and something about the origins of what is called modern physics.

How To Understand the Mathematics Used in Physics

A common complaint is that people understand the ideas of physics without being able to use the mathematics. This is a little bit like saying that you can understand a song without knowing the words. This seems to stem from frustration with learning the math at some point in our schooling. The ability to apply what mathematics we have learned is not practiced at the time we learn it.

The question becomes, "Must we accept the fact that we do not understand the equations that we encounter?" The answer is no. As a scientist, your job is to study and do science, not to invent new mathematics. Every mathematical tool you gain will help you, since the language of science is mathematics. If you learn no other mathematical tool, you must gain the tool of being able to look at an equation and have it reveal to you what it means.

The first point of confusion is the equals sign itself. In most equations you have a left side and a right side with the equals sign () in the middle. What does this mean? It means that the numerical value of the left and right sides are the same, and that is all it means. It does not define the left hand side with the right hand sign. That requires the symbol or .

I assume that you can perform basic arithmetic and some algebra. The real secret to gaining an understanding of how to interpret equations is to plot a graph of the equation. Find out what symbols are really constants, and replace each constant with a 1. We can put back the true values later. Here is an example of a complicated equation from quantum mechanics called the particle-in-a-box energy solution of the Schrödinger equation,

The first thing we need to do is define the symbols and decide if they are variables or constants.   is a positive integer greater than 0 and is determined when we choose the problem. Thus, it is a variable.  is called Planck's constant and is a constant.  is the symbol for mass and is a constant.  is the length where the wave function becomes zero. This is unique to a given system and is, thus, a constant.   is the energy for a given value of and is, thus, a variable.

Now we set all of the constants to 1,

Now we plot this equation.

While it is tempting to draw a smooth curve connecting the dots, we can't because can only have integer values. If we insert the values of the constants, the plot looks like this,

The shape is the same, only the numbers are different. This is very important. Equations with similar relationships between their variables all have the same shape and, thus, the same behavior.

In this way a theoretical physicist gains deep insights into a variety of problems by being able to recognize that similar equations have similar graphs, and similar graphs have similar equations.

Created by Mathematica  (March 26, 2004)