Quantum Physics

A.1 MATHEMATICAL NOTATIONMany mathematical symbols are used throughout this book. You are no doubtfamiliar with some, such as the symbol to denote the equality of two quantities.The symbol denotes a proportionality. For example, y x 2 means that y isproportional to the square of x.The symbol means is less than, and means is greater than. For example,x y means x is greater than y.The symbol means is much less than, and means is much greater than.The symbol indicates that two quantities are approximately equal to each other.The symbol means is defined as. This is a stronger statement than a simple .It is convenient to use the notation x (read as “delta x”) to indicate the changein the quantity x. (Note that x does not mean “the product of and x.”) For example,suppose that a person out for a morning stroll starts measuring her distanceaway from home when she is 10 m from her doorway. She then moves along astraight-line path and stops strolling 50 m from the door. Her change in positionduring the walk is x 50 m 10 m 40 m or, in symbolic form,x x f x iIn this equation x f is the final position and x i is the initial position.We often have occasion to add several quantities. A useful abbreviation for representingsuch a sum is the Greek letter (capital sigma). Suppose we wish to adda set of five numbers represented by x 1 , x 2 , x 3 , x 4 , and x 5 . In the abbreviated notation,we would write the sum aswhere the subscript i on x represents any one of the numbers in the set. For example,if there are five masses in a system, m 1 , m 2 , m 3 , m 4 , and m 5 , the total mass ofthe system M m 1 m 2 m 3 m 4 m 5 could be expressed asFinally, the magnitude of a quantity x, written x , is simply the absolute value ofthat quantity. The sign of x is always positive, regardless of the sign of x. Forexample, if x 5, x 5; if x 8, x 8.A.2 SCIENTIFIC NOTATIONMany quantities that scientists deal with often have very large or very small values.For example, the speed of light is about 300 000 000 m/s and the ink required tomake the dot over an i in this textbook has a mass of about 0.000 000 001 kg. Obviously,it is cumbersome to read, write, and keep track of numbers such as these.We avoid this problem by using a method dealing with powers of the number 10:10 1 10x 1 x 2 x 3 x 4 x 5 5M 5m ii 110 0 110 2 10 10 10010 3 10 10 10 1 000x ii 110 4 10 10 10 10 10 00010 5 10 10 10 10 10 100 000APPENDIX AMathematical ReviewA.1