This Quantum World/Appendix/Mathematical tools

Mathematical toolsEdit

Elements of calculusEdit

A definite integralEdit

Imagine an object   that is free to move in one dimension — say, along the   axis. Like every physical object, it has a more or less fuzzy position (relative to whatever reference object we choose). For the purpose of describing its fuzzy position, quantum mechanics provides us with a probability density   This depends on actual measurement outcomes, and it allows us to calculate the probability of finding the particle in any given interval of the   axis, provided that an appropriate measurement is made. (Remember our mantra: the mathematical formalism of quantum mechanics serves to assign probabilities to possible measurement outcomes on the basis of actual outcomes.)


 


We call   a probability density because it represents a probability per unit length. The probability of finding   in the interval between   and   is given by the area   between the graph of   the   axis, and the vertical lines at   and   respectively. How do we calculate this area? The trick is to cover it with narrow rectangles of width  

 


The area of the first rectangle from the left is   the area of the second is   and the area of the last is   For the sum of these areas we have the shorthand notation

 

It is not hard to visualize that if we increase the number   of rectangles and at the same time decrease the width   of each rectangle, then the sum of the areas of all rectangles fitting under the graph of   between   and   gives us a better and better approximation to the area   and thus to the probability of finding   in the interval between   and   As   tends toward 0 and   tends toward infinity ( ), the above sum tends toward the integral

 

We sometimes call this a definite integral to emphasize that it's just a number. (As you can guess, there are also indefinite integrals, which you will learn more about later.) The uppercase delta has turned into a   indicating that   is an infinitely small (or infinitesimal) width, and the summation symbol (the uppercase sigma) has turned into an elongated S indicating that we are adding infinitely many infinitesimal areas.

Don't let the term "infinitesimal" scare you. An infinitesimal quantity means nothing by itself. It is the combination of the integration symbol   with the infinitesimal quantity   that makes sense as a limit, in which   grows above any number however large,   (and hence the area of each rectangle) shrinks below any (positive) number however small, while the sum of the areas tends toward a well-defined, finite number.