This Quantum World/Appendix/Mathematical tools

Mathematical tools

Elements of calculus

A definite integral

Imagine an object ${\mathcal {O}}$ that is free to move in one dimension — say, along the $x$ 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 $\rho (x).$ This depends on actual measurement outcomes, and it allows us to calculate the probability of finding the particle in any given interval of the $x$ 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 $\rho (x)$ a probability density because it represents a probability per unit length. The probability of finding ${\mathcal {O}}$ in the interval between $x_{1}$ and $x_{2}$ is given by the area $A$ between the graph of $\rho (x),$ the $x$ axis, and the vertical lines at $x_{1}$ and $x_{2},$ respectively. How do we calculate this area? The trick is to cover it with narrow rectangles of width $\Delta x.$

The area of the first rectangle from the left is $\rho (x_{1}+\Delta x)\,\Delta x,$ the area of the second is $\rho (x_{1}+2\,\Delta x)\,\Delta x,$ and the area of the last is $\rho (x_{1}+12\,\Delta x)\,\Delta x.$ For the sum of these areas we have the shorthand notation

$\sum _{k=1}^{12}\rho (x+k\,\Delta x)\,\Delta x.$

It is not hard to visualize that if we increase the number $N$ of rectangles and at the same time decrease the width $\Delta x$ of each rectangle, then the sum of the areas of all rectangles fitting under the graph of $\rho (x)$ between $x_{1}$ and $x_{2}$ gives us a better and better approximation to the area $A$ and thus to the probability of finding ${\mathcal {O}}$ in the interval between $x_{1}$ and $x_{2}.$ As $\Delta x$ tends toward 0 and $N$ tends toward infinity ($\infty$), the above sum tends toward the integral

$\int _{x_{1}}^{x_{2}}\rho (x)\,dx.$

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 $d$ indicating that $dx$ 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 $\textstyle \int$ with the infinitesimal quantity $dx$ that makes sense as a limit, in which $N$ grows above any number however large, $dx$ (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.