2.2 Functions as Vectors

The second mathematical idea that is crucial for quantum mechanics is that functions can be treated in a way that is fundamentally not that much different from vectors.

A vector $\vec{f}$ (which might be velocity $\vec{v}$, linear momentum ${\skew0\vec p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m\vec{v}$, force $\vec{F}$, or whatever) is usually shown in physics in the form of an arrow:

Figure 2.1: The classical picture of a vector.

However, the same vector may instead be represented as a spike diagram, by plotting the value of the components versus the component index:

Figure 2.2: Spike diagram of a vector.

(The symbol $i$ for the component index is not to be confused with ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$.)

In the same way as in two dimensions, a vector in three dimensions, or, for that matter, in thirty dimensions, can be represented by a spike diagram:

Figure 2.3: More dimensions.

For a large number of dimensions, and in particular in the limit of infinitely many dimensions, the large values of $i$ can be rescaled into a continuous coordinate, call it $x$. For example, $x$ might be defined as $i$ divided by the number of dimensions. In any case, the spike diagram becomes a function $f(x)$:

Figure 2.4: Infinite dimensions.

The spikes are usually not shown:

Figure 2.5: The classical picture of a function.

In this way, a function is just a vector in infinitely many dimensions.

Key Points

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Functions can be thought of as vectors with infinitely many components.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

This allows quantum mechanics do the same things with functions as you can do with vectors.

2.2 Review Questions

1. Graphi­cally compare the spike diagram of the 10-di­mensional vector $\vec{v}$ with components (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of the function $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.5 $x$.

   Solution funcvec-a

2. Graphi­cally compare the spike diagram of the 10-di­mensional unit vector ${\hat\imath}_3$, with components (0,0,1,0,0,0,0,0,0,0), with the plot of the function $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. (No, they do not look alike.)

   Solution funcvec-b