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 $\vec p = m\vec v$ , 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.
$\begin{figure} \centering % \htmlimage{extrascale=3,notransparent}{} \se... ...0,0){$f_y$}} \put(-33,26){\makebox(0,0){$m$}} \end{picture} \end{figure}$

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.
$\begin{figure} \centering % \htmlimage{extrascale=3,notransparent}{} \se... ...0)[b]{2}} \put(43,1.2){\makebox(0,0)[b]{$i$}} \end{picture} \end{figure}$

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

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.
$\begin{figure} \centering % \htmlimage{extrascale=3,notransparent}{} \se... ...[b]{30}} \put(143,1.2){\makebox(0,0)[b]{$i$}} \end{picture} \end{figure}$

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

**Figure 2.4:** Infinite dimensions.
$\begin{figure} \centering % \htmlimage{extrascale=3,notransparent}{} \se... ...r]{$f(x)$}} \put(43,2){\makebox(0,0)[b]{$x$}} \end{picture} \end{figure}$

The spikes are usually not shown:

**Figure 2.5:** The classical picture of a function.
$\begin{figure} \centering % \htmlimage{extrascale=3,notransparent}{} \se... ...r]{$f(x)$}} \put(43,2){\makebox(0,0)[b]{$x$}} \end{picture} \end{figure}$

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

Key Points

$\begin{picture}(13,5)(0,-3) \put(2,0){\makebox(0,0){$\circ$}}\put(4,0){\line(1,0... ...line(0,-1){2}}\put(11,-2){\line(1,0){2}}\put(13,0){\line(0,-1){2}} \end{picture}$

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

$\begin{picture}(13,5)(0,-3) \put(2,0){\makebox(0,0){$\circ$}}\put(4,0){\line(1,0... ...line(0,-1){2}}\put(11,-2){\line(1,0){2}}\put(13,0){\line(0,-1){2}} \end{picture}$

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 .
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 . (No, they do not look alike.)
Solution funcvec-b