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.
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\centering
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\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}{}
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...0)[b]{2}}
\put(43,1.2){\makebox(0,0)[b]{$i$}}
\end{picture}
\end{figure}

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

And just like vectors can be interpreted as spike diagrams, spike diagrams can be interpreted as vectors. So a spike diagram with very many spikes can be considered to be a single vector in a space with a very high number of dimensions.

In the limit of infinitely many spikes, 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 $f$ of a continuous coordinate $x$:

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}

For functions, 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 single vector in an infinite dimensional space.

Note that the $(x)$ in $f(x)$ does not mean “multiply by $x$.” Here the $(x)$ is only a way of reminding you that $f$ is not a simple number but a function. Just like the arrow in $\vec{f}$ is only a way of reminding you that that $f$ is not a simple number but a vector.

(It should be noted that to make the transition to infinite dimensions mathematically meaningful, you need to impose some smoothness constraints on the function. Typically, it is required that the function is continuous, or at least integrable in some sense. These details are not important for this book.)


Key Points
$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
A function can be thought of as a vector 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