1.2 Functions as Vectors

The second mathematical idea that is critical 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 1.1: 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 1.2: Spike diagram of a vector.

(The symbol $i$ 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 1.3: 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 1.4: Infinite dimensions.

The spikes are usually not shown:

Figure 1.5: The classical picture of a function.

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

Key Points

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

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

1.2 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 $f(x)=0.5 x$. Answer

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 $f(x)=1$. (No, they do not look alike.) Answer