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 (which might be velocity , linear momentum , force , or whatever) is usually shown in physics in the form of an arrow:
However, the same vector may instead be represented as a spike diagram, by plotting the value of the components versus the component index:
(The symbol for the component index is not to be confused with .)
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:
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 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 of a continuous coordinate :
For functions, the spikes are usually not shown:
In this way, a function is just a single vector in an infinite dimensional space.
Note that the in does not mean “multiply by .” Here the is only a way of reminding you that is not a simple number but a function. Just like the arrow in is only a way of reminding you that that 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.)
- A function can be thought of as a vector with infinitely many components.
- This allows quantum mechanics do the same things with functions as you can do with vectors.
Graphically compare the spike diagram of the 10-dimensional vector with components (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of the function 0.5 .
Graphically compare the spike diagram of the 10-dimensional unit vector , with components (0,0,1,0,0,0,0,0,0,0), with the plot of the function 1. (No, they do not look alike.)