Mathematical Tools for Physics

7—Operators and Matrices 173

7.3 Examples of Operators
The four cases that I started with, rotation in the plane, angular momentum of a rotating rigid body, polarization
of a crystal by an electric field, and the mass attached to some springs all fit this definition. Other examples:

5. The simplest example of all is just multiplication by a scalar:A(~v)≡c~vfor all~v. This applies to any vector
   space and its domain is the entire space.


6. On the vector space of all real valued functions on a given interval, multiply any functionf by1 +x^2 :
   (Af)(x) = (1 +x^2 )f(x). The domain of Ais the entire space of functions of x. This is an infinite
   dimensional vector space, but no matter. There’s nothing special about1 +x^2 , and any other function will
   do.


7. On the vector space of square integrable functions

[∫

dx|f(x)|^2 <∞

]

ona < x < b, define the operator

as multiplication byx. The only distinction to make here is that if the interval is infinite, thenxf(x)may

not itself be square integrable. The domain of this operator is thereforenotthe entire space, but only those

functions such thatxf(x)is also square-integrable. On the same vector space, differentiation is a linear

operator:(Af)(x) =f′(x). This too has a restriction on the domain: It is necessary thatf′also exist and

be square integrable.

8. On the vector space of infinitely differentiable functions, the operation of differentiation,d/dx, is itself a
   linear operator. It’s certainly linear, and it takes a differentiable function into a differentiable function.
   So where are the matrices? I started this chapter by saying that I’m going to show you the inside scoop on
   matrices and so far I’ve failed to produce even one.
   When you describe vectors you can use a basis as a computational tool and manipulate the vectors using
   their components. In the common case of three-dimensional vectors we usually denote the basis in one of several
   ways

ˆı, ˆ, ˆk, or ˆx, y,ˆ z,ˆ or ~e 1 , ~e 2 , ~e 3

and they all mean the same thing. The first form is what you see in the introductory physics texts. The second
form is one that you encounter in more advanced books, and the third one is more suitable when you want to
have a compact index notation. It’s that third one that I’ll use here; it has the advantage that it doesn’t bias you
to believe that you must be working in three spatial dimensions. The index could go beyond 3, and the vectors
that you’re dealing with may not be the usual geometric arrows. (And why does it have to start with one? Maybe
I want 0, 1, 2 instead.) These don’t have to be either perpendicular to each other or to be unit vectors.