Can someone explain the notion of "unbounded" operator as simple as possible?

Question

I've read about these operators in quantum mechanics, but I have never seen them in action. I think that is because I absolutely do not intuitively understand this concept. I've read some stuff online but they require too much background to understand. I only know the position and momentum operator, are these unbounded? How Can I prove it?

Can someone please explain as simple as possible as to what an unbounded operator is and what is the significant of this notion?

Answer 1

An unbounded (linear) operator is one that can produce arbitrarily large outputs from inputs that are of limited size.

An example is the derivative operator. Suppose we map polynomial functions $f(x)$ to their derivatives $f'(x)$. This is a linear operator, i.e. the derivative of $f(x) + g(x)$ is the sum $f'(x) + g'(x)$ of the respective derivatives and the derivative of a constant multiple $c f(x)$ is the same multiple $c f'(x)$ of the derivative.

Now various "norms" can be used to measure the size of functions. Since we are dealing with polynomials, which are continuous, let's measure the size by taking the maximum absolute value of $f(x)$ on the interval $[0,1]$:

$$ ||f|| = \max_{x \in [0,1]} |f(x)| $$

You can make a polynomial $f$ that "wiggles" between $y=+1$ and $y=-1$ a number of times, say $n$, but never gets any bigger than one in absolute value. Then $||f|| = 1$ but the norm of its derivative $||f'||$ must be at least $2n$. So by making $n$ big, you can get an arbitrarily large output (the derivative) from an input (the polynomial $f$) of limited size.

The derivative of polynomials is an unbounded operator with respect to this norm.

Answer 2

If you understand the notion of bounded functions, it's not too far a leap. For example, $x\mapsto \sin x$ is bounded. Now, replace $\Bbb R$ with a certain space of functions, and call functions that act on that space "operators" (we could still call them functions but for obvious reasons it would be confusing).

A bounded operator is one such operator whose images' "size" or "length", but more technically norm, isn't allowed to grow too much (as for how to define a concept of "length" for elements of a function space, that's for another discussion).

The only distinction to be made is that we don't ask for global constants, as is $1$ for the sine function; a different constant for different subsets of the space is good enough. An example of this in the reals would be $x\mapsto 2x$, which is not globally bounded, but in every $[x-\varepsilon,x+\varepsilon]$ it is.

There are some more particularities, such as "bounded operator" only referring to linear operators in most cases. Also, well, it should be obvious that I've presented something far from precise or rigorous. I mean this to be a sketch. I also have no idea of quantum mechanics or how this relates to it.

Oh, and an unbounded operator is just an operator that isn't necessarily bounded :)