Quantumania 3 - Linear Operators, A Digression

I have previously covered some basics of vectors and tensors without touching on linear operators, which is a big thing in QM.

An operator on vectors is a function that maps an input vector to a resultant vector. A linear operator is one that you can do inside (before) or outside (after) addition and scaling, and get the same result either way:

\[\hat{O}(\alpha \vec{v} + \beta \vec{w}) = \alpha \hat{O}(\vec{v}) + \beta \hat{O}(\vec{w})\]

See how (on the left) we can operate with $\hat{O}$ on the result of scaling (by the scalars $\alpha$ and $\beta$) and adding two input vectors $\vec{v}$ and $\vec{w}$, or (on the right) we can operate on the raw input vectors first, and then scale and add, and we get the same resultant vector either way. If this is true then $\hat{O}$ is linear. And this rule is recursive, in that we can break $\vec{w}$ down into two more vectors that we scale and sum to make $\vec{w}$ and so $\hat{O}$ can descend down through that new layer, and so on for as many layers of unfolding that we like. In other words, if the above rule works for scaling and summing two component vectors then it works over any number of components.

Such operators have the simple and extremely useful property that if we choose a basis:

\[\{ \vec{e_1}, \vec{e_2} \dots \vec{e_n} \}\]

such that any vector $\vec{v}$ in the space can be expressed as a scaled sum of our chosen basis vectors:

\[\vec{v} = \sum_{n} v_n \vec{e}_n\]

Then a linear operator such as $\hat{O}$ acting on $\vec{v}$ can “move inside” the scaling and summing:

\[\hat{O}(\vec{v}) = \hat{O}\left(\sum_{n} v_n \vec{e}_n\right) = \sum_{n} v_n \hat{O}(\vec{e}_n)\]

So expressed in this form, the operator seems to only ever act on the basis vectors, and therefore we can fully describe the behaviour of $\hat{O}$ by saying what effect it has on each basis vector. In a given basis, the basis vectors have the “one hot” coordinates of the standard basis, so for example in three dimensions:

\[\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \,\,\, \vec{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix} \,\,\, \vec{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix}\]

We can fully describe a linear operator that maps from a vector to a vector by giving the coordinates of the result of it acting on each of the above column vectors. An operator that makes no difference and leaves every basis vector unchanged, the identity operator, is just:

\[\hat{I} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\]

Those $N \times N$ numbers are like a control panel of little adjustable dials, allowing to make every possible linear operator, and they fully described what effect the operator has on any vector. Given some choice of basis, every matrix is a unique operator, and every operator has a unique matrix: they are isomorphic. We can read the matrix in two equivalent ways:

1. The $n$th vertical column tells us the final state of the $n$th basis vector after the operator is applied to it.
2. The $n$th row is a recipe for making the $n$th coordinate of an output vector by scaling the coordinates of an input vector.

The geometrical intuitive picture of what linear operators can and cannot do is equally important. They can reflect, rotate, scale, and shear. That’s it. But excitingly, they are composable: the result of applying two linear operators in succession is itself a linear operator, meaning that you can “flatten” any sequence of operations into a single matrix, though in general the order they are applied matters (this going to be a massively important point in QM.)