In this chapter, we derived the canonical commutation relation between the position and momentum operators, but we had previously constructed the derivative operator \(\mathbb{D}\) on a grid of positions in Chap. 2. In this problem, we will study the commutation relations of momentum and position on this grid and see if what we find is compatible with the continuous commutation relations.

(a) First, from the discrete derivative matrix \(\mathbb{D}\) in Eq. (2.10), construct the Hermitian momentum matrix on the grid, \(\mathbb{P}\). In this problem, we'll just explicitly consider \(3 \times 3\) matrices.

(b) Now, in this same basis, construct the \(3 \times 3\) Hermitian matrix that represents the position operator on this grid; call it \(\mathbb{X}\).

(c) What is the commutator of \(\mathbb{X}\) and \(\mathbb{P},[\mathbb{X}, \mathbb{P}]\) ? Is it what you would expect from the canonical commutation relation?

(d) Now, imagine taking \(\Delta x\) smaller and smaller, for the same interval in position. That is, consider larger and larger-dimensional momentum and position matrices. What do you expect the general form of the commutator \([\mathbb{X}, \mathbb{P}]\) is when \(\mathbb{X}\) and \(\mathbb{P}\) are \(N \times N\) matrices? Does this have a sensible limit as \(\Delta x \rightarrow 0\), or \(N \rightarrow \infty\) ?