In Quantum Mechanics, the translation operator $\hat{T}$ can be written as
$$\hat{T}(\boldsymbol{x}) = 1 - \dfrac{ix\cdot \hat{p}}{\hbar} - \dfrac{i(x\cdot \hat{p})^2}{2\hbar^2} - \dfrac{i(x\cdot \hat{p})^3}{6\hbar^3} + \ldots$$ with $$\hat{p} = -i\hbar \nabla $$ This question isn't really about the translation operator itself. I just wanted to mention it as an example. Something that is really bothering me in that Taylor expansion are the expressions $$\hat{p}^n = (-i\hbar)^n \nabla^n$$ More specifically I wanted to ask what $$\nabla^n $$ means? From what I know, $\nabla f = \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix}$ for a scalar function $f$. This makes sense. But then what is $\nabla^2 f$ supposed to be? From the kinetic energy operator I know that $\nabla^2 = \Delta$ should be the Laplacian-Operator. But this isn't how to product of two operator is defined. By the definition I should apply the nabla operator to $\nabla f = \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix}$ again: $$\nabla \nabla f = \nabla \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix} = \begin{pmatrix}\partial_x \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix} \\\partial_y \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix} \\ \partial_z \begin{pmatrix}\partial_x f \\\partial_y f \\ \partial_z f\end{pmatrix} \end{pmatrix}$$ which could probably be interpreted as the Jacobian matrix.
