Derive (in Cartesian coordinates) the quantum mechanical operators for the three components of angular momentum starting from the classical definition of angular momentum, \(l=r \times p\). Show that any two of the components do not mutually commute, and find their commutator.

In quantum mechanics, quantum mechanical operators play a pivotal role. They are mathematical objects associated with observable physical quantities such as position, momentum, and in our case, angular momentum. Unlike classical physics, where these quantities are simply numerical values, in quantum mechanics they are represented by operators that act on wave functions (the state of a quantum system). For example, the position operator in Cartesian coordinates is denoted by components \( \hat{x}, \hat{y}, \hat{z} \) and the momentum operator by \( -i\hbar\frac{\partial}{\partial x}, -i\hbar\frac{\partial}{\partial y}, -i\hbar\frac{\partial}{\partial z} \), with \(-i\) being the imaginary unit and \(\hbar\) the reduced Planck's constant.

The operators have to adhere to specific rules and principles. One of these principles is the commutation relation, which will tell us if two physical observables can be measured precisely at the same time. The use of these operators allows us to calculate and predict outcomes of quantum systems, making them fundamental to understanding quantum theory.

The commutation of angular momentum components is a crucial concept in quantum mechanics that refers to the inability of certain pairs of angular momentum components to be measured simultaneously with arbitrary precision. This is related to the Heisenberg uncertainty principle. For example, to check if the \(x\)- and \(y\)-components of angular momentum commute, we calculate the commutator \( [\hat{L}_x, \hat{L}_y] \).

A resulting commutator that equals zero indicates that the operators commute and thus can be measured exactly at the same time. However, for angular momentum components, as shown in the exercise's final result \( [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z \), we find that they do not commute. The non-zero commutator implies an intrinsic quantum mechanical property called 'quantum uncertainty' and forces a limit on the precision of measuring two angular momentum components simultaneously. This commutation property is one of the distinctive aspects that separates quantum mechanics from classical physics.

The use of Cartesian coordinates in quantum mechanics is very similar to their use in classical mechanics – it is a way to describe the position of points in space using three axes (\(x, y, z\)). In quantum mechanics, the position of a particle is described by the position operator with respect to these axes. When dealing with the angular momentum, expressed as \( \textbf{l} = \textbf{r} \times \textbf{p} \) in classical mechanics, we adopt these Cartesian components to define the quantum angular momentum operators.

However, it's important to note that position and momentum are no longer mere values but operators acting on wave functions. This change from classical to quantum description requires a new framework of mathematics, including operator algebra and eigenvalue equations, which are vital to performing physical and measurable predictions in quantum systems.

In the expression for momentum operators, the appearance of derivatives in quantum operator expressions is particularly noteworthy. These derivatives act on the wave function (a function describing the quantum state of a system) and help determine how this state changes with respect to space. In our angular momentum operator example, partial derivatives indicate how the wave function's spatial variation contributes to the overall angular momentum.

Specifically, for the angular momentum operator components \( \hat{L}_x, \hat{L}_y, \hat{L}_z \), derivatives with respect to the coordinates are taken, highlighting the orientation of angular momentum in space. Understanding these derivative operations is fundamental for analyzing quantum systems, especially when applying operators to wave functions in order to extract physical information like probability densities and expectation values.