(a) Repeat the previous problem but instead use only the four properties of the Poisson bracket and the facts that \(\left\{x, p^{x}\right\}=\left\{y, p^{y}\right\}=\left\{z, p^{z}\right\}=1\) while the other brackets of positions and momenta vanish.

(b) Write the quantum version of this algebra using the quantization scheme.

Data from previous problem

The angular momentum vector of a particle of mass m is written as \(\mathbf{L}=\mathbf{r} \times \mathbf{p}\). Find the Poisson brackets of any two components of the angular momentum vector in Cartesian coordinates. Do this using the explicit representation of the Poisson bracket as derivatives with respect to canonical coordinates and momenta. Show that the result is given by \(\left\{L^{x}, L^{y}\right\}=L^{z},\left\{L^{y}, L^{z}\right\}=L^{x}\), and \(\left\{L^{z}, L^{x}\right\}=L^{y}\) (i.e., cyclic permutations of \((x y z)\). This is known as the angular momentum algebra. You might find it useful to write \(L^{i}=\varepsilon^{i j k} x^{j} p^{k}\), and use identities involving the totally antisymmetric tensor \(\varepsilon^{i j k}\).