We had illustrated that rotations in three dimensions are non-commutative or non-Abelian through the example of rotating a coffee cup about two orthogonal axes in two different orders. While a very visceral representation of rotations, we would like to provide concrete mathematics to demonstrate the same property. To do this, we will need to explicitly construct the three-dimensional representation of the rotation group.

(a) First, let's just write down the three rotation matrices for arbitrary rotations about the \(x\)-, \(y\)-, and \(z\)-axes individually. Using the rotation matrix in two dimensions, Eq. (8.7), as a guide, write down the three, three-dimensional matrices that correspond to rotation by angle \(\theta\) about the \(x\)-, \(y\)-, and \(z-\) axes individually. Call these matrices \(\mathbb{U}_{x}(\theta), \mathbb{U}_{y}(\theta)\), and \(\mathbb{U}_{z}(\theta)\), respectively. Hint: A rotation about the \(x\)-axis, for example, mixes components of a vector in which plane?

(b) Now, for each of these rotation matrices, write them as the exponential of a Hermitian operator, as we did in Sec.8.2

. For example, write

\[\begin{equation*}\mathbb{U}_{x}(\theta)=e^{i \frac{\theta \hat{L}_{x}}{\hbar}}, \tag{8.155}\end{equation*}\]

for Hermitian \(\hat{L}_{x}\). That is, determine the elements of the three-dimensional representation of the Lie algebra of rotations.

(c) We explicitly compared what happens to a three-dimensional object when it is rotated about two orthogonal axes by \(\pi / 2\) in different orders. First, draw a picture of how an initial three-dimensional vector \(\vec{v}=\left(\begin{array}{lll}0 & 0 & 1\end{array}\right)^{\top}=\hat{z}\) is rotated in two ways: first \(\pi / 2\) about the \(x\)-axis, then \(\pi / 2\) about the \(y\)-axis, and then vice-versa. Call these resulting vectors \(\vec{v}_{y x}\) and \(\vec{v}_{x y}\), respectively. Note that the initial vector \(\vec{v}\) is just the unit vector along the \(z\)-axis. Your drawing should show three vectors: the initial vector \(\vec{v}\), and the two final vectors corresponding to action of the rotation in two different orders. What is the difference of the resulting vectors, \(\vec{v}_{y x}-\vec{v}_{x y}\) ?

(d) Now, using the result of part (a), calculate the commutator of the two rotation matrices \(\left[\mathbb{U}_{y}(\theta), \mathbb{U}_{x}(\theta)\right]\) with \(\theta=\pi / 2\). Act the commutator on the vector \(\vec{v}\) from part (c):

\[\begin{equation*}\left[\mathbb{U}_{y}(\theta), \mathbb{U}_{x}(\theta)\right] \vec{v} \tag{8.156}\end{equation*}\]

Does this equal the difference vector that you wrote down in part (c), \(\vec{v}_{y x}-\) \(\vec{v}_{x y}\) ?