(Solved): 5.1- Identity for rotations in terms of Euler angles Let (,,) be Euler angles, that is, R^( ...

5.1- Identity for rotations in terms of Euler angles Let $$(?,?,?)$$ be Euler angles, that is, $$\hat{R}(?,?,?)=\exp \left(?i?{\hat{J}}_{z_2}\right)\exp \left(?i?{\hat{J}}_z\right)\exp \left(?i?{\hat{J}}_y\right)$$ where the subscripts $$(x,y,z)$$ denotes axes in the lab frame, $$z_2$$ is the $$z$$ axis after the rotations by $$?$$ and $$?$$, and $$?=1$$. (a) Show that if $${\hat{J}}_{z_1}=\exp \left(?i?{\hat{J}}_y\right){\hat{J}}_z\exp \left(i?{\hat{J}}_y\right),\text{ then }\exp \left(?i?{\hat{J}}_{z_1}\right)=\exp \left(?i?{\hat{J}}_y\right)\exp \left(?i?{\hat{J}}_z\right)\exp \left(i?{\hat{J}}_y\right)\text{, }$$ where $$?$$ is some arbitrary angle. Hint: First show that $${\hat{J}}_{z_1}^n=\exp \left(?i?{\hat{J}}_y\right){\hat{J}}_z^n\exp \left(i?{\hat{J}}_y\right)$$ and then expand the exponential $$\exp \left(?i?{\hat{J}}_z\right)$$. (b) Use the result of (a) to show that if $${\hat{J}}_{z_2}=\exp \left(?i?{\hat{J}}_z\right)\exp \left(?i?{\hat{J}}_y\right){\hat{J}}_z\exp \left(i?{\hat{J}}_y\right)\exp \left(?i?{\hat{J}}_z\right)$$ then $$\exp \left(?i?{\hat{J}}_{z_2}\right)=\exp \left(?i?{\hat{J}}_z\right)\exp \left(?i?{\hat{J}}_y\right)\exp \left(?i?{\hat{J}}_z\right)\exp \left(i?{\hat{J}}_y\right)\exp \left(i?{\hat{J}}_z\right).$$ (c) Use the result of (b) to show that $$\hat{R}(?,?,?)=\exp \left(?i?{\hat{J}}_z\right)\exp \left(?i?{\hat{J}}_y\right)\exp \left(?i?{\hat{J}}_z\right)$$ where all rotations are about the lab frame axes.