Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture and explain how these formulations give rise to identical physical predictions. Derive an equation of motion for an operator in the Heisenberg picture, assuming the operator is independent of time in the Schrödinger picture.

State clearly the form of the unitary operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$ (a unit vector) for a general quantum system. Verify your statement for the case in which the system is a single particle by considering the effect of an infinitesimal rotation on the particle's position $\hat{\mathbf{x}}$ and on its spin $\mathbf{S}$.

Show that if the Hamiltonian for a particle is of the form

$$H=\frac{1}{2 m} \hat{\mathbf{p}}^{2}+U\left(\hat{\mathbf{x}}^{2}\right) \hat{\mathbf{x}} \cdot \mathbf{S}$$

then all components of the total angular momentum are independent of time in the Heisenberg picture. Is the same true for either orbital or spin angular momentum?

[You may quote commutation relations involving components of $\hat{\mathbf{x}}, \hat{\mathbf{p}}, \mathbf{L}$ and $\mathbf{S}$.]