4) a) Verify the expression given in Zettili equation 5.122. Note, to do this you can express the
exponential as a Taylor series ($1 + x + \frac{x^2}{2!} ...$).
b) Now suppose that a spin-1/2 particle is subjected to the operator $e^{iat\sigma_x}$. If the particle is
initially in a spin state $|\frac{1}{2}, -\frac{1}{2}\rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}$, find an expression for the spin state at later times, and
show that the expectation value \(\langle \vec{S} \rangle\) as a function of time is a vector of fixed length that rotates at
a constant rate around the x axis. What is the fixed length?
c) By taking the time derivative of the time-dependent spin state, show that $a\sigma_x$ must be equal
to $CH$, e.g. a constant C times the Hamiltonian acting on the particle's spin state. Find C. Note,
expressed this way, the exponential operator is the time evolution operator.
