Consider the Pauli Matrices $I = \pmatrix{1\ 0 \\0 \ 1}$, $\sigma_{1} = \pmatrix{0\ 1 \\ 1\ 0}$, $\sigma_{2} = \pmatrix{0\ -i\\ i\ 0}$, $\sigma_{3} = \pmatrix{1\ 0\\ 0\ -1}$,
$\bf{Part 1:}$ Compute the matrix exponential $e^{-itX}$ for X being each of the Pauli matrices.
I did this and am pretty sure I got the correct answers.
$\bf{Part 2:}$ Define Hamiltonian operator $H = -B\sigma_{1}$ for fixed $B$ a real number and use it to evolve the state $\psi(0) = \pmatrix{1 \\ 0}$ via Schrodinger Unitary evolution. Describe the resulting path in state space.
So I'm confused here. From my understanding, I need to find $U = e^{-itH} = e^{iB\sigma_{1}}$ and simply apply this matrix to $\psi(0)$ to get the evolution as a function of time? Is this correct?
From part 1, I calculated $e^{-it\sigma_{1}} = \cos(t)\pmatrix{1 \ 0 \\ 0\ 1} -i\sin(t)\pmatrix{0\ 1\\ 1 \ 0}$. Modifying this, I get $U = e^{iB\sigma_{1}} = \cos(Bt)\pmatrix{1 \ 0 \\ 0\ 1} + i\sin(Bt)\pmatrix{0\ 1\\ 1 \ 0}$. Thus, $\psi(t) = U\psi(0) = \pmatrix{\cos(Bt) \\ i\sin(Bt)} $. Is this correct?
$\bf{Part 3:}$ Compute the vector $J(t) = (<\sigma_{1}> ,<\sigma_{2}>, <\sigma_{3}>)$ of expectation values for the family of states $\psi(t)$.
I'm confused as to which values to put in for the calculation. To calculate each expectation in the vector do I simply do $<\psi(t)| \sigma_{i} | \psi(t)>$ where $\psi(t)$ is as calculated in part 2 and $\sigma_{i}$ are the pauli matrices themselves? I'm confused that in part 2, we used $\sigma_{1}$ to define the Hamiltonian H and now we use the $\psi(t)$ based off of this particular pauli matrix? Or do I need to recalculate part 2 for each pauli matrix and use that particular $\psi(t)$ in the corresponding expectation calculation?