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To solve this problem, we need to determine the state of the system at a later time \( t \) given the initial state and the Hamiltonian. The Hamiltonian for an electron at rest in a magnetic field \( B_0 \) along the \( z \)-direction is given by: \[ H = -\frac{\hbar \gamma B_0}{2} \sigma_z \] where \( \sigma_z \) is the Pauli matrix: \[ \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \] The initial state of the system is given by: \[ |\psi(0)\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] This is the spin-up state along the \( z \)-axis. ### Step 1: Solve the Schrödinger Equation The time evolution of the state is governed by the Schrödinger equation: \[ i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle \] Given the Hamiltonian \( H \), we ___ _________ ___ ________ ______ ______ _____ ________ __________.
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