Past exam of the mathematical methods I course of the University of Oxford / 2016 / 11 / b / Solution

At $t=0$ the power series gives $\mathbf{v}(0)={\mathbf{v}}_0$. Equate the coefficient of $t^n$ after differentiating the series and substituting into the linear system of ordinary differential equations:Mathematical induction yields ${\mathbf{v}}_n=(-iH{)}^n{\mathbf{v}}_0/n!$. Summing gives the matrix exponential solution $\mathbf{v}(t)=e^{-iHt}{\mathbf{v}}_0$.

For the specified diagonal Hermitian matrix, its powers and exponential act separately on the two components. ThusThis unitary matrix preserves ${\mathbf{v}}^{†}\mathbf{v}$: the two components acquire opposite phases of unit modulus. The coefficient recurrence applies to any constant finite square matrix $H$; unitarity here additionally uses its Hermitian property.