Two-state quantum system

Example

A two-state quantum system has the state space $\C^2$, and its Hamiltonian is a $2 \times 2$ Hermitian matrix $H$. Every such matrix can be written $H = a I + b_1 \sigma_x + b_2 \sigma_y + b_3 \sigma_z$ with $a \in \R$ and $b = (b_1, b_2, b_3) \in \R^3$, where $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices. Since $(b_1\sigma_x + b_2\sigma_y + b_3\sigma_z)^2 = \abs{b}^2 I$, the eigenvalues of $H$ are $a \pm \abs{b}$.

Rabi oscillation: let $\Omega > 0$ and $H = \frac{\hbar\Omega}{2} \sigma_x$, and start in the state $e_0 = (1, 0)$. Because $\sigma_x^2 = I$, the evolution operator is $e^{-iHt/\hbar} = \cos(\Omega t/2) I - i \sin(\Omega t/2) \sigma_x$. So the state at time $t$ is $\cos(\Omega t/2) e_0 - i \sin(\Omega t/2) e_1$ with $e_1 = (0, 1)$. The probability of finding the system in $e_1$ is $\sin^2(\Omega t/2)$, which swings between $0$ and $1$.