  We show that the stochastic Schr\"odinger equation (SSE) provides an ideal
way to simulate the quantum mechanical spin dynamics of radical pairs. Electron
spin relaxation effects arising from fluctuations in the spin Hamiltonian are
straightforward to include in this approach, and their treatment can be
combined with a highly efficient stochastic evaluation of the trace over
nuclear spin states that is required to compute experimental observables. These
features are illustrated in example applications to a flavin-tryptophan radical
pair of interest in avian magnetoreception, and to a problem involving
spin-selective radical pair recombination along a molecular wire. In the first
of these examples, the SSE is shown to be both more efficient and more widely
applicable than a recent stochastic implementation of the Lindblad equation,
which only provides a valid treatment of relaxation in the extreme-narrowing
limit. In the second, the exact SSE results are used to assess the accuracy of
a recently-proposed combination of Nakajima-Zwanzig theory for the spin
relaxation and Schulten-Wolynes theory for the spin dynamics, which is
applicable to radical pairs with many more nuclear spins. An appendix analyses
the efficiency of trace sampling in some detail, highlighting the particular
advantages of sampling with SU(N) coherent states.
