to the Bloch vector precessing around with angular frequency . Without loss of generality, assume the field is uniform points in , so that the time evolution operator is given as It can be seen that such a time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of Larmor precession) The above method can be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic field in the previous case) if the interaction is given by an appropriate coupling term that is analogous to the magnetic moment. The precession of the state vector (which need not be a physical spinning as in the previous case) can be viewed as the precession of the state vector on the Bloch sphere. The representation on the Bloch sphere for a state vector will simply be the vector of expectation values . As an example, consider a state vector that is a normalized superposition of and , that is, a vector that can be represented in the basis as The components of on the Bloch sphere will simply be . This is a unit vector that begins pointing along and precesses around in a left-handed manner. In general, by a rotation around , any state vector can be represented as with real coefficients and . Such a state vector corresponds to a Bloch vector in the xz-plane making an angle with the z-axis. This vector will proceed to precess around . In theory, by allowing the system to interact with the field of a particular direction and strength for precise durations, it is possible to obtain any orientation of the Bloch vector, which is equivalent to obtaining any complex superposition. This is the basis for numerous technologies including quantum computing and MRI. Evolution in a time-dependent field: Nuclear magnetic resonance Nuclear magnetic resonance (NMR) is an important example in the dynamics of two-state systems because it is involves the exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field B0 (the "holding field") and then applying a weak, transverse field B1 that oscillates at some radiofrequency ωr. Explicitly, consider a spin-1/2 particle in a holding field and a transverse rf