field B1 rotating in the xy-plane in a right-handed fashion around B0: As in the free precession case, the Hamiltonian is , and the evolution of a state vector is found by solving the time-dependent Schrödinger equation . After some manipulation (given in the collapsed section below), it can be shown that the Schrödinger equation becomes where and . As per the previous section, the solution to this equation has the Bloch vector precessing around with a frequency that is twice the magnitude of the vector. If is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that , in the rotating frame the state vector will precess around with frequency , and will thus flip from down to up releasing energy in the form of detectable photons. This is the fundamental basis for NMR, and in practice is accomplished by scanning until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use is made of the rotating wave approximation in deriving such results. Relation to Bloch equations The optical Bloch equations for a collection of spin-1/2 particles can be derived from the time dependent Schrödinger equation for a two level system. Starting with the previously stated Hamiltonian , it can be written in summation notation after some rearrangement as Multiplying by a Pauli matrix and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields Adding this equation to its own conjugate transpose yields a left hand side of the form And a right hand side of the form As previously mentioned, the expectation value of each Pauli matrix is a component of the Bloch vector, . Equating the left and right hand sides, and noting that is the gyromagnetic ratio , yields another form for the equations of motion of the Bloch vector where the fact that has been used. In vector form these three equations can be expressed in terms of a cross product Classically, this equation describes the dynamics of a spin in a magnetic field. An ideal magnet consists of a collection of identical spins behaving independently, and thus