Rotation operator approach to spin dynamics and the Euler geometric equations

Abstract

The rotation operator approach proposed previously is applied to spin dynamics in a time-varying magnetic field. The evolution of the wave function is described, and that of the density operator is also treated in terms of a spherical tensor operator base. It is shown that this formulation provides a straightforward calculation of accumulated phases and probabilities of spin transitions and coherence evolutions. The technique focuses, not on the rotation matrix, but on the three Euler angles and its characteristic equations are equivalent to the Euler geometric equations long known to describe the motion of a rigid body. The method usually depends on numerical calculations, but analytical solutions exist in some situations. In this paper, as examples, a hyperbolic secant pulse is solved analytically, and a Gaussian-shaped pulse is calculated numerically.

A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR

“…Jaynes transcribed from, the use of the Euler angles in the recently proposed the usual SO(3) matrix representation of the rotation group rotation-operator approach (21)(22)(23) has furnished yet anto a spinor representation, whose SU(2) transformation maother example of the link between spin and rigid-body kinetrix elements are the Cayley-Klein (åCK) parameters for matics. To reinforce this link, the kinematic relations for the the classical description of rotations.…”

“…To reinforce this link, the kinematic relations for the the classical description of rotations. In view of the isomorEuler angles (the Euler kinematic equations) obtained using phism between SU (2) and the group of quaternions, Q(4), the rotation-operator approach (21)(22)(23) are derived here Jaynes, thus, effectively anticipated much of the recent NMR from a classical point of view. In the case of the rotation work exploiting quaternions (5-7) and spinorized Bloch angle/axis {F, n } parametrization, the corresponding kineequations (8-11).…”

“…by Sanctuary and colleagues (21)(22)(23) for the description of spin dynamics in a time-varying magnetic field, different parametrizations of the three-dimensional rotation group can Å exp[ig(t)I z ]exp[ib(t)I y ]exp[ia(t)I z ] be used to discuss the links between spin kinematics and rigid-body rotational motion. There are two possible ways to…”

“…[12] brief review of the rotation-operator approach in the Euler angle {a, b, g} parametrization used by Sanctuary and colleagues (21)(22)(23) to obtain the Euler kinematic equations, followed by a more detailed treatment of the rotation-operaThese equations for the three time-dependent Euler angles tor approach in the rotation angle/axis {F, n } parametrizaa(t), b(t), and g(t) determine the explicit form of the Wigtion. In the latter case, the corresponding kinematic equations ner rotation operator D[a(t), b(t), g(t)] of Eq.…”

“…ing a suitable recombination and transformation, the Euler In the rotation-operator approach adopted by Sanctuary kinematic equations so obtained can be recast into the Riccati and colleagues (21)(22)(23), the time-dependent Zeeman interequation, whose analytic solution can then easily be used action Hamiltonian H(t) can be written as for the analytic solution of the Euler kinematic equations (22,23).…”