β Scribed by Siminovitch, David J.
No coin nor oath required. For personal study only.
From both the classical and the quantum-mechanical point of view, the description of rotations in NMR is essential for explaining the response of spin systems to radio frequency pulses. Of the many methods used for this description, quaternionic formalisms offer unique advantages. Quaternions are defined and used to introduce Ro-{ } drigues quaternions, whose elements in a rotation angle α axis β½, n parameterization are ( ) { } defined by the Euler α Rodrigues ER parameters cosβ½ / 2, n sinβ½ / 2 . The utility of Ro-drigues quaternions for handling the calculus of rotations via a simple composition rule is ( ) ( ) emphasized. The traditional use of SO 3 and SU 2 matrix representations of the rotation group in a classical description of NMR is discussed, noting in particular the isomorphism ( ) between SU 2 and the group of Rodrigues quaternions. It is this isomorphism that underlies the equivalent utility of the Cayley α Klein parameters or the ER parameters in the { } classical description of rotations. The rotation angle α axis β½, n and the Euler angle { } β£,β€,β₯ parameterizations of rotations are compared by deriving and solving the corre-{ } sponding kinematic relations for the ER parameters and the Euler angles β£ , β€, β₯ . The linearity of the ER parameter kinematic relations is noted and used to obtain a straightforward solution in the constant-field case.
π SIMILAR VOLUMES
A quantum-mechanical context is used to discuss the advantages offered ( ) by a Euler α Rodrigues ER parameterization of the rotation implicit in the propagators of 1 1 spin-and spin-1 systems. To appreciate these advantages, we begin with a spin-2 2 { } { } propagator, using either a rotation angl