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Chapter 1
88
RESONANT PRECESSION OF THE SPIN-1/2-CURRENT-DENSITY FUNCTION GIVES RISE TO THE BOHR MAGNETON
The Stern Gerlach experiment described below demonstrates that the magnetic moment of the electron can only be parallel or antiparallel to an applied magnetic field. In spherical coordinates, this implies a spin quantum number of 1/2 corresponding to an angular momentum on the z-axis of . However, the Zeeman splitting energy corresponds to a magnetic moment of B and 2 implies an electron angular momentum on the z-axis of —twice that given by Eq. (1.128). Consider the case of a magnetic field applied to the orbitsphere. As shown in Figure 1.23, the orbitsphere comprises an angular momentum component of 2 along the z-axis and two angular momentum components in opposite directions in the xy-plane. The magnetic moment 4 corresponding to the angular momentum along the z-axis results in the alignment of the z-axis of the orbitsphere with the magnetic field while one of the vectors in the xy-plane causes precession about the applied field. The precession arises from 4 angular momentum components to a Larmor excitation by a corresponding resonant photon that couples to one of the 4 conserve the angular momentum of the photon such that the precession direction matches the handedness of the Larmor photon. An example given in Figure 1.25 regards a right-hand polarized photon that excites the right-handed Larmor precession by angular momentum component as shown. The precession frequency is the Larmor frequency coupling to the corresponding 4 e , and the magnetic flux B [15]. The energy of the precessing given by the product of the gyromagnetic ratio of the electron, 2m electron corresponds to Zeeman splitting—energy levels corresponding to the parallel or antiparallel alignment of the electron magnetic moment with the magnetic field and the excitation of transitions between these states by flipping the orientation along the field by a further resonant photon of the Larmor frequency. Thus, the energy of the transition between these states is that of projection on the z-axis and the resonant photon. The angular momentum of the precessing orbitsphere comprises the initial 2 the initial vector component in the xy-plane that then precesses about the z-axis with the Larmor photon. As shown in the 4 Excited States of the One-Electron Atom (Quantization) section, conservation of the angular momentum of the photon of gives rise to of electron angular momentum that gives rise to a contribution to the angular momentum along the magnetic2 field or z-axis. The parameters of the photon standing wave for the Zeeman effect are given in the Magnetic Parameters of the Electron (Bohr Magneton) section and Box 1.1. projection on the z-axis (Eq. The angular momentum of the orbitsphere in a magnetic field comprises the static 2 (1.128)) and the vector component in the xy-plane (Eq. (1.127)) that precesses about the z-axis at the Larmor frequency. The 4 precession at the Larmor frequency as well as the excitation of a spin-flip transition is equivalent to the excitation of an excited state as given in the Excited States of the One-Electron Atom (Quantization) section. Consider the first resonant process. A resonant excitation of the Larmor precession frequency gives rise to a trapped photon with of angular momentum along a precessing S -axis. In the coordinate system rotating at the Larmor frequency (denoted by the axes labeled X R , YR , and Z R in and S of magnitude are stationary. The angular momentum along X R Figure 1.25), the X R -component of magnitude 4 4 with a corresponding magnetic moment of
B 4
(Eq. (28) of Box 1.1) causes S to rotate in the YR Z R -plane to an angle of
3 such that the torques due to the Z R -component of and the orthogonal X R -component of are balanced. Then the Z R 2 4 corresponds to the ratio of component due to S is cos . The reduction of the magnitude of S along Z R from to 3 2 2