Angular momentum polar component

The diagonal elements of the density matrix contain the populations of each of the BO states, whereas off-diagonal elements contain the relative phases of the BO states. The components of the density matrix with a = a describe the vibrational and rotational dynamics in the electronic state a, while the rotational dynamics within a vibronic state are described by the density matrix elements with a = a and va = v ,. The density matrix components with na = n a, describe the angular momentum polarization of the state Ja, often referred to as angular momentum orientation and alignment [40, 87-89]. The density matrix may be expanded in terms of multipole moments as ... 

Thus () is an eigenvalue of Lz with eigenvalue The angle-dependent part of the wave equation is seen to contain wave functions which are eigenfunctions of both the total angular momentum as well as the component of angular momentum along the polar axis. 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

Since the spin operator commutes with the momentum operator, it is possible to speak of states of definite momentum p and spin component /x. The components of the polarization vector may be chosen in such a way that e = XP- The two possible polarizations correspond to only two values of the component of spin angular momentum y,. The third value is excluded by the condition of tranversality. If the z-axis is directed along p, then x0 s excluded. The two vectors Xi and X2> corresponding to circular polarization are equivalent, respectively to Xi and X-i- Thus, the value17 of the spin component y = 1 corresponds to right circular polarization, while /z = — 1 corresponds to left circular polarization. 

Polarization functions may be optionally chosen to be of pure or Cartesian form (by another keyword). In the former case, one includes the expected number of angular-momentum components (i.e., five d orbitals, seven f orbitals, etc.), whereas in the latter case some additional component(s) of lower angular momentum are included (e.g., a Cartesian d set includes five d orbitals plus one s orbital, a Cartesian f set includes seven f orbitals plus three p orbitals, and so forth). 

If we substitute Eqs. (4.27) and (4.28) into Eq. (4.25), average over the z components of the initial total angular momentum, and sum over the z components of the final total angular momentum, we obtain [84] (note that the proof in Ref. 84 is specialized to the case of linearly polarized light, m = 0)... 

Of course, predictable differences between He(2 5) and He(23S) occur regarding the polarization of Penning ions and electrons caused by total spin conservation. The fact that a component of spin angular momentum is conserved in Pgl has been demonstrated by the observation of the transfer of spin polarization from optically pumped He(23S) atoms to the Penning ions of cadmium, zinc, and strontium.72 The polarization was detected by measuring the polarization of the light emitted from the excited 2Ds/2 ions in 2Ds/2- 2D2/2 transitions. If a component of spin angular momentum is conserved, we may write the Pgl process as... 

Assuming that the ion is formed with equal probability in states of different mt, the population of the different m, sublevels of 2Ds/2 is in the ratios 0 1 2 3 4 5, which gives rise to polarization of the observed light emitted in the 2D5/2—>2DJ/2 transition.72 It may be noted that the exchange mechanism of Pgl" implies the conservation of a component of spin angular momentum. 

Conservation of a component of spin angular momentum in Pgl was also demonstrated by showing that Pgl electrons formed by ionization of argon are polarized by optically oriented He(235) in a flowing afterglow.73... 

Figure 2.12 Definition of the components of angular momentum in cartesian and in spherical polar coordinates.

It has the same form as the cartesian components and the solution, = ke tmf describes rotation about the polar axis in terms of the orbital angular momentum vector LZ1 specified by the eigenvalue equation... 

In photoelectron diffraction experiments monoenergetic photons excite electrons from a particular atomic core level. Angular momentum is conserved, so the emitted electron wave-function is a spherical wave centered on the source atom, with angular momentum components / 1, where / is the angular momentum of the core level. If the incident photon beam is polarized, the orientation of the emitted electron wave-function can be controlled. These electrons then propagate through the surface and are detected and analyzed as in LEED experiments. A synchrotron x-ray source normally produces the intense beams of variable energy polarized photons needed for photoelectron diffraction. 

In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, p is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. (J)=0 if J<0 and 4(J)=1 or all other values of J,... 

The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. The position of an arbitrary point r is described by three coordinates r, 0, 0, as shown in Fig. 6.2. 

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