Angular momentum wave relationship

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. 

Several theories have been developed to explain the rainbow phenomena, including the Lorenz-Mie theory, Airy s theory, the complex angular momentum theory that provides an approximation to the Lorenz-Mie theory, and the theory based on Huy gen s principle. Among these theories, only the Lorenz-Mie theory provides an exact solution for the scattering of electromagnetic waves by a spherical particle. The implementation of the rainbow thermometry for droplet temperature measurement necessitates two functional relationships. One relates the rainbow angle to the droplet refractive index and size, and the other describes the dependence of the refractive index on temperature of the liquid of interest. The former can be calculated on the basis of the Lorenz-Mie theory, whereas the latter may be either found in reference handbooks/literature or calibrated in laboratory. 

The exploitation of the community of the transformation properties of irreducible tensors and wave functions gives us the opportunity to deduce new relationships between the quantities considered, to further simplify the operators, already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements. Indeed, it is possible to show that the action of angular momentum operator Lf on the wave function, considered as irreducible tensor tp, may be represented in the form [86] ... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

It is significant that in both cases Planck s constant appears in the specification of the dynamic variables of angular momentum and energy, associated with wave motion. The curious relationship between mass and energy that involves the velocity of a wave, seems to imply that the motion of mass points also has some wavelike quality. Only because Planck s constant is almost vanishingly small, dynamic variables of macroscopic systems appear to be continuous. However, when dealing with atomic or sub-atomic systems... 

The quantity mvr is the orbital angular momentum of the electron, and equation (1.9) shows that it must be an integer multiplied by hl2%. This same relationship had been earlier suggested arbitrarily by Bohr, and it is of great interest that the wave concept provides a simple interpretation of this relationship. 

The addition of two angular momenta (formula of the type (10.4)) may be directly generalized to cover the case of an arbitrary number of momenta. However, in such a case it is not enough to adopt the total momentum and its projection for the complete characterization of the wave function of coupled momenta. Normally, the quantum numbers of intermediate momenta must be exploited too. Moreover, these functions depend on the form (order) of the coupling between these momenta. The relationships between the functions, belonging to different forms of coupling of their momenta, may be found with the aid of transformation matrices. 

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