Angular wave number

The angular wave number m may be either a positive or a negative integer depending on the sense of rotation of harmonic angular waves. 

The right hand side of Eq. (4.81) shows a maximum in its dependence on n at (m + n)/r2j = T1/R2, which verifies the suggested strong interaction of modes with similar angular wave number yielding Eq. (4.78). The exponential in Eq. (4.81) indicates that the interaction of modes decreases with exp(—21cd), as is true in the case of half-spaces. 

The multiple reflection terms / 2 in Eq. (4.89) are added in accordance with our findings on the selective interaction of modes showing similar angular wave number, yielding dispersion relation (4.84). An explicit proof of Eq. (4.89) for arbitrary orders I may be based on the fact that reflection fields which cyclicly reproduce themselves after I reflections can be represented in terms of fields which gain a phase factor exip(—2nin/l at each reflection (Floquet s theorem). This was reported by Langbein in 1971 [95,99]. 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

Equation (1.8) represents a plane wave exp[i(A x — mt)] with wave number k, angular frequency m, and phase velocity m/A, but with its amplitude modulated by the function 2 cos[(AA x — Amt)/2]. The real part of the wave (1.8) at some fixed time to is shown in Figure 1.2(a). The solid curve is the plane wave with wavelength X = In jk and the dashed curve shows the profile of the amplitude of the plane wave. The profile is also a harmonic wave with wavelength... 

In general, the angular frequency (o(k) is a function of k, so that the angular frequencies in the composite wave (jc, t), as well as the wave numbers, vary from one plane wave to another. If (o k) is a slowly varying function of k and the values of k are confined to a small range AA , then (x) k) may be expanded in a Taylor series in k about some point ko within the interval AA ... 

Thus, the wave packet P(jc, 0 represents a plane wave of wave number ko and angular frequency mo with its amplitude modulated by the factor B(x, i). This modulating function B x, i) depends on x and t through the relationship [x — (dm/dA )o/]. This situation is analogous to the case of two plane waves as expressed in equations (1.7) and (1.8). The modulating function B(x, t) moves in the positive x-direction with group velocity given by... 

In the evaluation of the integral in equation (1.24), the wave number k is regarded as a function of the angular frequency m, so that in place of (1.13) we have... 

As before, the wave packet is a plane wave of wave number ko and angular frequency mo with its amplitude modulated by a factor that moves in the positive x-direction with group velocity given by equation (1.16). Following... 

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

Consider a plane monochromatic wave with angular frequency and wave number k which is propagating in the z direction in a nonabsorbing medium. In discussions of polarization it is customary to focus attention on the electric field E ... 

From Eqs. (6) and (8) one verifies that the angular separation of two specific wave numbers is the same in single and double pass. Equations (6) and (8) are used to determine scan rates or scanning observation intervals as discussed later. 

Here, is the angular frequency of a matter wave, such as that of an electron, is its wave number magnitude, and m ) is the rest mass of the particle corresponding to the matter wave. The rest mass could be the photon s rest mass, estimated to be less than 10 68 kg. 

The appearance of a vibration-rotation band is shown in Fig. 4.9. Because A/=0 is forbidden, we have a gap at the wave number of the band origin o0. (We are considering only diatomic molecules in 2 electronic states for electronic states in which the electronic orbital angular-momentum quantum number A is nonzero, transitions with AJ=0 are allowed, giving a Q branch in each band. An example is NO, which has a 2n electronic ground term, -branch transitions also occur for vibration-rotation bands of polyatomic molecules see Chapter 6.) Under low resolution, an infrared band of a diatomic molecule looks like Fig. 4.10. 

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

The angular part of the wave function determines the shape of the electron cloud and varies depending upon the type of orbital involved (s. p, tl, or /) and its orientation in space. However, for a given type of orbital, such as s or />., the angular wave function is independent of the principal quantum number Or energy level. Some... 

The first Tour letters originate in spectroscopic notation (see page 26) and the remainder follow alphabetically. In the previous section we have seen the various angular wave functions and the resulting distribution or electrons. The nature or the angular wave Function is determined by the value or the quantum number i... 

Here to is the angular frequency ( = 2nf) and k is the wave number ( = ca/c + ia), which contains information about the ultrasonic properties of the material, i.e., the velocity and... 

In the free electron counterterm the bound state wave functions a) and n) are expanded in terms of free electron states. Unlike [14,15,17] where the plane-wave-type functions were used for this purpose, the spherical-wave-type functions were employed in [18]. The summation over n in Eq. (16) is replaced by the integration over the continuous quantum number p, defining the energy of the free electron Ep = /p2 + to2, and the summation over the standard angular quantum numbers An important difference between the standard numerical... 

Here pj7m) denotes the spherical-wave free-electron function with the usual notations for Dirac angular quantum numbers. The numbers jlm are fixed by the overlap with the bound-electron wave function a) = njlm) where n is the principal quantum number. Integration over p is interpreted as integration over energies Ep = /p2 + m2. 

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