Quantum number, azimuthal radial

In deducing this we began by introducing two quantum conditions and hence two quantum numbers, the radial quantum number W and the azimuthal quantum number h. As the orbit has only one period, however, it is found unnecessary to use both quantum numbers in finding the energy levels, as the latter involve only the sum of the two quantum numbers. This we call the principal quantum number, as in the unperturbed motion it alone determines the positions of the terms. 

From equations (8), (9) and (10) it is evident that the path of the electron in the ith region is a segment of the KepleT ellipse defined by the segmentary radial and the azimuthal quantum numbers n and k, so that it can be described by the known equations... 

The shell theory has had great success in accounting for many nuclear properties (3). The principal quantum number n for nucleons is usually taken to be n, + 1, where nr, the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nr + / +1 / is the azimuthal quantum number.) Strong spin-orbit coupling is assumed,... 

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

Fig. 17.1 Illustrations of whispering gallery modes (WGM) in a spherical optical resonator. The WGM modes are classified in terms of their radial quantum number p as well as by their angular momentum quantum number / and the azimuthal quantum number m that can have (21+ 1) values, meaning that the resonance frequency ( ,/ has a (2/ + 1) degeneracy...

The radial quantum number (n ) was introduced to specify the eccentricity of elliptic orbits and an azimuthal quantum number (k) to specify the orientation of orbits in space. The three quantum numbers are related by... 

Here E denotes the energy of the bound electron after deducting the rest energy, and Eq is the rest energy mc is the radial quantum number (Sommerfeld) is identical with Bohr s azimuthal quantum number /c, and corresponds therefore to the H + 1 of wave mechanics. Since, however, as we have just seen, two terms with dilierent I but the same j always coincide when we take the spin into account, discrimination between the terms by means of the quantum number is identical with discrimination by means of j we therefore have = i + I- The principal quantum number is then found as the sum n=- n>r + The constant a is given by... 

In this equation we have introduced a new quantum number n, called the total quantum number, as the sum of the azimuthal quantum number k and the radial quantum number nr ... 

Equations similar to equation (6.16) have been studied by mathematicians, and acceptable solutions found. The mathematics involved are quite lengthy and wiU not be given here they can be found in standard textbooks on quantum mechanics. Two quantum numbers are needed to specify a particular radial wavefunction. The first one is the azimuthal quantum number. I, and the second one is a new quantum number , often referred to as the principal quantum number. All the solutions have the same mathematical structure, which can be expressed by the equation ... 

Fig. 1.6 Density distributions of the radial functions, R i (n and / tire principal and azimuthal quantum numbers), for the electronic motion in the hydrogen atom...

The notation of a state often contains its sequence number n + 1 as well, e.g., Is, Ip, Id, 2s, 2p, etc. (Frequently + 1 is called the radial quantum number. The number n or n + 1 counts the radial nodes, thus it differs from the principal or azimuthal quantum number used for the classification of the states of the hydrogen atom.)... 

The standard procedure (Rose 1961), for heavy atoms, is to solve the wave equation in a spherical potential with relativistic kinematics. In open-shell systems the charge density is spherically symmetrized by averaging over all azimuthal quantum numbers. The wave equations to be solved in practice are, therefore, the coupled first-order differential equations for the radial components of the Dirac equation (Rose 1961)... 

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