Quantum number INDEX

In addition to size, an atomic orbital also has a specific shape. The solutions for the Schrodinger equation and experimental evidence show that orbitals have a variety of shapes. A second quantum number indexes the shapes of atomic orbitals. This quantum number is the azimuthal quantum number (1). 

The integer n, called the principal quantum number, indexes the individual energy levels. These are identical to the energy levels predicted by the Bohr theory. Here, however, quantization comes about naturally from the solution of the Schrodinger equation, rather than through an arbitrary assumption about the angular momentum. 

The index for the orbital ( ). (r) can be taken to include the spin of the electron plus any other relevant quantum numbers. The index runs over the number of electrons, each electron being assigned a unique set of quantum... 

It is important in defining the monodromy matrix, which quantifies changes in the unit cell in Figs. 4 and 5, to specify the lengths of the unit cell sides that define the basis. The monodromy theorem—that the monodromy index is equal to the number of pinch points on the pinched torus [40]—applies in a basis in which the cell sides represent unit changes in the relevant quantum number. 

Each quantized property can be identified, or indexed, using a quantum number. These are integers that specify the values of the electron s quantized properties. Each electron in an atom has three quantum numbers that specify its three variable properties. A set of three quantum numbers is a shorthand notation that describes a particular energy,... 

The most important quantized property of an atomic electron is its energy. The quantum number that indexes energy is the principal quantum number (n). Eor the simplest atom, hydrogen, we can use Equation to calculate... 

Summarizing, the principal quantum number (n) can have any positive integral value. It indexes the energy of the electron and is correlated with orbital size. As U increases, the energy of the electron increases, its orbital gets bigger, and the electron is less tightly bound to the atom. 

Among atomic orbitals, s orbitals are spherical and have no directionality. Other orbitals are nonspherical, so, in addition to having shape, every orbital points in some direction. Like energy and orbital shape, orbital direction is quantized. Unlike footballs, p, d, and f orbitals have restricted numbers of possible orientations. The magnetic quantum number (fflj) indexes these restrictions. 

Kier and Hall noticed that the quantity (S -S) jn, where n is the principal quantum number and 5 is computed with Eq. (2), correlates with the Mulliken-Jaffe electronegativities [19, 20]. This correlation suggested an application of the valence delta index to the computation of the electronic state of an atom. The index (5 -5)/n defines the Kier-Hall electronegativity KHE and it is used also to define the hydrogen E-state (HE-state) index. 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

It is customary to use the index J for the rotational quantum number. Equation (10.33) then becomes... 

The above equations have to be complemented with rules for dealing with the singularities emerging when PHP has eigenvalues at energy E, which is the case primarily when the Wp) states are continuum states. In such a case, we denote /p) = c,E 1) and ) = Ic.ii"), with c standing for all quantum numbers related to a continuum channel (the channel index). The notation E = E - iO, serves to remind of the incoming boundary conditions used for example, for the c,E ) states one has... 

We will see in Chapter 7 that our model predicts the existence of states indexed by the quantum numbers and m but fails to predict the factor of two introduced by the spin quantum number s. The beauty of this prediction is that it is close to the experimental data—off only by a measly factor of two — even though the assumptions are quite meager. We discuss spin in Chapter 10. Readers who have seen these predictions come out of the analysis of the Schrodinger equation should note that the predictions of Chapter 7 use neither the concept of energy nor the theory of observables. In other words, we will make these powerful predictions from symmetry considerations alone. 

The convention is that r ranges from - 7 to 7 as E increases. The index t is a bookkeeping number, rather than a true quantum number. The degeneracy of the asymmetric-top energy levels is 27+ 1, corresponding to the 27 +1 values of M, which do not affect the energy. Each asymmetric-top wave function is a linear combination of the 27+1 symmetric-top wave functions with the same value of 7 and of M as i Let us consider some examples. For 7 = 0, the only value of K is 0, and the secular equation (5.78) is... 

The excited-state wavepacket spontaneously emits photons while undergoing transitions to any of the electronically-ground vibrational wavefunctions t (where we have lumped the final state quantum numbers i>/, jf in a single index f). The rate of emission from a given 4% component of the excited wavepacket to a given ground state is given in terms of the Einstein A-coefficient [9],... 

To avoid confusion, let us explain the difference between the probability Wlh determined by eqn. (19) and the probability Wfn nj), determined by eqn. (38). Wt is the probability of transition from the state with the complete set of quantum numbers for the nuclear motion (the index i characterizes this set). Wfn rif) is the probability of tunneling with the fixed quantum numbers Ti, and nt of one quantum oscillator while by all the other (classical) oscillators the averaging has been carried out. 

The index number refers to the principal quantum number and corresponds to the K shell designation often used for the electron of the normal hydrogen atom. The principal quantum number 2 corresponds to the L shell, 3 to the M shell, and so on. The notation s (also p, cl, f to come later) has been carried over from the early days of atomic spectroscopy and was derived from descriptions of spectroscopic lines as sharp, principal, diffuse, and fundamental, which once were used to identify transitions from particular atomic states. 

Index k runs here through odd values 1,3,...,2 . Expression (19.79) is diagonal with respect to seniority quantum number v, i.e. 

It is worth emphasizing that coefficients (20.29) and (20.30) do not depend on orbital quantum numbers. These numbers define only the parity of summation index k, following from the conditions of the nonvanishing of radial integrals in (20.27) and (20.28). For the direct term, parameter k acquires even values, whereas for the exchange part the parity of k equals the parity of sum h + h-If one or two subshells are almost filled, then the following equalities are valid ... 

N electron/hole excitations is characterized by the set of excitation energies Cfe P2 where the index k labels the quantum number of excitation p. k labels its quantum number. Assuming a simple model for electron tunneling matrix elements [7]... 

A PWC function is defined as an (N — l)-electron bound parent state (atom or ion) with well-defined spin, parity, angular momentum and energy Ia = (Sa, na, La, Ea), coupled first to the spin-angular part of a single-particle state for the Nth electron, with definite orbital angular momentum la, to form a state with definite parity n, spin S, angular momentum L, and their projections L and M (for brevity, we indicate these global quantum numbers with the collective index T)... 

The main idea of the method is to represent the wave function of a particle as a linear combination of some known localized states ipa(r, a), where a denote the set of quantum numbers, and a is the spin index (for example, atomic orbitals, in this particular case the method is called LCAO - linear combination of atomic orbitals)... 

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