Quantum number, azimuthal total

The wave functions for a state of a hydrogenlike atom described by the quantum numbers n (total quantum number), l (azimuthal quantum number), and m (magnetic quantum number) are usually expressed in terms of the polar coordinates r, 8, and . The orbital wave function is a product of three functions, each depending on one of the coordinates ... 

Solution of these equations leads naturally to the principal quantum number n and to two more quantum numbers, / and m The total energy of the electron is determined by n, and its orbital angular momentum by the azimuthal quantum number l. The value of the total angular momentum is /(/ + ) 2h. The angular momentum vector can be oriented in space in only certain allowed directions with respect to that of an applied magnetic field, such that the components along the field direction are multiples of fi the multiplying factors are the mi quantum... 

To ensure that all microstaics have been written, the total number N. of microstates associated with an electronic configuration, A, having. r electrons in an orbital set with an azimuthal quantum number, I, is... 

The angular momentum (or azimuthal) quantum number / can take values from zero up to a maximum of n. It determines the total angular momentum of the electron about the nucleus. 

In parhelium, then, the total spin moment s is 0 hence the total orbital moment is identical with the total angular momentum j = 1. This implies that the whole of the terms of parhelium are singlets, i.e. that to every azimuthal quantum number I there belongs only a single term with the inner quantum number j equal to 1. 

We add a brief remark on the notation for the terms, A few lines above we have used capital letters S, P, D,. . , instead of small letters (p. 126). This is customary in the case of several electrons for the purpose of indicating the total orbital moment. It is also customary to define the multiplet character, that is to say, the number of terms belonging to a particular multiplet (all with the same principal and azimuthal quantum numbers), by attaching this number to the term symbol as a left-hand upper index also, the multiplicity written down is always the one which occurs when I is large, viz. 25+1. In fact,... 

Hence the total angular momentum of the orbit was restricted by the old quantum theory to values which are integral multiples of the quantum unit of angular momentum h/2ir. The quantum number k is called the azimuthal quantum number. 

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 ... 

The terminology and symbolism used to specify the various quanmm numbers are not too informative. The numbers are known as the principal (n), azimuthal (1), magnetic (mi), spin (s) and magnetic spin (m ), quantum numbers. The first three are integers, such that, for one set of eigenfunctions, is a positive number, I is always less than n and w has a total of (2/+ 1) allowed values, clustered about zero. For n = 2 and / = 1 it follows that m/ has the three possible values +1, 0 and -1. The quantum numbers s and m have half-integer values. All electrons have 5=5 and Wj = 5. 

We consider the case where atom A with electrons I, 2 is in the ground state and atom B with electrons 3, 4 is in an excited state. Let v)a(L 2), b (3, 4) and 953(3, 4) be the atomic wave functions of FS" state, of 2 P state and of 2 5 state, respectively. The superscript A indicates the azimuthal quantum number 0, 1 with respect to the molecular axis of A—B. Since A and B are distant from each other, we express the total electronic wave function (Tab) of the system as the antisymmetrized product of respective atomic wavefunctions. We can write... 

We have replaced the rotational quantum number J by /, since this is the usual notation in atomic systems. The quantum number / is called the azimuthal quantum number and characterizes the total angular momentum of the atom,... 

There are (21 + 1) orbitals in a subshell with azimuthal quantum number 1, and a total of n orbitals in total in a shell with principal quantum number n. 

In electron configurations with the same main and azimuthal quantum numbers, the highest total spin configuration is the most stable. 

The total effective potential determining the electronic motion via the Kohn-Sham equations is expected to be spheroidal as well. Therefore all spherical shells n, /, m are expected to split into spheroidal subshells m, p,k. Here m is the preserved azimuthal quantum number. For time-reversal symmetry only its magnitude m counts p is the parity and k just enumerates the levels of a certain symmetry. The reduced spheroidal symmetry lifts the spherical degeneracy as depicted in Figure 1.11 for Na in the size-range of N from 3 to 18. 

Each atomic orbital (AO) is specified by a principal quantum number n, which is simply the total number of nodes plus one, and by an azimuthal quantum number /, equal to the number of angular nodes. These quantum numbers come out of the solution of the Schrodinger equation for the individual hydrogen AOs. For the ground-state orbital (Fig. 1.4b), n = 1 and I = 0 for the orbital of Fig. 1.5(d), n = 2 and / = 1. Orbitals with / = 0 are called s orbitals those with / = 1 are p orbitals those with / = 2 are d orbitals. Each orbital is further specified by the value of n. Thus the orbital of Fig. 1.2(b) is the Is orbital that of Fig. 1.4(c) is the 2s orbital that of Fig. 1.4(d) is a 2p orbital. Note that we said the Is orbital and the 2s orbital, but a 2p orbital. The s orbitals are unique, but orbitals with a given value... 

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