Electron quantum number

In the following, it shall always be assumed that the zeroth-order solution is known, that is, we have a complete set of eigenvalues and wave functions, labeled by the electronic quantum number n, which satisfy... 

Given the following sets of electron quantum numbers, indicate those that could not occur, and explain your answer... 

The electrons that occupy the levels of a Fermi gas have energies < and may be considered as confined to a (Fermi) sphere of radius kF in k-space. For large volumes the free-electron quantum numbers may be treated as continuous variables and the number of states in a range dk = dkxdkydkz, is... 

However, while one-electron quantum numbers n and might be ascribed to an atom even in an approximate sense, the two other numbers m and ms generally cannot be associated with an atomic state in any valid approximation. They however are quite often used not as physical quantities but as mere counters. 

In an analysis of a molecular spectrum, the primary task is, for purpose of characterisation, to assign each narrow spectral feature to a transition between two molecular states specified with rotational, vibrational and electronic quantum numbers or other indices. Nearly as important as the former, another task is to... 

In rigorous quantum mechanics, something like an electronic base function parametrically dependent on nuclear configuration space cannot be. Such dependence would imply that the electronic quantum number of the base function depends upon the particular selected region of nuclear configuration space. 

The important result is that an electronic quantum number doesn t depend at all on the domain of Q that we might have selected to look at equation (4) is valid for all Q-values the quantum number results from boundary conditions and general frame-invariance. 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

Eq. (10) is block diagonalized as a function of an electronic quantum number, Sjg (Kronecker symbol) factor. Equations (9) and (10) are related. These equations for each electronic subspace differ only by a local linear transformation, then, one may write ... 

The caveat is that integration over electronic configuration space is performed first. Any physical base quantum state with respect to the Coulomb Hamiltonian is a robust species. Observe that no structural features are implied yet. Thus, separability via electronic quantum numbers is achieved although the general quantum states E(q,Q,t) are not separable. 

Electronic Molecular Spectra.—In general the absorption and emission spectra of molecules involve change in the electronic quantum numbers as well as in the vibrational and rotational quantum numbers. These molecular spectra are complex, and their interpretation is diffi-... 

The electronic quantum numbers of diatomic molecules are discussed in Sections 1.19 and 4.11. The selection rule for A can be shown to be... 

Having considered the possible changes in electronic quantum numbers, we now discuss the allowed changes in v and J. To do so, we must deal with the complete integral (7.6). The bracketed integral in (7.6) is a function of the internuclear distance R calling this integral Pel, we have... 

To a first approximation each of several electrons in such a partly filled shell may be assigned its own private set of one-electron quantum numbers, n, /, m, and s. However, there are always fairly strong interactions among these electrons, which make this approximation unrealistic. In general the nature of these interactions is not easy to describe, but the behavior of real atoms often approximates closely to a limiting situation called the L-S or Russell-Saunders coupling scheme. 

Antisymmetry of the atomic wave function is the mathematical expression of the requirements of the Pauli exclusion principle, demanding that two or more atomic electrons cannot simultaneously occupy the same state, i.e. cannot possess the same set of one-electron quantum numbers. In a particular case of two-electron atoms the wave function of an atom... 

In each product function, the same set of one-electron quantum numbers is arranged in the same order (usually in the standard order 1,2,..., N) but the electron coordinates ri,r2,r3,... have been rearranged into some new order r i,ry2,ry3,. The summation in (10.8) is over all N possible permutations P = jij2h jN of the normal coordinate ordering 12 3. .. N, and p is the parity of the permutation P (p = 0 if P is obtained from the normal ordering by an even number of interchanges, and p = 1 if an odd number of interchanges is involved). 

Wave functions (13.1) form an orthonormal set, but their normalization factors are defined only up to a sign. The fact is that the wave function (13.1) is antisymmetric not only under coordinate permutations, but also under permutations of one-electron quantum numbers. Thus, to fix the sign of a wave function requires a way of ordering the set of quantum numbers (a) = ai, 0C2,..., ajy. There exists, however, a convenient formalism that allows us to include the constraints imposed by the requirement that the wave functions be antisymmetric in a simple operator form. This formalism became known as the second-quantization method. This chapter gives a detailed description of the fundamentals of the second-quantization method. 

For the majority of interesting cases of electronic configurations (s-, p-, d- and /-electrons), quantum numbers v and J are sufficient to classify... 

The dependence of photoabsorption cross-section on many-electron quantum numbers (sets of quantum numbers of a chain of electronic shells) is mainly determined by the submatrix element of the transition operator. Their non-relativistic and relativistic expressions for the most widely considered configurations are presented in Part 6. When exciting an atom by X-rays the main type of transitions are as follows ... 

These features of lines of various spectra (X-ray, emission, photoelectron, Auger) are determined by the same reason, therefore they are discussed together. Let us briefly consider various factors of line broadening, as well as the dependence of natural line width and fluorescence yield, characterizing the relative role of radiative and Auger decay of a state with vacancy, on nuclear charge, and on one- and many-electron quantum numbers. 

The electronic state function Pa(r, R) depends not only on electron coordinates r but also on the nuclear coordinates R. The subscript a denotes a set of electronic quantum numbers. Because the mass of the electrons is much smaller than the mass of the nuclei, the electron motion follows the motion of the nuclei adiabatically, so it is customary to adopt the Bom-Oppenheimer approximation, as a result of which the state function may be written as a product of electronic and nuclear state functions ... 

This is the exact solution to the problem in a fixed frame related to the stationary geometry ofthe n-th electronic state. The electronic quantum number in Een( ) is sufficient to remind of the existence of a priviledged frame (and possibly specific symmetry constraints). In the stationary case, the hamiltonians He(p a) and... 

The quantum states of the molecular hamiltonian can be labelled with the linear momentum of each rigid configuration for the sources of static Coulomb potential, Pi and P2 or koi = fl and k02 = P2/, respectively. The total angular momentum of each cluster, Ji and J2 and their projections Jiz and J2z together with the vibrational and electronic quantum numbers provide a set of labels helping to characterize the quantum states of the system ... 

Nomenclature. The spectroscopic nomenclature is directly equivalent to that used for spectroscopy X-ray, and is related to the various quantum numbers such as the principal quantum number n, the electronic quantum number 1, the total angular momentum quantum number j, and the spin quantum number s, which can take... 

For atomic systems, it is often said that each electron is defined by four quantum numbers n, l, me, and ms. Actually, there is a fifth quantum number, s, which has the value of 1 /2 for all electrons. Quantum number ms can be either 1/2 or -1/2, corresponding to spin function a (spin up) and p (spin down), respectively. Spin functions a and p form an orthonormal set,... 

Averaging it over the ground state of the B- h group (( yields the following one-electron operator acting on the electron quantum numbers of the A-th group ... 

PWE procedure [14,15,16,17] and the version used in [18] is that in [18] only terms diagonal in p,j,l,m are retained in the counterterm after expansion of the bound states (a and n). The motivation is that the interaction with the vacuum cannot change the exact free-electron quantum numbers. 

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