Electrons indistinguishability

A major difficulty is the problem of electron indistinguishability. The natural choice of the unperturbed Hamiltonian is the sum of the Hamiltonians for the separated molecules, but this is not symmetric with respect to permutations of electrons on one molecule with electrons on the other. The order of a term in the perturbation expansion then becomes undefined,129 and although this difficulty can be overcome,130 the application to large systems is probably not in sight. 

Figure 6.1 Schematic representation of one of the channels of the Is-1 Ne+ Auger decay one of the valence electrons (2s) is filling the core vacancy while another one (2p) is ejected into continuum. The same final state results also from the 2p —> Is recombination and 2s ionization (not shown here). The former ( direct ) and the latter ( exchange ) contributions interfere due to electron indistinguishability.

There are two ways of combining [Pg.49]

The product /J 1 ) j/,(2) implies that electron 1 is resident in orbital a and electron 2 is in orbital b and the product j/ (2) /, (l) implies that electron 2 is resident in orbital a and electron 1 is in orbital 6. Both products by themselves violate the rule of electron indistinguishability. The linear combinations of the two ways of placing two electrons in the two orbitals represent the method of allowing for the indistinguishability of the two electrons. 

We can symmetrize the a(l)j8(2) and j8(l)a(2) wavefunctions to obtain two new functions that reflect electron indistinguishability. We do this by using the two as a basis set from which we construct linear combinations, in the same way that we combined the 2s and 2p orbitals to create two sp hybrid orbitals in Eqs. 2.5 and 2.6 ... 

The quantum theory requires electron indistinguishability, and thus, it is also possible that each electron is associated with the opposite nucleus. Consequently, the wavefunction... 

To obtain this expression, we have omitted all atomic orbital configurations for which two or more electrons occupy the same atomic orbital with the same spin quantum numbers. Such configurations are forbidden by the Pauli exclusion principle. A derivation of the above linear combination that takes proper account of electron indistinguishability is provided in Refs. 9a, b. 

A symmetry that holds for any system is the permutational symmetry of the polyelectronic wave function. Electrons are fermions and indistinguishable, and therefore the exchange of any two pairs must invert the phase of the wave function. This symmetry holds, of course, not only to pericyclic reactions. 

The total Hamiltonian operator H must commute with any pemiutations Px among identical particles (X) due to then indistinguishability. For example, for a system including three types of distinct identical particles (including electrons) like Li2 Li2 with a conformation, one must satisfy the following commutative laws ... 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry Because the N Electrons are Indistinguishable Eermions... 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

The electron-optical performance of the EPMA system is indistinguishable from that of a conventional scanning electron microscope (SEM) thus, EPMA combines all of the imaging capabilities of a SEM with quantitative elemental analysis using both energy- and wavelength-dispersive X-ray spectrometry. ... 

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