Wave-mechanical position matrix

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 quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

To ensure that the heretical ideas of wave mechanics remain permanently suppressed it was necessary to show that matrix mechanics provided a complete and infallible description of the atomic world. The famous debates against Einstein were obviously conducted by Bohr to defend this position. It is generally agreed that Bohr prevailed in this confrontation, and his stance, no longer considered in dispute, was accepted as the orthodox interpretation of all quantum theory. 

Shortly after de Broglie s theory appeared, new mathematical treatments, known collectively as quantum mechanics, were introduced to describe the behaviour of electrons in atoms. Erwin Schrodinger (1887-1961) developed wave mechanics, and Werner Heisenberg (1901-1976) used a different approach called matrix mechanics. Heisenberg s treatment led to the uncertainty principle, which stated that it is impossible simultaneously to determine precisely both the position and the velocity of the electron. Other workers in the field were Max Born (1882-1970), Pascual Jordan and Paul Dirac (1902-1984). Although the approaches of Schrodinger and Heisenberg seemed very different, it was ultimately realised that there is a fundamental unity between them. 

The Ci k = 2, 4, 6) are adjustable parameters, Si is the spin operator for the i-th electron and S is the total spin operator. The c parameters must be negative if the contraction of the radial wave functions corresponds to parallel spins. Positive values for C6 have been reported however (Newman et al. 1982) and were explained to be due to a charge-transfer mechanism and to covalent effects. Only three additional parameters have to be introduced. The caleulations can be done in practice by replacing in the expression for the crystal-field matrix element (eq. 49) the double reduced matrix element yi "T SL ),hy... 

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