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8/21/03 |
Unit 5Statistical Quantum Mechanics |
At the beginning of the twentieth century classical statistical mechanics was well developed and thoroughly understood. Furthermore, it was known to be extremely general in its application, suitable for essentially any system of a large number of particles. Therefore, it was extremely successful in predicting many of the general thermodynamic properties of macroscopic objects in terms of their microscopic components. Nevertheless, classical statistics failed numerous times to predict correctly certain properties one would expect to be correctly predicted.
One of the reasons for such failures is that classical particles, at least in principle, can all be distinguished from one another. But nature seems unable to make such distinctions in the case of certain fundamental particles. In other words, all electrons are the same, all hydrogen atoms are the same, etc.
Quantum statistics guarantees this indistinguishability of particles by requiring that the probability density of the system remain unchanged when two identical particles are interchanged. Since the probability density is the square of the wavefunction, this means that either the wavefunction itself is unchanged by a particle interchange or else the wavefunction reverses its sign when two particles are interchanged. In the first case, we say the wavefunction is symmetrical with respect to an interchange of particles, that the particles are bosons, and that the particles obey Bose-Einstein statistics. In the second case, we say the wavefunction is antisymmetrical with respect to particle interchange, that the particles are fermions, and that they obey Fermi-Dirac statistics.
With the application of these two types of quantum statistics to the appropriate physical situation, many if not all of the failures of statistical mechanics were rectified. Today, quantum statistics is as firmly founded upon experimental observation as classical statistics was at the beginning of the twentieth century.
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