Physical Systems

Abstract

Matter around us and within us consists of electrons and atomic nuclei, which are governed by the laws of quantum mechanics Relativistic effects arise only in the fine details (cf. III, § 1), so the forces of primary relevance are electrostatic and, for cosmic bodies, gravistatic (nonrelativistic). Moreover, the precise nature of the atomic nuclei is of little consequence on the macroscopic scale, so they can be considered as point charges. In order to understand the gross features of matter we shall study a Hamiltonian

$${H_{{\text{mat}}}} = \sum\limits_{i = 1}^M {\frac{{|{p_i}{|^2}}}{{2{m_i}}} + \sum\limits_{i >j} {\frac{{({e_i}{e_j} - k{m_i}{m_j})}}{{|{x_i} - {x_j}|}}} } $$

(4.1)

for ordinary matter. The first important issue to confront is that of why macroscopic bodies behave classically; in what sense is the thermodynamic limit N → ∞ equivalent to the classical limit h →* 0? There are a variety of ways to pass to the limit N → ∞. In this section we begin by letting the nuclear charge Z and the nuclear masses both tend to infinity, while continuing to neglect gravity. This will permit a rather explicit mathematical treatment, as the action is determined by an average field, and the single-particle model becomes exact. The same will be true in § 4.2 when we deal with cosmic bodies, for which gravitation predominates. However, macroscopic bodies on the scale of humans are far from these limits: nuclear charges are for the most part small, and yet gravitation is of little importance. In this intermediate range of normal matter it would be too much to hope for an explicit solution. Section 4.3 will discuss this case, but the results will be confined to general existence theorems and rather crude bounds on the values of observables of physical interest.

Among the best examples of large quantum systems are atoms and molecules with highly charged nuclei. Classical features arise in the limit Z → ∞, N → ∞, except that the Fermi statistics continue to have an important effect.