Freely jointed molecular chain: Dynamical variables, quantization, and statistical mechanics (original) (raw)

A freely rotating linear chain, formed by N (N~2) atoms with N-1 "bonds" of fixed lengths, is studied in three spatial dimensions. The classical (c) theory of that constrained system is formulated in terms of the classical transverse momentuma, , and angular momentum lJ, associated to the jth "bond" (j =1,. .. , N-1). The classical Poisson brackets of the Cartesian components ofa, , and 1, , are shown to close an algebra. The quantization of the chain in spherical polar coordinates is carried out. The resulting "curved-space" quantization yields modified angular momenta I,. Quantum-mechanical transverse momenta (e,) are constructed. The commutators of the Cartesian components of e, and 1j satisfy a closed Lie algebra, formally similar to the classical one for Poisson brackets. Using e~'s and 1,-'s, the quantum theory is shown to be consistent by itself and, via the correspondence principle, with the classical one. Several properties of e, and the modified 1, are given: some sets of eigenfunctions (modified spherical harmonics, etc.) and uncertainty relations. As an example, the case of N =3 atoms in two spatial dimensions is worked out. The peculiar properties of the chain regarding distinguishability at the quantum level play an important role in justifying the absence of a "Boltzmann counting" factor [(N-1)! ] in its classical statistical distribution. The physical limitations and the methodological virtues of the model at the classical and quantum levels, and its relationship to previous works by different authors, are discussed.