Freely jointed molecular chain: dynamic variables, quantization, and statistical mechanics

Abstract

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 momentum –a_j,c and angular momentum l_j,c associated to the jth ‘‘bond’’ (j=1,...,N-1). The classical Poisson brackets of the Cartesian components of -a_j,c and l_j,c 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 l_j. Quantum-mechanical transverse momenta (e_j) are constructed. The commutators of the Cartesian components of e_j and l_j satisfy a closed Lie algebra, formally similar to the classical one for Poisson brackets. Using e_j’s and l_j’s, the quantum theory is shown to be consistent by itself and, via the correspondence principle, with the classical one. Several properties of ej and the modified l_j 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)!]^−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.