A ONE-DIMENSIONAL STATISTICAL-MECHANICS MODEL WITH EXACT TRANSFER INTEGRAL SOLUTION

A simple one-dimensional model with scalar variables - 1 less-than-or- equal-to s(n) less-than-or-equal-to + 1, coupled according to a Hamilt onian, H = -JSIGMA(n)\s(n)-s(n+1)\, is presented. This Ising-like mode l is an interesting example of a graduate-level statistical mechanics problem where the eigenfunctions and eigenvalues of an integral operat or, the transfer integral, can be determined exactly. This can be cont rasted with the determination of eigenfunctions and eigenvalues of dif ferential operators, such as the quantum mechanical finite-depth squar e well. In this problem transcendental equations leading to the eigenv alues and eigenfunctions are also obtained. The largest eigenvalue is identified and the free energy is determined. The thermodynamic proper ties depend on the sign of J, unlike classical Ising or Heisenberg mod els.