On a Minimum Property of the Free Energy

a minimum ; the value of the integral is the corresponding eigenvalue El of the Hamiltonian H. It is also well known that this property leads to a powerful method of approximating, at least qualitatively, 4,1 and Er, by minimizing (1) among a restricted class of functions. No similarly simple minimum property exists for the higher eigenvalues . One can obtain the nth eigenvalue by minimizing (1) with the subsidiary condition that >G be orthogonal to ¢l, 42, y n-r• However, this procedure becomes progressively more cumbersome as n increases and besides it is of no use unless 4,1, 02, • • • , 4'n-1 are very exactly known. For problems of thermal equilibrium one is often concerned with the free energy, rather than with the individual eigenvalues . It is the purpose of this note to draw attention to a simple minimum property of the free energy which may be considered as a generalization of the variation principle for the lowest eigenvalue. The free energy has the following property: If v1, W2 , • • • , pn, • • • are an arbitrary set of orthogonal and normalized functions, and