A practical method for finding free energy barriers for transitions in high-dimensional classical and quantum systems is presented and used to calculate the dissociative sticking probability of H2 on a metal surface within the transition state theory. The reversible work involved in shifting the system confined to a hyperplane from the reactant region towards products is evaluated directly. Quantum mechanical degrees of freedom are included by using Feynman path integrals with the hyperplane constraint applied to the centroid of the cyclic paths. An optimal dividing surface for the rate estimated by the transition state theory is identified naturally in the course of the reversible work evaluation. The free energy barrier is determined relative to the reactant state directly so that an estimate of the transition rate can be obtained without requiring a solvable reference model for the transition state. The method has been applied to calculations of the sticking probability of a thermalized hydrogen gas on a Cu(110) surface. The two hydrogen atoms and eight surface Cu atoms were included quantum mechanically and over two hundred atoms in the Cu crystal where included classically. The activation energy for adsorption and desorption was determined and found to be significantly lowered by tunneling at low temperature. The calculated values agree quite well with experimental estimates for adsorption and desorption. Dynamical corrections to the classical transition state theory rate estimate were evaluated and found to be small.