The upper bound on the total transition probability p(t) previously obtained by Aspinall and Percival in 1967 is generalized to systems with an infinite or arbitrary finite number of states and to arbitrary time-dependent trial states |ψ_tr(t) > . The bound for transitions from |ψ_tr(t) > iswhere |ψ_tr(- ∞) > coincides with the initial state, l = H -iℏ d/dt and H is the time-dependent Hamiltonian. A bound on total inelastic cross sections is obtained for collisions between heavy systems with sufficiently short-range interaction potentials. The method is applied in its simplest form to H-H and p-H collisions, but gives results of little physical significance. A reason is given and possible remedies suggested. A variational method for time-dependent states is proposed.