Bounds on transition probabilities and cross sections
L Aspinall, Ian Colin Percival · Journal of Physics B Atomic and Molecular Physics · 1968
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 ) > is where |ψ tr (- ∞) > coincides with the initial state, l = H -iℏ d/d t 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.