  We discuss a technique for solving the Landau-Zener (LZ) problem of finding
the probability of excitation in a two-level system. The idea of time reversal
for the Schrodinger equation is employed to obtain the state reached at the
final time and hence the excitation probability. Using this method, which can
reproduce the well-known expression for the LZ transition probability, we solve
a variant of the LZ problem which involves waiting at the minimum gap for a
time t_w; we find an exact expression for the excitation probability as a
function of t_w. We provide numerical results to support our analytical
expressions. We then discuss the problem of waiting at the quantum critical
point of a many-body system and calculate the residual energy generated by the
time-dependent Hamiltonian. Finally we discuss possible experimental
realizations of this work.
