  The absence of a characteristic momentum scale in the pseudo-potential
description of atomic interaction in ultracold (two-component Fermi) gases is
known to lead to divergence in perturbation theory. Here we show that they also
plague the calculation of the dynamics of the total energy following a quantum
quench. A procedure to remove the divergence is devised, which provides finite
answers for the time-evolution of the total energy after a quench in which the
interaction strength is ramped up according to an arbitrary protocol. An
important result of this analysis is the time evolution of the asymptotic tail
of the momentum distribution (related to Tan's contact) and the contact for a
linear interaction ramp are obtained, as a function of the interaction ramp
time in the crossover from the sudden quench to the adiabatic limit are
reported. In sudden quench limit, the contact, following a rapid oscillation,
reaches a stationary value which is different from the equilibrium one. In the
adiabatic limit, the contact grows quadratically in time and later saturates to
its equilibrium value for the final value of the scattering length.
