The problem of zero point energy in classical trajectory computations is discussed and illustrated by an example of dissociation when the zero point energy is used to provide the required energy. This is not possible in quantal dynamics. A proposed route to the alleviation of the problem, based on using classical-like trajectories which mimic the solution of the (expectation values) of Heisenberg equations of motion, is discussed. In general, one cannot simultaneously correct for all possible expectation values, so the remedy is at best partial. The variable whose expectation value and variance is to be handled correctly is examined in detail for a one-dimensional anharmonic potential, and is identified with the logarithmic derivative of the wave function in the Wentzel-Kramers-Brillouin (WKB) approximation. The multidimensional case is also discussed and it is pointed out that the zero point energy problem can be particularly severe for systems which exhibit a locally unstable classical motion.