The authors have investigated the non-stationary self-focusing of Gaussian laser pulse in cold quantum plasma. In case of high dense plasma, the nonlinearity in the dielectric constant is mainly due to relativistic high intense interactions and quantum effects. In this paper, we have introduced a ramp density profile for plasma and presented graphically the behavior of spot size oscillations of pulse at rear and front portions of the pulse. It is observed that the ramp density profile and quantum effects play a vital role in stronger and better focusing at the rear of the pulse than at the front in cold quantum plasmas.