Thickness dependence of magnetic anisotropy energy (MAE) and spin polarization in Fe and Co multilayers on a Pt(001) substrate are investigated with ab initio techniques. As the thickness of the thin films is increased, an oscillatory behavior of the MAE of the systems is observed. For Fe multilayers, this even results in the periodic switching of the easy axis from out of plane to in plane. Capping the multilayers with Pt, in most cases, leads to a strong enhancement of the MAE. Our calculations give clear evidence that such oscillatory behavior of MAE can be attributed to spin-polarized quantum well states (QWS) existing in thicker multilayers. The spin polarization of the QWS is also shown to have a profound thickness dependence. Finally, we show that both MAE and spin polarization of the QWS are sensitive to external electric field and can be tailored therewith.