Need help choosing bookshelf thickness
It will depend on your book load, and whether your "hard maple" is really sugar maple or not.

There is, of course, an easy way to deal with the calculations these days. Do be sure to read all the "notes" below the calculator. Shelf thickness (or "depth" in beam speak) cubed is indeed the correct factor, and why even small changes in thickness make large differences in stiffness.

http://www.woodbin.com/calcs/sagulator.htm

For lesser woods, 48" is a heck of a long unsupported span for a thin shelf. Actual sugar maple at 8 x 3/4" will do 50 lbs per foot within the "acceptable" range and typical book load is given as 20-40 lbs. Mind, that is based on the strength of "clear, straight grained samples" and "initial sag only." At 35 lbs/ft that shelf reaches the suggested target of 0.02 in/ft to allow for additional sag from sustained loading.

Silver maple is "borderline" at 40 lbs/ft. red, black, and bigleaf are "acceptable" at the same figure. All are somewhat less stiff than sugar maple.

IMPE you should be able to get 13/16" planed from 4/4 rough. If you want the appearance of a thinner shelf you can cut a long taper on the bottom front edge, which will reduce strength somewhat, but not as much as thinning the whole shelf width. 

At a higher level of difficulty in the "absurdly thin non-sagging shelf" direction you can mount steel rods, say 16" in from the ends and drill holes in the shelf (the difficulty is in getting the wall-mounted rods and the holes in the shelf to agree) for "invisible supports."