There's nothing worse than arriving home from the hardware store with shiny new tools, only to find that they won't work with your current tools. If you purchase a 9/16-inch drill bit, and come home to find your drill is equipped with a 3/8-inch chuck, simple math will tell you the bit just won't fit into the chuck. You're temporally out of luck. But all is not lost. There are a few things you can do to fix this problem.
Most drill manufacturers offer replacement chucks that you can add onto your existing drill. Once you have a 1/2-inch chuck--which will work fine with a 9/16-inch bit--you must remove the old chuck. Open the chuck as far as it will go. Determine the type of fastener that holds it in place inside the spindle. It's usually a hex type of Allen screw, but sometimes you get lucky and it's a simple Phillips screw. Either way, turn the screws clockwise to loosen them (they are reverse threaded). Insert a large, L-shaped Allen wrench into the spindle and tighten it securely. Gently tap the hammer on the Allen wrench several times to remove the chuck.
Replace the new 1/2-inch chuck following the above steps in reverse.
Locate a drill bit that's 9/16 inches at its cutting end, with a smaller spindle that will fit a 3/8 inch drill. These are fairly common, and will work well on all but the most extreme jobs.
Buy a 1/2-inch drive drill in the first place. The difference in cost between the two is negligible. The 1/2-inch drill can do everything the 3/8-inch drill can do, but not the other way around.