I have a motor with a 4mm shaft. I've purchased a propeller that's designed to fit a 1/2 inch shaft. How can I securely couple the two?
If the motor speed matches the speed range needed for the prop, then make an adapter with relevant locking screws or equivalent, balanced as necessary.
something like :
Following on from Solar Mike's answer, I'm not sure you'll be able to find something off-the-shelf with the diameters you have (mix of imperial and metric units), so you may need to get something custom made instead. I have sketched the following (not knowing what shaft lengths you had, I made a few assumptions - adjust to suit your needs):
The outer diameter is 1/2 inch, while the ID is 4mm. I have toleranced it as H7 hole so that it's a snug fit on the 4mm shaft. The adapter is secured onto the 4mm shaft with 2 M3 grub screws at 90 degrees, you can also use Loctite 638 if you want to be sure. I have added a 1mm chamfer on the end of the shaft adapter.
Use gears, mechanical losses will only be 1 % and that's negligible.