Can a rifle scope be mounted on a shotgun? Of course it can. Mechanically speaking, the same type of mounts found on most rifles are also found on many shotguns and so simply mounting such a scope would not be a challenge. Now that we have that question answered lets move on to a more important question: Will it work? And the answer to that question is: It depends.
While shotguns and rifles are both considered "Longarms" their function is quite different. A shotgun uses shells which are loaded with small pellets that are intended to be fired outwards in a spreading pattern. This makes the shotgun ideal for close range (45 yards or less) shooting at fast moving targets such as running rabbits or flying ducks. Although there are "Slug" loads for shot guns which can make them effective weapons for deer sized game out to a hundred yards, the vast majority of shotgun use involves the use of "Birdshot" loads for fast moving close in targets.
Rifles on the other hand, fire a single bullet. In the case of high powered rifles, this single projectile can have a muzzle velocity that is almost three times that of a shotgun and, depending upon the specific type of rifle and the skill of the shooter, can be effective and accurate on large game out to over 400 yards.
Rifle scopes are designed to place precision shots over distances that are usually greater then those ranges for which shotguns are commonly used. The nature of such shooting will usually allow the hunter to brace himself and take the time to get a precise sight picture before firing. The magnification power of rifle scopes intended for hunting may vary from 4 on up to 12 power. Variable power scopes are also popular, with the most commonly types being the 3x9 and the 4x12 magnifications.
There are two main drawbacks in trying to use a rifle scope on a shotgun. The first being eye relief. Eye relief is the precise distance your eye must be from the scope lens to obtain a clear view. The second is the field of vision of the scope. The field of vision is the actual size of the area which appears in the scope picture. The higher the magnification gets, the smaller the field of vision becomes. Keeping in mind that shotguns are generally used for the taking of smaller fast moving game that is close in, it would be very difficult to first find the game in the relatively small field of vision a rifle scope provides and even more difficult to then get the cross hairs on target and a shot off before the quarry moved out of range.
There is another type of scope that is actually designed to be used effectively on either shotguns or rifles. These are the relatively new "Illuminated reticle" and "Holographic" type scopes that use a small battery to illuminate a dot which, when properly aligned and placed on the target, will provide a hit. The low magnification of these scopes (Usually only one or two power) provides a wide field of vision, therefore making it possible for quick target acquisition. The US military has adopted this scope for mounting on M-16 rifles. While these scopes do not provide the same long range pinpoint accuracy provided by standard rifle scopes they are very good for firing quick shots and hitting man sized targets out to a hundred yards or a bit more. In regards to eye relief, these scopes are not nearly as finicky as standard rifle scopes and that, coupled with their low magnification power also make them suitable for shotgun shooting. Quality in such scopes will vary and with optics, you usually get what you pay for. A cheap Chinese made scope can be had for as little as $30.00 while those scopes which have been approved for military use may run considerably more. An example of this is the Eurotech Holographic rifle sight which is currently "On sale" at several sites for $629.00.
So, can you mount a rifle scope on a shotgun? Sure, but if you want it to be effective, it will have to be the right kind of a rifle scope.