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The motion of a particle bounded to a plane curve C is discussed in detail from the point of view of quantum mechanics, considering a sequence of increasingly stronger potentials which limit the motion of the particle to an infinitely thin neighbourhood of C. This procedure gives a well defined result provided that one takes some simple precautions (explained in the text) in the choice of the constraining potentials. The result obtained in this way presents an interesting property which has no classical analogue: in its motion along the curve the particle 'sees' a longitudinal potential which is a function of the curvature. The consequences of this result, as well as some further information about more complex situations, are presented using the simplest language and a minimum of mathematical sophistication in order to render this paper legible and useful to both teachers and students.