Abstract: This paper discusses the expression of the position-momentum uncertainty relation for the case of a free particle moving in a one-dimensional finite region, which is subjected to the box normalization treatment. It is found that for such a specific physical situation, although the physical thesis that the position coordinate and the momentum of a particle cannot be exactly measured simultaneously certainly holds, the mathematical form of the usual position-momentum uncertainty relation needs some modifications. In this paper the concrete form of the position-momentum uncertainty relation for such a situation has been obtained, which deepens our understanding of the uncertainty principle in quantum mechanics. From the viewpoint of the cognition of the momentum uncertainty of a free particle subjected to the box normalization, a discussion of the physical cognition of the infinite volume limit is also made.

Key words: uncertainty relation, periodic boundary condition, infinite volume limit