  The Dirac delta function can be defined by the limitation of the rectangular
function covering a unit area with decrease of the width of the rectangle to
zero, and in quantum mechanics the eigenvectors of the position operator take
the form of the delta function. When discussing the position measurement in
quantum mechanics, one is prompted by the mathematical convention that uses the
rectangular wave function of sufficiently narrow width to approximate the delta
function in order to making the state of the position physical. We argue that
such an approximation is improper in physics, because during the position
measurement the energy transfer to the particle might be infinitely large. The
continuous and square-integrable functions of both sharp peak and sufficiently
narrow width can then be better approximations of the delta function to
represent the physical states of position. When the slit experiment is taken as
an apparatus of position measurement, no matter what potential is used to model
the slit, only the ground state of the slit-dependent wave function matters.
