A thought experiment is described for the process of position measurement, using ordinary quantum mechanics in detail. The treatment is nonrelativistic and essentially one-dimensional, and involves the perfectly-reflecting collision of two particles represented by Gaussian wave packets. From simultaneous momentum-time measurements on the measuriing particle (number 2) information concerning the position and momentum of the measured particle, number 1, is obtained both for a single measurement interaction and an ensemble. Momentum measurements on particle 2 are correlated with those on particle 1, a fact reflected in the use in quantum mechanics of the joint space of both particles. The only position of particle 1 which is meaningful for measurement is the collision position, x c . We have not solved the problem of finding an operator for x c , but we offer as a preliminary to the discussion of this problem a special semiclassical definition for the position measurement in terms of the intersection of the paths (with appropriate quantum spreads) of particle 2 before and after collision. By this definition x c is inferred from the initial conditions and the measured momentum and time of particle 2. The time measurement necessitates using a current density in the subspace of particle 2. It is found that the probability of observing x 1 to be at a given x c in a given range dx c is not in general approximated by the quantity ψ 1 ∗ψ 1 dx c as asserted by ordinary quantum mechanics, but rather by an expression ψ c ∗ψ c dx c which has a maximum for the same value as x 1 but has a variance depending on the initial states of both particle 1 and particle 2. An important result is that the uncertainty product of the third kind [see Part I, Ann. Phys. 46, 577 (1968)] Δ 2 x c Δ 2 p r1 , where Δ 2 p r1 is the variance of the momentum of particle 1 after collision, can be made less than h ̵ 2 4 . On the other hand, Δ 2 x c Δ 2 p r1 does have a lower limit for a model in which particle 2 is replaced by a gamma ray. This calculation suggests that the gamma-ray microscope example of Bohr and Heisenberg was not appropriate for illustrating the uncertainty principle. In the nonrelativistic case there is enough latitude in the choice of parameters that Δ 2 x c and Δ 2 p r1 can be made to depend on the independently specifiable variances of different particles; e.g., we can cause Δ 2 x c to be roughly equal to the intial-position variance of particle 2, and Δ 2 p r1 to be roughly equal to the initial-position variance of particle 1. We show that the projection postulate or the “reduction of the wavepacket” is not needed in this example, and that the state of the measured system after the interaction that yields a meaningful measurement of x c is a mixture. Our results raise the question of whether the physical implications of quantum mechanics go beyond the logical content of currently available axiomatic formulations of the theory.