Measuring the Momentum of a Quantum System using Position Measurements A basic way to measure the momentum of a charged particle is to know that (classically) it follows a circular trajectory in a uniform magnetic field perpendicular to the plane of its trajectory, with a radius $R$ given by $$R=\frac{p}{eB}$$ in the nonrelativistic regimes, where p is the momentum of the particle, e its charge, and B the magnetic field strength. 
Reading in my quantum mechanics textbooks, the basic idea thus to measure momentum is by sending a stream of particles through a slit as shown in the figure below, which guarantees (classically) that the momentum is essentially in the $+\hat{z}$ direction. Beyond is a region of uniform magnetic field, followed by a set of detectors. If the intensity of the source is very weak, flashes will appear on the detectors one by one; and using the expression for $R$ we can associate with each flash a value of momentum in the $+\hat{z}$ direction depending on the position of the flash, $p = ReB$.

My confusion however comes in that in order to measure the momentum, we have measured the position of the particle as well! If we can see the point at which the detector flashed, we have certainly determined the position. Thus again, we know the position, and we also know the momentum. This seems to violate Heisenberg's Uncertainty principle! What am I missing here?
 A: The problem is that the equation you wrote cannot be correct at a fundamental level. In fact the equation is not compatible with the canonical commutation relation
$$
\big [ x_\alpha , p_\beta \big ] = i \hbar \delta_{\alpha,\beta} \, . \ \ \ \ (1)  
$$
In other words $\mathbf{p}$ cannot be a function of $\mathbf{x}$ otherwise the variables would commute. 
Not surprisingly Eq. (1) is at the basis of Heisenberg's uncertainty principle. 
A: 
Thus again, we know the position, and we also know the momentum. This seems to violate Heisenberg's Uncertainty principle! What am I missing here?

Your argument misses that all measurement values have uncertainties. Actual measurement will determine both position and momentum magnitude at the moment of impact with some uncertainties, and these cannot be both made arbitrarily small.
The formula
$$
p = ReB
$$
assumes that the particle exits the slit radially into the cylinder. This is not exactly true, because wide slit allows for different angles of momentum already outside the cylinder, and narrow slit, although it limits direction of momentum outside, causes diffraction, where momentum when particle is different from the momentum when outside. The formula above does not take into account different possible directions of momentum inside the cylinder right after the particle passes the slit, so it is not accurate enough to challenge the uncertainty relations.
