Optimal Measuring Apparatus

An upper limit for the accuracy of the measurement of a simple quantity which does not commute with a conserved quantity is obtained in terms of the "size" of the apparatus. The "size" of the apparatus is defined as the mean square value ${\hslash }^2{M}^2$ of the conserved quantity for the apparatus which is, in the example chosen, the $z$ component of the angular momentum. The measured quantity is the projection of a spin in a perpendicular direction. It is found that the probability of an unsuccessful measurement is at least $\frac{1}{8M^2}$.