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A precise theory of the simultaneous meaSllrement of a pair of conjugate observables is necessary for obtaining the classical limit from the quantum theory, for determining the limitations of coherent quantum mechanical amplifiers, etc. The uncertainty principle, of course, does not directly address this problem, since it is a statement about the variances of two hypothetical ideal measurements. We will adopt the approach that there exist instantaneous inexplicable ideal measurements of a single observable. Just as von Neumann1 uses an ideal measurements together with all interaction to explain an indirect observation, we use ideal measurements together with iuteractions to explain the simultaneous measurements of an observable and its conjugate.