We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $hat{A}$ and $hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Ne
umann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {em probe} variable $Q_2$ conditioned on the first {em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $delta a$ around $a^{(n)}$. The width of this statistical distribution as function of $delta a$ constitutes an {em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.
We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a sp
ecific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.
