variances of and can be calculated explicitly: The product of the standard deviations is therefore For all , the quantity is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when , in which case Constant momentum Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 according to where we have introduced a reference scale , with describing the width of the distribution—cf. nondimensionalization. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are Since and , this can be interpreted as a particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is such that the uncertainty product can only increase with time as Additional uncertainty relations Systematic and statistical errors The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation . Heisenberg's original version, however, was dealing with the systematic error, a disturbance of the quantum system produced by the measuring apparatus, i.e., an observer effect. If we let represent the error (i.e., inaccuracy) of a measurement of an observable A and the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A, then the inequality proposed by Ozawa — encompassing both systematic and statistical errors — holds: Heisenberg's uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the systematic error. Using the notation above to describe the error/disturbance effect of sequential measurements (first A, then B), it could be written as The formal derivation of the Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years. Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors and . There is increasing experimental evidence that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality. Using the same formalism, it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements (A and B at the same