Two Kinds of Uncertainty? Variances and Resolution in Quantum Bound States

  In (1), which deals with entropic uncertainty relationships, it is pointed out that  x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values.

  We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if  the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. 

  We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.