Quantum uncertainty: what to teach?

Abstract

We present a new secondary school teaching method of quantum uncertainties of two-state systems. Intending to be a material teachable in schools, only two-state systems described by real numbers can be considered. An elementary argumentation based on school statistics leads to the identification of the uncertainty of a physical quantity in such systems with the standard deviation of two random variables. We provide a qualitative picture on the state-dependence of the uncertainty, leading to a pictorial representation in the form of four petals of a flower. When considering the product of uncertainty of two essentially different physical quantities we conclude that the general feature: “if the measurement of one of the quantities is certain, the other remains uncertain”, cannot be faithfully expressed by means of an inequality, the product has no lower bound different from zero. The application of techniques used by school materials for teaching quantum physics leads to an exact formula for the state-dependence of the uncertainty valid in any two-state system described by real numbers, in full harmony with the qualitative picture. We compare the two-state case with the celebrated Heisenberg position-momentum uncertainty relation and show that these are both specific facets, but only the Heisenberg relation can be expressed by an inequality. The latter hardly provides any hint on the uncertainties of physical quantities in two-state systems. We conclude that the two-state approach is worth teaching in schools also in relation to the uncertainty relation, even if the Heisenberg relation is not part of the curriculum.