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.

Dirac’s Approach to Quantum Mechanics in Physics Teacher Education: From Linear to Circular Polarisation

“…[9,10] Since this is a university course, this paper extends the secondary school material to university level by providing the appropriate physical background knowledge. This paper extends the results of a previous pilot experiment in which a part of the extension to two-dimensional matrix formalism using real numbers was presented and secondary school statistics was introduced for teacher trainees [11,12]. Another focus is a step-by-step exploration of the linear algebra behind quantum phenomena and the introduction of complex states via a physical phenomenon: circular polarisation.…”

“…The seminar augmenting the main lecture on quantum mechanics includes 13 lessons (90 minutes each) at ELTE. In the pilot project reported here, the first 2 lessons cover the secondary school material [2,[11][12][13]. Students explore the phenomena of polarisation through experiments with polarisers, the same as in secondary school.…”

“…They then explore the elementary features of two-state quantum mechanics using only real numbers, just as secondary school students do, but in a much shorter time. As the papers [11,12] show, the term "eigenvalue" is used in addition to "property" (the term "property" is a secondary school analogue of eigenvalue in [2]). The third and fourth lessons go beyond the secondary school level, students explore the linear algebra behind quantum phenomena using two-dimensional vectors and matrices as reported in [11].…”

“…If only a single photon falls on a polariser, then the probability of transmission is cos 2 ϑ (the Malus law) where ϑ is the angle between the polarisation direction of photon and the transmission axis of the polariser. [2] These experiments are supplemented by the previous pilot projects [11][12]: measured values can be assigned to the permitted outcomes with a polariser: λ1 = 1 and λ2 = 0 correspond to the event of free passing through and full absorption. These assigned values are called eigenvalues.…”

“…To characterise the possible outcomes when measuring a given quantity (polarisation) A, students perform statistical calculations as e.g., paper [11][12] suggests. Provided the probability p1 = cos 2 ϑ (p2 = sin 2 ϑ) of measuring eigenvalue λ1 (λ2) is known in a sequence of repeated identical experiments, the expected value ⟨A⟩ of quantity A in a given state is…”