Entangled quantum particles in an infinite square well: knowledge of the whole versus knowledge of the parts

Entangled states of composite quantum systems exhibit one of the most distinct and non-classical features of the quantum mechanical description of Nature, first pointed out by Schroedinger: 'Maximal knowledge of a total system does not necessarily imply maximal knowledge of all its parts'. We provide an elementary illustration of this fundamental aspect of quantum physics by considering a system of two particles in an infinite, one-dimensional square potential well. In contrast to standard introductory presentations of quantum entanglement, our present considerations do not require density matrix formalism, nor explicit use of the tensor product structure for the description of composite quantum systems.