Playing with elastic bands to understand quantum mechanics

Sourav Banerjee

This chapter concerns the basic understanding of quantum mechanics from an engineering perspective. Understanding the topological behavior of elastic and acoustic waves demands physical knowledge and the mathematical treatments developed for quantum mechanics. This chapter thus presents quantum mechanics and its relation to the understanding of acoustic and elastic wave propagation for quantum analog applications. It is assumed in this chapter that the readers are familiar with the fundamentals of modern physics and the construct of an atom with nucleus and cloud of electrons. Assuming this familiarities, first the Schrödinger equation is derived. The derivation employed the concepts of relativistic particles and their properties of wave-particle-duality. Schrödinger wave equation and an elastic wave equation are solved in a one-dimensional system, simultaneously, to show the respective equivalency. Specific features such as the uncertainty principle of angular momentum and the relation to the Pauli matrices are derived for the eigenspinors. Next the Schrödinger equation is solved in a periodic crystal with the Bloch solution, and sample energy bands are presented., It is explained how, in these energy bands at a specific energy level, the Dirac equation for relativistic particle can help find the Hamiltonian operator. It is explained how using the Hamiltonian for spinors could help understand the band structure with their respective spin behavior at Dirac cones. These features and understanding will be useful for quantum analog applications of acoustic and elastic waves.

K.G. Nichols, E. V. Vernon

Jean Bricmont

Massimiliano Sassoli de Bianchi

Abstract From the beginning of his research, the Belgian physicist Diederik Aerts has shown great creativity in inventing a number of concrete machine-models that have played an important role in the development of general mathematical and conceptual formalisms for the description of the physical reality. These models can also be used to demystify much of the strangeness in the behavior of quantum entities, by allowing to have a peek at what’s going on behind the “quantum scenes,” during a measurement. In this author’s view, the importance of these machine-models, and of the approaches they have originated, have been so far seriously underappreciated by the physics community, despite their success in clarifying many challenges of quantum physics. To fill this gap, and encourage a greater number of researchers to take cognizance of the important work of so-called Geneva-Brussels school, we describe and analyze in this paper two of Aerts’ historical machine-models, whose operations are based on simple breakable elastic bands. The first one, called the spin quantum-machine, is able to replicate the quantum probabilities associated with the spin measurement of a spin-1/2 entity. The second one, called the connected vessels of water model (of which we shall present here an alternative version based on elastics) is able to violate Bell’s inequality, as coincidence measurements on entangled states can do.

Tarun Biswas

Quantum mechanics courses focus mostly on its computational aspects. This alone does not provide the same depth of understanding as most physicists have of classical mechanics. The understanding of classical mechanics is significantly bolstered by the intuitive understanding that one acquires through the playing of games like baseball at an early age. It would be good to have similar games for quantum mechanics. However, real games that involve quantum phenomena directly are impossible. So, computer simulated games are good alternatives. Here a computer game involving three interacting quantum particles is discussed. It is hoped that such games played at an early age will provide the intuitive background for a better understanding of quantum mechanics at a later age.