Energy quantization in nanoscale materials is manifested in a “blue-shift” of the emission spectrum of nanoparticles of decreasing size. The phenomenon is known as the “quantum size effect,” namely, the increasing gaps between energy levels for a spatially confined particle. The effect is demonstrated by solving the one-dimensional Schrödinger equation for the “particle in a box” model. The confining potential translates into boundary conditions, which result in energy quantization, where the corresponding standing wave solutions demonstrate remarkable differences from the classical description. Different energy levels are obtained by changing boundary conditions to periodic, for a “particle on a ring,” where the phenomenon energy level degeneracy is introduced. Extending the discussion to multidimensional “boxes” enables one to analyze the energy spectrum and the density of states nanostructures including quantum dots, wires, and wells, with references to the devices based on a two-dimensional electron gas, and to quantized conductance through point contacts.

We learn the language of the wave function, which is the representation of the quantum state vector in position space. We introduce the position and momentum operators and learn the rules for translating bra-ket formulae to wave function formulae. We use these new tools to solve the infinite square potential energy well problem and the finite square well problem.