Section 4.1 Introduction
We have seen that particles have wave-like properties, and that the de Broglie relation, connecting the properties of momentum and wavelength, has far-reaching consequences for the behavior of atoms and other microscopic systems. The main consequence is that the idea of a particle as a point-like object having a precise position at any given time has to be replaced with a probability density describing a distribution of positions where the particle is likely to be found.
In the previous chapter, we introduced the idea of a wavefunction \(\psi(x)\) from which we can determine the probability density \(P(x) = |\psi(x)|^2\text{.}\) We also discussed Heisenberg's uncertainty principle, that states that there is a minimum combined spread in a particles position and momentum, a principle that also leads to a minimum kinetic energy for any confined particle. And at the end of the previous chapter, we introduced the Schrödinger equation which enables us to determine wavefunctions for particles in regions of given potential energy.
In this chapter, we show how Schrödinger's Equation predicts the quantization of energies for confined particles, i.e., only certain, well-defined energies are allowed. 1 Quantization of energies is a profound and distinctly quantum principle (i.e., not predicted at all from classical mechanics) which has extensive applications in modern technology. We will illustrate quantization of particle energies with a simple system — the particle in a box (also known as the infinite square well potential) — for which solutions of Schrödinger's Equation are readily obtained. We will discuss the properties of this system, which has recently led to the development of quantum dots for use in numerous applications.
We will also discuss how quantum systems absorb and emit light, a result that explains not only the colors that we see from many physical systems but which has also become a common tool for identifying the material constituents of a range of systems, from biological systems in microscopic studies up to stars and star-forming regions millions (and billions) of light years from the Earth.