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problem. We imagined an electron confined to a very small one-dimensional box, as shown in Figure 8.1. The particle in a box is a useful problem because the math is simple enough to find the quantized energy levels without great difficulty. A formula was obtained that showed that the energy states of the particle in a box come in discreet steps that depend on a quantum number n, where n is an integer that starts at 1 and can take on any integer value. However, it was pointed out that this is a very artificial example of quantum confinement. In nature, there are no truly one-dimensional systems. Furthermore, the walls of the box are infinitely high and completely impenetrable. This is also physically unrealistic. As discussed in connection with the photoelectric effect in Chapter 4, if a photon has sufficient energy to overcome the binding energy of electrons to the atoms in a piece of metal, the interaction of the photon with an initially bound electron can eject the electron from the metal (see Figure 4.3).

However, for a number of reasons it is very useful to examine the particle in a box. First, we found that the energy levels are quantized (see Figure 8.6). In contrast to classical mechanics, the energy an electron can have when confined to a box the size of an atom or molecule is not continuous. Rather, the energy comes in discrete steps. A photon of the right energy can excite an electron from one energy level to another (see Figure 8.7). The energy of the photon must match the difference between the energy of the level the electron starts in and the energy of the one it ends up in. But in contrast to real systems, no amount of energy can cause the electron to come flying out of the box because the walls are infinitely high. This is a way of saying that an electron would have to have infinite energy to get out of the box. The box is an infinitely deep well and the electron is trapped in it; no finite amount of energy can overcome the infinite binding energy.

Another important feature of the particle in the box is the nature of the wave functions. The wave functions are probability amplitude waves that are related to where the electron is inside of the box (see Figure 8.4). The square of these wavefunctions (Figure 8.5) gives the probability of finding the electron in some region of space. The probability amplitude waves have nodes. As the quantum number increases, the number of nodes increases, as can be seen in Figure 8.5. Nodes are places where the probability of finding a particle, such as an electron, is zero.

Atoms are real three-dimensional physical systems in contrast to the one-dimensional particle in the box. The three-dimensional nature of atoms is a major difference, but as discussed in Chapter 10, some of the most important features of the quantum mechanical description of atoms are qualitatively similar to the particle in the box results. Atoms have quantized energy levels. They have wavefunctions that have an increasing number of nodes as the quantum number increases. Many other things are very different. The quantum states of atoms have associated with them several quantum numbers, and because atoms are three dimensional, their wavefunctions have three-dimensional shapes. These properties of atoms will be discussed in detail beginning in Chapter 10 with the simplest atom, the hydrogen atom. But first, we will look at some of the early observations that indicated that classical mechanics was not going to be able to describe atoms.

THE SOLAR BLACK BODY RADIATION SPECTRUM

We have introduced the experimental method of spectroscopy, that is, taking a spectrum of the light that comes out of a system or the spectrum of the light that is absorbed by a system. A spectrum is just a recording of the intensity of the various colors of light. We measure the amount of light at each wavelength (color). When we refer to colors, we don’t mean only the colors we can see, the visible spectrum, but also longer wavelengths (lower energy), the infrared, and shorter wavelengths (higher energy), the ultraviolet.