Absolutely Small - Michael D. Fayer [43]
Quantum chemistry is mathematically very intense. Even the quantum mechanical calculation of the simplest atom, the hydrogen atom, is mathematically complex. The hydrogen atom consists of a single electron bound to a single proton. The proton, which is the nucleus of a hydrogen atom, is positively charged and the electron is negatively charged. It is the attraction of the negative electron to the positive proton that holds the hydrogen atom together. The detailed calculation of the energy levels of the hydrogen atom will not be presented, but in the next chapters, the results of the calculations are discussed in some detail. The calculations give the hydrogen atom energy levels and the hydrogen atom wavefunctions. The wavefunctions, that is, the probability amplitude waves for the hydrogen atom, are the starting point for understanding all atoms and molecules. Atoms and molecules are complicated because they are absolutely small three-dimensional systems, and it is necessary to deal with how protons and electrons interact with each other.
THE PARTICLE IN A BOX—CLASSICAL
There is a very simple related problem called the particle in a box. We can solve this problem without complicated math. The solutions to the particle in a box problem make it possible to illustrate many of the important properties of bound electrons, such as quantized energy levels and the wavelike nature of electrons in bound states. Before analyzing the nature of an electron in an atomic-sized one-dimensional box, the classical problem of an ideal one-dimensional racquetball court will be discussed so that the differences between a classical (big) system and a quantum mechanical (absolutely small) system can be brought out.
Figure 8.1 is an illustration of a perfect “box.” It is one dimensional. The walls are taken to be infinitely high, infinitely massive, and completely impenetrable. There is no air resistance inside the box. In the figure, the inside of the box is labeled as Q = 0, and outside of the box, Q = ∞. Earlier we said that a free particle is a particle with no forces acting on it. For a force to be exerted on a particle, there must be something for the particle to interact with. For example, a negatively charged particle like an electron can interact with a positively charged proton. The attractive interaction between the oppositely charged particles will exert a force on the electron. In the description of electron steering in a CRT (see Figure 7.3), an electric field exerted a force on the electrons, causing them to change direction. A measure of the interaction of a particle with some influence on it, such as an electric field, is called a potential and has units of energy. In the figure, the potential is labeled Q. Inside the box, Q = 0, just as in a free particle. This means the particle is not interacting with anything inside the box. There are no electric fields or air resistance. However, outside the box, Q = ∞. An infinite potential for the particle means the particle would have to have infinite energy to be located in the regions outside the box. Q = ∞ is just the formal way of stating that the walls are perfect. There is no possibility of the particle penetrating the walls or going over the top of the walls no matter how much energy it has. Therefore, if a particle is put inside the box, it is going to stay inside the box. There is no way for it to escape. In this sense, the particle is bound inside the box. It can be in the space of length L, but nowhere