Absolutely Small - Michael D. Fayer [48]
with n being any integer, 1, 2, 3, etc. The integer n is called a quantum number.
We have obtained a very important result, the energies for an absolutely small particle in an absolutely small box. The results are closely related to electrons in atoms or molecules. As can be seen in the formula, the energies are not continuous because n can only take on integer values; the other parameters are constants for a particular system. We say that the energy is quantized. It can only have certain values, which are determined by the physical properties of the system and the quantum number.
A Discreet Set of Energy Levels
There is a discreet set of energy levels for a given mass, m, and a given box length, L. As the quantum number n takes on values, 1, 2, 3, etc., the energies are
Figure 8.6 is an energy level diagram for the first few energy levels of the particle in a box. The energy is plotted in units of h2/8mL2. To get an actual energy, it is only necessary to plug in particular values for m and L in the energy level formula. The plot shows the energy increasing as the square of the quantum number n. The dashed line locates where the energy is zero. In the quantum particle in a box, the lowest energy level does not have zero energy, in contrast to a classical particle in a box. In the classical racquetball court, the energy that the ball can have is continuous. By hitting the ball a little harder or slightly softer, the ball’s energy can be changed any amount up or down. Here, the quantum racquetball can only take on energies that have distinct values, as shown in Figure 8.6. As we discussed at the beginning of our analysis of the quantum particle in a box, the lowest energy is not zero. If the quantum particle in a box could have zero energy, it would violate the Uncertainty Principle.
FIGURE 8.6. Particle in a box energy levels. The quantum number is n. E is the energy, which increases as the square of the quantum number. The energy is plotted in units of h2/8mL2, so that it is easy to see how the energy increases. The dashed line is zero energy. The lowest energy level does not have E = 0, in contrast to a classical particle in a box.
PARTICLE IN A BOX RESULT RELATED TO REAL SYSTEMS
The particle in a box is a very simple example of a general feature of absolutely small systems. The energy of such systems is not necessarily continuous. The particle in the box is not a physically realizable system because it is one dimensional and it has “perfect” walls. However, atoms and molecules are real systems. The energy levels of atoms and molecules have been studied in great detail, and their quantized energy levels have been measured and calculated. Just as the energy levels of the particle in the box depend on the properties of the system, that is, the mass of the particle and the length of the box, the energy levels of atoms and molecules depend on the properties of the atoms and molecules.
Molecules Absorb Light of Certain Colors
Although the particle in a box is not a physically realizable system, features of this problem are also found in atoms and molecules. In the photoelectric effect, the incident photon energy is so great that electrons fly out of the piece of metal (see Chapter 4). For high enough energy, a photon incident on a molecule can also result in electron emission. However, for lower energy photons, when light shines on an atom or molecule, it can be absorbed without electron emission. The atom or molecule will have its internal energy increased because it has the additional energy of the photon. Molecules (and atoms) are composed of charged particles, electrons that are negatively charged, and atomic nuclei that are positively charged. In the visible and ultraviolet range of wavelengths of light, that is, wavelengths shorter than 700 nm, the frequency of light is very high. The oscillating electric field of the light interacts with the charged particles of the molecules. Electrons are very light, and therefore,