Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 8. Quantum Racquetball and the Color of Fruit

IN THE PREVIOUS CHAPTERS, the fundamental concepts of quantum theory were introduced and explained. The examples given, however, looked only at the behavior of free particles. It was shown that electrons could behave as particles in the discussion of how a CRT works, but they behaved as waves in the description of electron diffraction from crystal surfaces. A free particle can have any energy. Its energy, which is kinetic, is determined by its mass and its velocity. As the velocity increases, the energy increases. A tiny increase in velocity produces a tiny increase in energy. A large increase in velocity produces a large increase in energy. The steps in energy can be any size; they are continuous. Bound electrons were discussed briefly in connection with the photoelectric effect. It was pointed out that if the energy of the incoming photon is insufficient to overcome the binding of electrons in the metal, no electrons will be ejected from the metal. Electrons bound to nuclei of atoms are responsible for the properties of atoms and molecules. It was also mentioned that Planck explained black body radiation, which will be discussed in detail later, by postulating that bound electron energies can only change in discreet steps. To understand the properties of the atomic and molecular matter that surrounds us in everyday life, it is necessary to treat bound electrons with quantum theory.

The essential feature of electrons bound to an atom or molecule is that their energy states are discreet. We say that the energies an electron can have are quantized, that is, an electron bound to an atom or molecule can only have certain energies. The energy goes in steps, and the steps are certain discreet sizes. The energy states are like a staircase. You can stand on one stair, or you can stand on the next higher stair. You cannot stand halfway between two stairs. These discreet or quantized energies are frequently called energy levels. Unlike a staircase, the energy levels are not generally equally spaced.

An important area of modern quantum theory research is the calculation of the electronic quantum states of molecules. This field is called quantum chemistry. Such calculations yield the quantized energies of electrons in molecules (energy levels), and they also calculate the structures of molecules. Molecular structure calculations give the distances between atoms and the positions of all atoms in a molecule within limits set by the uncertainty principle. Thus, quantum mechanical calculations are able to determine the size and shapes of molecules. Such calculations are important for understanding the basic principles of the bonding of atoms to form molecules and to design new molecules. As quantum theory continues to develop and the ability to solve complex math problems on increasingly fast and sophisticated computers continues to advance, larger and larger molecules can be investigated using quantum chemistry. One area of great importance is the quantum theory design of pharmaceuticals. Molecules can be designed to have the right size and shape to “fit” into a particular location in a protein or enzyme.

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 else.

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FIGURE 8.1. A perfect one-dimensional box. The walls are infinitely high, infinitely thick, infinitely massive, and completely impenetrable. There is no air resistance in the box. In the box, Q, the potential energy is zero, and outside the box, it is infinite. The box has length, L.

Figure 8.2 shows a racquetball bouncing off of the walls in a perfect one-dimensional classical (big) racquetball court. As discussed, the walls are perfect, and there is no air resistance. In addition, the ball is perfect, that is, it is perfectly elastic. When a ball hits a wall, it compresses like a spring and springs back, which causes the ball to bounce off the wall. A real ball is not perfectly elastic. When the ball compresses, not all of the energy that went into compressing the ball goes back into pushing the ball off of the wall. Some of the energy that went into compressing the ball goes into heating the ball. However, here we will take the ball to be perfectly elastic. All of the kinetic energy of the ball that compresses it when it hits the wall goes into pushing the ball off of the wall. Therefore, the speed of the ball just before it hits the wall is equal to the speed of the ball after it bounces off.

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FIGURE 8.2. A ball in a perfect one-dimensional racquetball court. There is no air resistance, and the ball is perfect. When the ball strikes the wall at L, it bounces off, hits the wall at 0, and keeps bouncing back and forth. Because the court is perfect, the ball is perfect, and there is no air resistance; once the ball starts bouncing, it keeps bouncing back and forth indefinitely.

In this perfect racquetball court, the ball bounces off of the walls without losing any energy and there is no air resistance or gravity. Therefore, the ball will bounce back and forth between the walls indefinitely. It will hit the wall at position L, bounce off, and then hit the wall at position 0, bounce off, and just continue bouncing back and forth. Because the potential inside the box is zero (see Figure 8.1), there are no forces acting on the ball. Therefore, its energy is purely kinetic energy, 040; m is the mass of the ball and V is its velocity. If the ball is hit a little bit harder, it will go a little bit faster, that is, V will be a little bit bigger. Ek will be a little bigger. If the ball is hit somewhat more gently, the ball’s velocity will be a bit smaller, and Ek will be a little smaller. In this perfect racquetball game, the energy can vary continuously. Ek can go up or down by any amount, the amount depending only on how hard you hit the ball.

Another important feature of classical racquetball is that it is possible to stop the ball and put it on the floor. In this situation, the ball has no velocity, V = 0. If V = 0, then Ek = 0. If V = 0, then the momentum is zero because p = mV. So we know the momentum exactly. If the ball is placed on the floor with V = 0, then the position is known. If the position is called x (see Figure 8.2), x can take on values between 0 and L. It cannot have any other values because the ball is inside the court (the box) and can’t get out because of the perfect walls. The ball can be placed at a certain position x on the floor of the court. Therefore, its position is known exactly. This is a characteristic of a macroscopic racquetball court, even a perfect one. It is a classical system, and it is possible to know simultaneously both the momentum, p, and the position, x, precisely.

A racquetball court is 40 feet long (about 12 m), and a ball is 2.25 inches in diameter and weighs 1.4 ounces (about 0.04 kg). Clearly, racquetball is a game describable by classical mechanics. You can watch the ball bounce back and forth by observing it with light without changing it.

PARTICLE IN A BOX—QUANTUM

What are the differences if we now consider quantum racquetball? The court is still perfect, but now its length is 1 nm (10-9 m) rather than 12 m. Furthermore, the particle now has the mass of an electron, 9.1 × 10-31 kg rather than 0.04 kg. This is the quantum particle in a box problem.

We can immediately say that the lowest energy of a quantum particle in a 1-nm-length box cannot be zero. In the classical racquetball court, the velocity V could be zero, which means the momentum, p = mV is zero. In addition, the position x could have a perfectly well-defined value. For example, the ball could be standing still (V = 0) exactly in the middle of the court, which would be x = L/2. Then, for our classical racquet ball, Δp = 0 and Δx = 0. The product, ΔxΔp = 0, is not in accord with the Heisenberg Uncertainty Principle, which is okay because this is a classical system. However, the absolutely small particle in the nanometer size box is a quantum particle, and it must obey the Uncertainty Principle, that is, ΔxΔp ≥ h/4π. If V = 0 and x = L/2, we know both x and p. The result would be ΔxΔp = 0, the same as the classical racquetball. This is impossible for a quantum system. Therefore, V cannot be zero. The particle cannot be standing still at a specific point. If V cannot be zero, then Ek can never be zero. The Uncertainty Principle tells us that the lowest energy that a quantum racquetball can have cannot be zero. Our quantum racquetball can never stand still.

ENERGIES OF A QUANTUM PARTICLE IN A BOX

What energies can a quantum particle in a nanometer-size box have? This question can be answered without a great deal of math, but we need to think about waves again. In Chapter 6, we discussed the wavefunctions for free particles. The wavefunction for a free particle with a definite momentum p is a wave that extends throughout all space. So an electron with a perfectly well-defined momentum is a delocalized wave over all space. The probability of finding a free electron is equal everywhere. Such an electron has a well-defined kinetic energy, Ek = 1/2mV2, because it has a well-defined momentum, p = mV.

An electron in a nanometer-size box is something like our free particle in the sense that inside the box, Q = 0. Inside the box there is no potential, which means that there are no forces acting on the particle. This is just like the free particle; there are no forces acting on a free particle. However, there is a major difference between a particle in the box and a free particle, the walls of the box. An electron in a box is located only inside the box. Its wavefunction cannot be spread over all space because of the perfect nature of the box. The particle is inside the box and can’t ever be outside the box. The wavefunction gives the probability amplitude of finding the particle in some region of space. This is the Born interpretation of the wavefunction. If our electron can only be found inside the box and never outside of the box, there must be finite probability of finding the particle inside the box but zero probability of finding the particle outside the box. If the probability of finding the particle outside the box is zero, then the wavefunction must be zero for all locations outside the box.

The result of the reasoning just presented is that the wavefunction for a particle in a box is like a free particle wavefunction, but the wavefunction must be zero outside the box. In his interpretation of the nature of the quantum mechanical wavefunction, Born placed certain physical constraints on the form wavefunctions can have. One of these is that a good wavefunction must be continuous. This condition means that the change in the wavefunction with position must be smooth. An infinitesimal change in position cannot produce a sudden jump in the probability. This is really a simple idea. If the probability of finding a particle in some very small region of space is, for example, 1%, then moving over an unimaginably small amount can’t suddenly make the probability of finding the particle 50%. This is clear from the illustrations of wave packets in Figure 6.7. The probability changes smoothly with position. Therefore, we can say something else about the wavefunctions for a particle in the box in addition to the fact that they are waves with finite amplitudes inside the box and zero amplitude outside the box. Because the wavefunction must be continuous, right at the walls of the box the wavefunction inside the box must have zero amplitude so that it will match up with the wavefunction’s zero amplitude outside of the box.

Figure 8.3 illustrates a discontinuous wavefunction (not allowed) inside a box. The wavefunction is called ϕ (Greek letter phi). The vertical axis gives the amplitude of the wavefunction. The dashed line shows where zero is. Wavefunctions, which are probability amplitude waves, can oscillate positive and negative. The wave-function shown in Figure 8.3 has values at the walls that are not 0. However, the wavefunction must be zero outside the box, that is, for values of x less than 0 and greater than L it must be zero. As drawn, the wavefunction jumps suddenly from nonzero values at the walls to zero values immediately beyond the walls outside the box. Therefore, the wavefunction as drawn in Figure 8.3 is not a good wavefunction because it is not continuous. This function cannot represent a quantum particle in the box.

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FIGURE 8.3. A wavefunction inside the box that is discontinuous. The wavefunction is called ϕ. The vertical axis is the amplitude of the wavefunction. The dashed line shows where the wavefunction is zero, which must be outside the box. The wavefunction has a nonzero value at the walls and then must drop discontinuously (not smoothly) to zero outside the box.

Wave Function Must Be Zero at the Walls

For the wavefunctions representing the particle in the box to be physically acceptable functions, their values at the walls must be zero so that there is no discontinuity at the walls. This is not a difficult condition to meet. Figure 3.1 illustrates a wave in free space. It oscillates positive and negative. Every time it goes from positive to negative or negative to positive, it crosses through zero. In fact, the zero points are separated by one-half of a wavelength. So what we need to do to get good particle in a box wavefunctions is pick waves with wavelengths such that they fit in the box with their zero points right at the walls. Figure 8.4 shows three examples of waves that are acceptable particle in the box wavefunctions. The one on the bottom, labeled n = 1, is composed of a single half wavelength. It starts on the left with an amplitude of 0, goes through a maximum, and then is zero again at the wall at position L. The next wave up, labeled n = 2, is one full wavelength. Again, it starts at the left wall with amplitude zero, goes through a positive peak, back through zero, a negative peak, and is zero at the wall at position L. The wave labeled n = 3 is one and one-half wavelengths. Any wave that is an integer number of half wavelengths, that is 1, 2, 3, 4, 5, etc. half wavelengths, and has a wavelength so that it starts at zero on the right and ends at zero on the left is okay.

The label n is the number of half wavelengths in the particular wavefunction. For n = 1, the wavelength, λ, is 2L because the box has length L, and n = 1 corresponds to a half wavelength. For n = 2, the wavelength is L because exactly one wavelength fits between the walls. For n = 3, 3 half wavelengths = L. That means 1.5λ = L. Then λ = L/1.5, so λ = 2L/3. Notice that there is a general rule here. λ = 2L/n, where n is an integer. For n = 1, λ = 2L. For n = 2, λ = 2L/2 = L. For n = 3, λ = 2L/3, and so forth.

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FIGURE 8.4. Three examples of wavefunctions, ϕ, inside the box that are continuous. They have been shifted upward for clarity of presentation. The vertical axis is the amplitude of the wavefunction. The dashed line shows where the wave function is zero, which it must be outside the box. The wavefunctions, which have zero values at the walls, are continuous across the walls.

Nodes Are Points Where the Wavefunction Crosses Zero

Nodes are another important feature of the wavefunctions. Nodes are points where the wavefunction crosses zero, going from positive to negative, or negative to positive. The n = 1 wavefunction has no nodes. The n = 2 wavefunction has one node right in the middle of the box. The n = 3 wavefunction has two nodes. Nodes are points (other than at the walls) where the probability of finding the particle is zero. In a classical system, as in Figure 8.2, the ball bounces back and forth. It can be at any location. But a particle in a quantum box has certain places (nodes) where the probability of finding it is zero. No matter how many measurements are made on identically prepared systems, we will never find the particle at a node.

Figure 8.4 shows the probability amplitude waves. As discussed, the probability of finding the particle in a certain region of space is proportional to the square of the wavefunction (actually the absolute value squared, but for our purposes, there is no difference). Figure 8.5 shows the square of the wavefunctions that are displayed in Figure 8.4. The square of the wavefunctions are always positive because the probability of finding a particle in some region of space cannot be negative. Where the amplitude is large, there is a large probability of finding the particle. As n increases, the number of nodes increases. As we will discuss in the next and later chapters, atomic and molecular wavefunctions also have nodes.

A question that is frequently asked is, how does a particle get through a node? For example, for n = 2, there is a node exactly in the middle of the box. In a classical system if we had a ball on the left side of the box and it was traveling to the right, but we said it could never be in the center of the box, we would be confident that the ball could never get to the right side of the box. However, we cannot think classically about an absolutely small particle, such as an electron in a molecular-sized box. It does not have a simultaneous definite position and momentum that can be described by an observable trajectory. A quantum particle, an electron, is described as a probability amplitude wave. Waves have nodes. Even classical waves have nodes. A quantum particle does not have to “pass through” a node because it is a delocalized probability amplitude wave. The idea of a trajectory in which to get from point A to point B a particle must pass through all the points in between just doesn’t apply to the proper wave description of electrons and other absolutely small particles.

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FIGURE 8.5. The squares of the first three wavefunctions, ϕ2, for the particle in a box. They have been shifted upward for clarity of presentation. The vertical axis is the amplitude of the wavefunction squared. The dashed line shows where the wavefunction is zero. The square of the wavefunctions are always positive because they represent probabilities. The wavefunctions shown in Figure 8.4 can be positive or negative.

The Energies Are Quantized

Now we will determine the possible energies that an absolutely small particle in a box can have. The classical ball in a racquetball court can have any energy, and the energy is continuous. We can determine what energies a particle, such as an electron, can have in a tiny box by using the rule for the possible wavelengths, λ = 2L/n, that allowed probability amplitude waves can have inside the box (see Figure 8.4). Here tiny means a box that is small in the absolute sense, that is, the wavelength is comparable to the size of the box. We will also need several other physical relationships that we have met previously. The other relations we need are the de Broglie wavelength, p = h/λ, where p is the momentum and h is Planck’s constant; the fact that the momentum is p = mV, where m is the mass and V is the particle velocity; and the kinetic energy of the particle, 044. Now let’s combine these formulas.

First, square p. Then,

p2 = m2V2.

If we now divide both sides of the equation by 2m, we see that the right side gives045 the kinetic energy, and the left side gives 046 . So we have the following expression for the kinetic energy,

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Using the de Broglie relation, we can replace p2 with p2 =h22. Putting this into the expression for the energy gives,

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Finally, we will use our rule, λ = 2L/n, for the possible wavelengths. Then λ2=4L2/n2. Substituting this expression for λ2 into the expression for the energy yields

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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

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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.

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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, it is easy for them to respond to the rapidly oscillating electric field of visible or ultraviolet light. The absorption of visible or ultraviolet light is caused by increasing the energy of the electrons in a molecule.

The question is, what wavelengths of light will be absorbed by a molecule? This is a very complex question for any given molecule. Large quantum theoretical calculations are performed to determine the absorption spectrum of a molecule. However, we can learn about important aspects of molecular absorption of light from the particle in a box problem. As an exceedingly simple model of a molecule, we will consider a single electron in a molecular-sized box. Later we will put in numbers. The electron will be in its lowest energy state, called the ground state, when no light shines on the electron in the box (the molecule). For the particle in the box, the lowest energy has the quantum number, n = 1. For n = 1, the energy is:

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When light shines on a molecule, a photon can be absorbed. If the photon is absorbed, the energy of the photon is lost from the total energy of all of the light. Energy must be conserved, which happens by an electron going into a higher energy state, that is, it goes from the ground state, the lowest energy level, to a higher energy level. However, this higher energy level cannot have any energy value because the energy levels of the particle in a box (and molecules) are quantized. The lowest energy state above the ground state has quantum number n = 2. This is called an excited state. The electron has been excited by absorption of a photon from the ground state to the first excited state. The energy of the first excited state, the n = 2 state, is:

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Energy must be conserved. This is true in classical mechanics, and it is true in quantum mechanics. We start with the electron in the ground state. When the photon is absorbed, the electron is in an excited state. Therefore, to conserve energy, the photon energy must equal the difference between the excited state energy and the ground state energy. Only a photon with this energy can be absorbed by the system. The photon energy determines the wavelength of the light. Therefore, only certain colors of light can be absorbed.

Figure 8.7 illustrates absorption of a photon. The arrows show two allowed paths for photon absorption. These are called transitions. The transitions from n = 1 to n = 2, and n = 1 to n = 3 are shown in the figure. For a photon to be absorbed, the photon energy must equal the difference in energy of two of the quantum levels. If the photon energy does not match the difference in energy between two levels, it cannot be absorbed.

The difference in the energy, ΔE, between the n = 2 first excited state energy level and the n = 1 ground state energy level is:

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FIGURE 8.7. Particle in a box energy levels. The quantum number is n. E is the energy plotted in units of h2/8mL2. The arrows indicate absorption of photons that can take an electron from the lowest energy level, n = 1, to higher energy levels, n = 2, n = 3, etc. For a photon to be absorbed, its energy must match the difference in energy between two energy levels.

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This is the energy that the photon must have to cause the electron to make a transition from the ground state to the first excited state. We can use Planck’s relation, E = hν, for the photon energy to see that the energy ΔE corresponds to a certain frequency of light. Also, because λν = c, the wavelength times the frequency equals the speed of light, we can determine the wavelength (color) of the light that will be absorbed.

The Color of Fruit

Let’s put in numbers. h = 6.6 × 10-34 J-s. The electron mass, me = 9.1 × 10-31 kg. For the length of the box, let’s take L to be that of a medium-sized molecule, that is, L = 0.8 × 10-9 m (0.8 nanometers, 0.8 nm). Then,

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Converting this energy to a frequency by dividing by h gives ν = 4.25 × 1014 Hz, which corresponds to the wavelength of the light that will be absorbed, λ = 7.06 × 10-7 m = 706 nm. Light with wavelength 706 nm is very deep red. It is on the red edge of colors that you can see by eye. What happens if the box (molecule) is smaller, say 0.7 nm instead of 0.8 nm? The energy of the light that is absorbed becomes higher and, therefore, the wavelength of light that is absorbed becomes shorter as the box becomes smaller. The energy absorbed goes inversely as L2 (L2 is in the denominator), which means as the box gets smaller, the energy levels get further apart, and the energy difference increases as the square of the box length. So for a 0.7 nm box, the absorbed wavelength is λ = 540 nm, which is green light. If the box is even smaller, say 0.6 nm, then λ = 397 nm, which is very blue, on the blue edge of the light that can be seen with the unaided eye.

These simple results are also generally true for molecules, although there are a lot of details that come into play. However, for a sequence of molecules that have essentially the same type of structure (types of atoms, etc.) the bigger the molecule, the further to the red it will absorb. The results presented for a particle in a box show very qualitatively why things have different colors. Small molecules absorb light in the ultraviolet part of the spectrum. We can’t see ultraviolet light, so absorption by small molecules does not result in a color. We see colors by the light that bounces off an object. The colors that are absorbed don’t bounce off. Large molecules absorb in the visible part of the spectrum, and these molecular absorptions give objects color.

Cherries are red and blueberries are blue because they have different molecules in them that absorb different colors of light strongly. These molecules have quantized electronic transitions. They can only absorb light from their ground electronic states to their excited states at wavelengths that are determined by their quantized energy levels. In the particle in a box, the transition energies of an electron are determined solely by the length of the box and the mass of the electron. For molecules, the quantized transition energies, and therefore wavelengths and colors, are determined both by the sizes of the molecules and by the details of the molecular structures, that is, the shape of the molecule, the types of atoms that make up the molecule, and how the atoms are arranged. Dyes are molecules that have specific strong absorptions in the visible. Dyes are used to give our clothes different colors. Brightly colored plants, green leaves and red roses, contain a wide variety of molecules with different sizes and shapes that absorb distinct colors of light strongly. The sizes and shapes of these molecular absorbers give the plants their brilliant colors. If the molecules absorb green and red strongly, then blue will bounce off of an object, and it will look blue. If blue and green are absorbed strongly, mainly red will bounce off, and an object will look red. The colors that are absorbed are determined by the quantized energy levels of the molecules in the object.

We observe color constantly in everyday life. Color is one of many phenomena that we encounter that is inherently quantum mechanical. There are many others. For example, when you turn on an electric stove, the element gets hot. Why electricity moving through metal produces heat (Chapter 19) is another everyday quantum phenomenon. Why is carbon dioxide a greenhouse gas (Chapter 17)? What is a trans fat (Chapter 16)? It is necessary to go into the quantum mechanics of molecular structure to understand such systems. In the following chapters, the quantum description of atoms and molecules will be developed and applied to a number of common issues and problems. The necessary machinery for understanding atoms and molecules is developed in Chapters 9 through 14. These chapters supply a great deal of interesting information about how atoms and molecules behave and provide the bridge between the general ideas of quantum theory that we have just developed and understanding many phenomena that surround us.