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

### Chapter 9. The Hydrogen Atom: The History

**IN CHAPTER 8, WE DISCUSSED** the particle in a box 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. A system may be a container of gas molecules, the leaf of a plant, or molecules in a liquid such as the molecules that make wine red. We use complicated dye molecules to color our clothes because the size and structure of a molecule determines which wavelengths of light it will absorb.

In Chapter 4, black body radiation was briefly discussed. A hot object gives off light. A fairly hot piece of metal will glow red. This can be seen in the wire elements of an electric space heater or an electric stove. As the temperature is increased the color will move toward the blue. It was mentioned that stars are well described as black bodies, and the color of a star can be used to determine its temperature. Planck developed a formula that yields the black body spectrum for a given temperature. Figure 9.1 shows the solar spectrum calculated using Planck’s formula that is the closest to the experimentally measured solar spectrum. The frequency is plotted in wave numbers (cm^{-1}). Multiplying the frequency in cm^{-1} by the speed of light in centimeters per second (3×10^{10} cm/s) gives the frequency in Hz, the conventional frequency units. The top axis in Figure 9.1 is the wavelength in nanometers (nm). 500 nm is green light. 400 nm is very blue light. 666 nm is very red light. 333 nm is in the ultraviolet region of the optical spectrum and cannot be seen with the eye. 1000 nm is in the infrared part of the spectrum and also cannot be seen with the eye. These wavelengths can be detected with electronic photodetectors. Originally, they were detected with photographic film. The vertical axis is the irradiance. It is the number of watts (joules per second) that would fall on an area of a square meter in a little slice of frequency 1 cm^{-1} wide. Basically the plot is the amount of energy per second of a particular color that falls on a square meter.

**FIGURE 9.1.** *The black body spectrum of the sun calculated using the Planck formula for black body radiation from a hot object. The curve is a good representation of the solar spectrum without some of the fine details. The lower axis is the frequency in wave numbers (see text). The top axis is the wave length in nanometers. The green light is 500 nm. Very blue is 400 nm; very red is 666 nm. The vertical axis is the amount of light (see text).*

The shape of the spectrum plotted in Figure 9.1 is almost identical to the actual solar spectrum. The calculated spectrum is obtained by adjusting the temperature in Planck’s formula until the best match is obtained to the experimental spectrum. The temperature that gives the best spectrum is 5780 K, where K is the unit of temperature in degrees Kelvin. This is the unit of absolute temperature developed by William Thomson, the first Baron Kelvin (Lord Kelvin, 1824-1907). The Kelvin scale is used in physics and chemistry because it has a well-developed physical meaning for 0 K, the absolute zero of temperature. At 0 K, all atomic motions associated with kinetic energy (heat), the energy of moving particles, stop. To obtain the temperature in degrees centigrade (C) subtract 273. Then in centigrade, the sun’s temperature is 5507 C. To convert to Fahrenheit (F), multiply by 9/5 and add 32 to the temperature in centigrade. Therefore, the surface temperature of the sun is 9945 F. The Fahrenheit temperature scale is named for Daniel Gabriel Fahrenheit (1686-1736).

**Dark Lines in the Solar Spectrum**

It is remarkable that the Planck formula developed using the first quantum concept, that the energies of electrons “oscillating” in a metal are not continuous, can be used to determine the temperature of stars. The calculated spectrum shown in Figure 9.1 is continuous because a hot object produces a continuous distribution of colors (energies of light). While the experimental measurement has the shape shown in Figure 9.1, it also has some very sharp features in it that are not part of the sun’s black body spectrum. Figure 9.2 is an illustration of the solar spectrum with thin dark lines that represent a lack of light at certain frequencies. The spectrum shown in Figure 9.1 is the light that comes from the sun. But the dark lines are certain narrow bands of color that do not make it to earth. The light is being absorbed between the sun and the earth. The lines are called absorption lines or bands. These same lines are very prominent in the spectra of light coming from stars other than our own sun.

**FIGURE 9.2.** *The visible portion of the solar spectrum. The continuous range of colors is the black body spectrum. The dark lines or bands are colors that do not reach Earth, so they appear as colors missing from the solar spectrum. The wavelengths of the lines and the spectrum are given in nm, nanometers, which are billionths (10*^{-9}*) meters.*

The same wavelengths that are seen as dark lines in the solar spectrum can also be seen as distinct colors from an arc lamp filled with hydrogen gas. A hydrogen arc lamp or discharge lamp is a sealed glass cylinder filled with hydrogen gas with electrodes at either end. When a sufficiently high voltage is connected to the lamp, positive connected to one electrode and negative to the other electrode, electricity arcs through the lamp like a small continuous lightning bolt. The colors (wavelengths) in the visible coming out of the lamp are the same as wavelengths of the black lines shown in Figure 9.2.

**The Hydrogen Line Spectrum**

The first attempt at understanding the line spectrum of hydrogen in the visible region was made in 1885 by the Swiss schoolteacher and mathematician, Johann Balmer (1825-1898). Balmer noted that the frequencies of the lines, f, in the visible part of the spectrum were related by the formula

The symbol ∝ means proportional to, so there is a multiplicative constant that is discussed below. In this equation, n is an integer greater than 2, that is, 3, 4, 5, etc. The spectral lines in the visible are called the Balmer series.

Later, lines were discovered in the ultraviolet and the infrared. These are called the Lyman series and the Paschen series, respectively, after their discoverers Theodore Lyman (1874-1954), a U.S. physicist and spectroscopist, and Louis Karl Heinrich Friedrich Paschen (1865-1947), a German physicist. In 1888, the Swedish physicist and spectroscopist, Johannes Rydberg (1854-1919) presented a formula that described all of the spectral lines seen in emission from a hydrogen arc lamp or in the absorption spectrum of solar or stellar light. The Rydberg formula for the frequency of the hydrogen atom spectral lines is

n_{1} is an integer beginning at 1. n_{2} is another integer that must be greater than n_{1}. n_{1} = 1 gives the Lyman series. n_{1} = 2 gives the Balmer series. n_{1} = 3 gives the Paschen series. The constant, R_{H}, is called the Rydberg constant for the hydrogen atom. It has the value, R_{H} = 109,677 cm^{-1}. Here the constant is given in wave numbers (cm^{-1}). When this value is used in the Rydberg formula, the frequency of a spectral line determined by the integers n_{1} and n_{2} is in wave numbers. To get it in Hz, the result is multiplied by the speed of light in cm/s, that is, 3 × 10^{10} cm/s. To find the wavelength of a spectral line, take the inverse of the frequency in wave numbers, that is, take 1 and divide it by the frequency in wave numbers. For example, if n_{1} = 2 and n_{2} = 3, then

the frequency in wave numbers. The inverse of this number is 6.56 × 10^{-5} cm = 656 × 10^{-9} m. 10^{-9} m is a nanometer, so the wavelength is 656 nm. This is the red line in the Balmer series shown in Figure 9.2.

In connection with Figure 8.7, we already discussed discreet optical transitions between quantized energy levels for the particle in a box. Figure 8.7 shows transitions between the particle in a box states for n = 1 going to n = 2 and n = 1 going to n = 3. So it should come as no surprise that the optical transitions of the hydrogen atom could involve discreet frequencies that depend on integers. However, in 1888, at the time of the Rydberg formula, it was still 12 years before the first use of the idea of quantized energy levels by Planck to explain black body radiation, and 37 years before true quantum theory took shape in 1925. The various series of spectroscopic lines that have energies related to integers through the Rydberg formula can be understood as optical transitions between discreet energy levels that are associated with the hydrogen atom. A few of the energy levels that give rise to the Lyman series and the Balmer series are shown in Figure 9.3. In the figure, the downward arrows indicate the emission of light that would come from a hydrogen arc lamp. The hydrogen atom starts in a higher energy level and ends up in a lower energy level. Energy is conserved by the emission of a photon. To conserve energy, the photon must have the energy difference between the initial higher energy level and the final lower energy level. In the Rydberg formula, the smallest value n_{1} can have is 1, and n_{2} must be bigger than n_{1} . The arrow labeled 2-1 represents emission from the n = 2 level to the n = 1 level. The next higher energy emission in the Lyman series is for emission from the n = 3 level to the n = 1 level. In the Rydberg formula, the next possible value for n_{1} is 2, and n_{2} must be bigger than n_{1}. Therefore, the lowest energy emission line in the Balmer series is labeled 3-2. The hydrogen atom begins in the n = 3 level and ends in the n = 2 level, and energy is conserved by emission of a photon with wavelength 656 nm. When light shines on hydrogen atoms, absorption occurs, which would be indicated in the diagram by up arrows. The Balmer series absorptions are shown in Figure 9.2.

**FIGURE 9.3.** *Schematic of some of the energy levels that give rise to the Lyman and Balmer series of hydrogen atom emission lines. The down arrows indicate that light is being emitted from a hydrogen arc lamp, for example. For absorption, shown by the black lines in Figure 9.2, the arrows would point up. The level spacings are indicative but not to the true scale.*

**BOHR’S HYDROGEN ATOM THEORY—NOT QUITE THERE**

The first detailed description of the energy levels of the hydrogen atom was developed by Niels Bohr (1885-1962) in 1913. Bohr won the Nobel Prize in Physics in 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them.” Bohr’s theory of the hydrogen atom is referred to as the old quantum theory. Bohr made many advances and in fact was able to calculate precisely the energy levels of the hydrogen atom, and therefore obtain the Rydberg relation and predict all of the hydrogen atom spectral lines. Bohr was also the first to propose two ideas we have already been using. He said an atomic system can only exist in certain states, which he called “stationary” states. Now, we usually refer to these as energy eigenstates. Each of these states has a well-defined energy, E. Transitions from one stationary state to another can occur by absorption or emission of light or other means that can give or take energy from the system, and the amount of energy must be equal to the difference in energy of the two states. This idea is the basis for Figures 9.3 and 8.7, where the arrows represent transitions between states that occur by absorption or emission of light.

Bohr also put forward what came to be known as the Bohn Frequency Rule. The frequency of light emitted or absorbed in making a transition from an initial state with energy E_{1} to a final state with energy E_{2} is the difference in the energies divided by Planck’s constant, that is,

ν is the frequency and h is Planck’s constant (h = 6.6 × 10^{-34} J-s). The vertical lines are the absolute value. For absorption E_{1} is less than E_{2}, so E_{1} - E_{2} would be negative. The absolute value means that you make the number positive even if the difference is negative. The frequency, ν, has to be a positive number. Multiplying both sides by h gives E, the energy difference between the energy levels (stationary states) as E = hν, which is the Planck relationship that Einstein used to explain the photoelectric effect discussed in Chapter 4.

What is a hydrogen atom and what are the failures of the Bohr method? A hydrogen atom is composed of two charged particles, a proton that has a positive charge, that is, a charge of +1, and an electron that has a negative charge, a charge of -1. When we say a charge of 1, it actually is shorthand for the charge that is on one proton. The charge in real units is 1.6 × 10^{-19} C, where C is the Coulomb, the unit of charge. Ernest Rutherford (1871-1937) did experiments in 1911 that showed that atoms were composed of a small, heavy positively charged nucleus and one or more electrons outside of the nucleus. Rutherford won the 1908 Nobel Prize in Chemistry “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances.” Rutherford’s findings applied to the hydrogen atom mean that the proton is the nucleus and a single electron is found outside the nucleus. Even for hydrogen where the nucleus is composed of a single proton, the nucleus is much heavier than an electron. The mass of a proton is m_{p} = 1.67 × 10^{-27} kg, while the mass of the electron is only m_{e} = 9.1 × 10^{-31} kg. A proton weighs about 1836 times as much as an electron.

In the Bohr model, the electron orbited the proton like a planet orbiting the sun. The lowest energy state of the hydrogen atom, n = 1, has the electron going around the proton in a circle. Higher energy states of the electron, with n greater than 1, could have different shapes. Some were still circles, but some were ellipses. This picture of the electron orbiting the proton should immediately set off danger warnings based on the material covered in earlier chapters. In Chapter 6, the Heisenberg Uncertainty Principle was discussed. We know that an absolutely small particle cannot have a classical trajectory. To have a trajectory, it is necessary to know the position and the momentum of a particle simultaneously over all time. But the Heisenberg Uncertainty Principle says that it is not possible to know both the position and the momentum precisely simultaneously. The uncertainty relation states that ΔxΔp ≥ h/4π, where h is Planck’s constant. Absolutely small particles are described in terms of probability amplitude waves, not trajectories. Of course in 1913, when Bohr came out with his mathematical treatment of the hydrogen atom, the nature of absolutely small particles was not known.

The failure of Bohr’s approach became apparent when it was applied to systems other than the hydrogen atom. While Bohr’s method could predict very accurately the energy levels, and therefore the spectrum of the hydrogen atom, it could not do so for the next simplest atom, the helium atom. Nor could it properly predict the properties of the simplest molecule, the hydrogen molecule, which is composed of two hydrogen atoms. The Bohr method could not account for the strength of the chemical bond that held the two hydrogen atoms together to form the hydrogen molecule. Although Bohr made giant steps in the right direction, the failures of his approach ultimately led to the development of true quantum theory in 1925.