The Bohr Model

In 1913, the year after the nucleus was discovered, Neils Bohr developed a somewhat ad hoc model that worked surprisingly well in explaining hydrogen. Bohr assumed that the electron in hydrogen could travel on only certain allowed orbits. There was a smallest, lowest energy orbit that is occupied by an electron in cool hydrogen atoms. The fact that this was the smallest allowed orbit meant that the electron would not spiral in and crush into the nucleus.
Using Maxwell’s theory, one views the electron as radiating light continuously as it goes around the orbit. In Bohr’s picture the electron does not radiate while in one of the allowed orbits. Instead it radiates, it emits a photon, only when it jumps from one orbit to another.
To see why heated hydrogen radiates light, we need a picture of thermal energy. A gas, like a bottle of hydrogen or the air around us, consists of molecules flying around, bouncing into each other. Any moving object has extra energy due to its motion. If all the parts of the object are moving together, like a car traveling down the highway, then we call this energy of motion kinetic energy. If the motion is the random motion of molecules bouncing into each other, we call it thermal energy.
The temperature of a gas is proportional to the average thermal energy of the gas molecules. As you heat a gas, the molecules move faster, and their average thermal

Figure 1: The allowed orbits of the Bohr Model.

energy and temperature rises. At the increased speed the collisions between molecules are also stronger. Consider what happens if we heat a bottle of hydrogen gas. At room temperature, before we start heating, the electrons in all the atoms are sitting in their lowest energy orbits. Even at this temperature the atoms are colliding but the energy involved in a room temperature collision is not great enough to knock an electron into one of the higher energy orbits. As a result, room temperature hydrogen does not emit light. When you heat the hydrogen, the collisions between atoms become stronger. Finally you reach a temperature in which enough energy is involved in a collision to knock an electron into one of the higher energy orbits. The electron then falls back down, from one allowed orbit to another until it reaches the bottom, lowest energy orbit. The energy that the electron loses in each fall, is carried out by a photon. Since there are only certain allowed orbits, there are only certain special amounts of energy that the photon can carry out.
To get a better feeling for how the model works, suppose we number the orbits, starting at orbit 1 for the lowest energy orbit, orbit 2 for the next lowest energy orbit, etc. Then it turns out that the photons in the red spectral line are radiated when the electron falls from orbit 3 to orbit 2. The red photon’s energy is just equal to the energy the electron loses in falling between these orbits. The more energetic blue photons carry out the energy an electron loses in falling from orbit 4 to orbit 2, and the still more energetic violet photons correspond to a fall from orbit 5 to orbit 2. All the other jumps give rise to photons whose energy is too large or too small to be visible. Those with too much energy are ultraviolet photons, while those with too little are in the infra red part of the spectrum. The jump down to orbit 1 is the biggest jump with the result that all jumps down to the lowest energy orbit results in ultraviolet photons. It appears rather ad hoc to propose a theory where you invent a large number of special orbits to explain what we now know as a large number of spectral lines. One criterion for a successful theory in science is that you get more out of the theory than you put in. If Bohr had to invent a new allowed orbit for each spectral line explained, the theory would be essentially worthless.

However this is not the case for the Bohr model. Bohr found a simple formula for the electron energies of all the allowed orbits. This one formula in a sense explains the many spectral lines of hydrogen. A lot more came out of Bohr’s model than Bohr had to put in.
The problem with Bohr’s model is that it is essentially based on Newtonian mechanics, but there is no excuse whatsoever in Newtonian mechanics for identifying any orbit as special. Bohr focused the problem by discovering that the allowed orbits had special values of a quantity called angular momentum.
Angular momentum is related to rotational motion, and in Newtonian mechanics angular momentum increases continuously and smoothly as you start to spin an object. Bohr could explain his allowed orbits by proposing that there was a special unique value of angular momentum—call it a unit of angular momentum.
Bohr found, using standard Newtonian calculations, that his lowest energy orbit had one unit of angular momentum, orbit 2 had two units, orbit 3 three units, etc. Bohr could explain his entire model by the one assumption that angular momentum was quantized, i.e., came only in units.
Bohr’s quantization of angular momentum is counter intuitive, for it leads to the picture that when we start to rotate an object, the rotation increases in a jerky fashion rather than continuously. First the object has no angular momentum, then one unit, then 2 units, and on up. The reason we do not see this jerky motion when we start to rotate something large like a bicycle wheel, is that the basic unit of angular momentum is very small. We cannot detect the individual steps in angular momentum, it seems continuous. But on the scale of an atom, the steps are big and have a profound effect. With Bohr’s theory of hydrogen and Einstein’s theory of the photoelectric effect, it was clear that classical physics was in deep trouble. Einstein’s photons gave a lumpiness to what should have been a smooth wave in Maxwell’s theory of light and Bohr’s model gave a jerkiness to what should be a smooth change in angular momentum. The bumps and jerkiness needed a new picture of the way matter behaves, a picture that was introduced in 1924 by the graduate student Louis de Broglie.