Taking the quantum leap, p.8
Taking the Quantum Leap, page 8
Since matter was assumed to be made of atoms, it was also natural to imagine that heated solid or liquid matter glowed because of the movements of the lighter electrons. These electrons were thought to oscillate back and forth inside of their respective atoms. And these oscillations, it was supposed, broadcast light waves just as Hertz had demonstrated in 1887 that his electrical oscillations broadcast radio waves. The only question was how to picture this. Remember, the Newtonian classical world view of physical processes still persisted in spite of the Planck-Einstein E = hf formula.
The size of the atom was known. It was less than two billionths of an inch in diameter. This diameter is so tiny that it is nearly inconceivable. To grasp how tiny an atom actually is, consider the following imaginative exercise. Suppose you had a golf ball in your hand. And suppose that you could inflate this golf ball—that is, blow it up like a balloon until it was large enough to enable you to see an atom within the golf ball. To be specific, supposed that you wished to blow the golf ball up until one of its atoms was as big as a normal sized golf ball. For that atom to become big enough to hold in your hand, the original golf ball would have to be inflated to the size of our earth! No wonder no one knew what an atom looked like or, in particular, how electrons fit inside of it.
By 1911, J. J. Thomson had become Sir Thomson. He had his own laboratory in England. He was the director of the world-famous Cavendish Laboratory. He also was the leader of a certain school of thought regarding the structure of atoms and the whereabouts of electrons within.
Thomson’s atom was pictured as a tiny raisin pudding. Embedded within this “pudding” were even tinier electron “raisins.” The number of electrons depended on the particular variety of atom. Hydrogen had just one electron raisin, one bit of negative electrical charge to balance out the observed positive charge and make the atom electrically neutral. By putting the atom into an electrical discharge, that single, negatively charged electron could be pulled out of the pudding and leave behind a positively charged atom “pudding.” The result was a hydrogen ion. Helium ions were observed to be doubly charged; thus, it was clear that a helium atom had to have two electrons within it to balance out its charge. And so on.
Another school of thought held that the atom looked more like a miniature solar system than a raisin pudding. Each electron in any given atom was pictured as a planet that moved in a closed orbit about a tiny nucleus at the atom’s center. Instead of a more or less random distribution of electron raisins embedded in the large and somewhat soft, tenuous background of a positively charged pudding, there was a well-organized series of electron planets, each in its own orbit, following a well-defined mechanical and repeatable movement. These electron planets moved like real planets. Each had its own respective “years.” In other words, there was a periodicity or frequency to their motions. The deciding feature between these two models had to do with the rest of the atom, the positive matter that held the electrons within.
The correctness of the pudding or planetary model of the atom could not be determined from the light emitted by atoms. Nor could anyone shine light on an atom and take a look. Atoms were much too small. The wave lengths of light were thousands of times longer than the diameters of atoms. Such details as the location of electrons or the distribution of the heavier, positively charged atomic matter would never be seen by using light waves. But there were other ways to explore an atom. You could throw other atomic particles at it and observe the scattering and atomic debris that would result from a collision. Just as the debris from a midair collision between airplanes can reveal the cause of the accident, atomic debris can reveal what the insides of the atom look like.
The question of whether matter in an atom was spread out like a pudding or gathered together in a tiny sunlike nucleus at the atom’s center was finally given an experimental test in 1911. Within a vacuum enclosure, a beam of helium ions was fired at a very thin foil of gold, and the truth was discovered. The helium ions scattered from the atoms within the gold foil with a pattern that suggested that the atoms of gold had nuclei. The pudding model was dropped.
The new atomic model was definitely planetary. The surprising thing of the planetary model was how small the nucleus appeared. If the golf ball-sized atom was once again inflated, this time to the size of a modern sports arena or football stadium, the nucleus of that atom would be the size of a grain of rice. Somehow the electrons whirled about, filling in the vast space within the tiny atomic world.
These experiments were carried out by Lord Ernest Rutherford and his assistant Ernest Marsden.1 Rutherford was also given his own laboratory in the industrial Midlands of Manchester, England. With the success of what is now called the Rutherford nuclear atom, Lord Rutherford led his group of scientists in an attempt to picture how the electron “planets” were able to maintain themselves in orbit and yet radiate energy in the form of light waves. Rutherford’s success was, I’m sure, not too palatable for his counterpart, Lord Thomson, down south.
Into this slight animosity a young innocent was about to step. His name was Niels Bohr.
Bohr’s Quantum Atom
Dr. Bohr had just completed his doctoral thesis in Copenhagen, Denmark, when he reported to work for J. J. Thomson at the “Cavendish.” Lord Thomson, Bohr’s first employer, probably felt less than enthusiastic over meeting the twenty-six-year-old Bohr. Besides possessing an incredible mind, Bohr was quite forthright and outspoken. Thomson’s model for an electron had been the subject of Bohr’s thesis, and Bohr immediately pointed out some mathematical errors in Thomson’s earlier work.
By the autumn of 1911, Bohr found himself, much to Thomson’s urging, on his way to Manchester to join Rutherford’s group. He quickly joined in with this newly excited group of physicists and began his own search for the electrons within atoms.
The simplest and lightest-known atom in the universe was hydrogen. It contained, according to Rutherford, a tiny nucleus and a single electron orbiting that nucleus. It was hoped that, if a successful model of this atom could be made, all other atoms would fall into line and be explained. So Bohr attempted to make a model of the hydrogen atom.
There was, however, a severe stumbling block in the way of the planetary picture of an atom. The problem was how could the electron keep a stable orbit? If the atom was as big as it appeared to be, its electron would necessarily be whirling around inside it with greatly accelerated changes in speed and direction, filling out the space like the tip of a whirling propeller blade fills out a circle. The electron would have to do this and not emit any energy. Certainly, it could not emit its energy continuously. To do so would be a disaster for the model. The reason for this is that the planetary model predicts a spiraling motion of the planet into the sun for any planet that gives up energy continuously. That would mean the electron would crash into its nucleus every time it emitted its light energy. The whole atom would be suddenly deflated and all matter would undergo a rapid collapse. It is amazing to consider how tiny atoms would be if the atomic electrons were gobbled up by their nuclei. A football stadium would be shrunk to a grain of rice. The earth would be shrunk to the size of a football stadium! All matter would thus appear with enormous density. (Neutron stars do appear in our universe with these densities. The force of gravity crushes the atoms together.) And all matter would be dead and lifeless. The light would be gone.
But if the electron could not emit energy continuously, how was it to radiate any light? Light emission took energy. The electron would have to radiate energy sometime or no light would ever be seen. The question was how to make up a planetary model in which the electron would only radiate energy sporadically or in a discontinuous manner. Thus Bohr attempted to visualize under what circumstances the electron would be “allowed” to radiate energy and under what circumstances it would be “forbidden” to do so. This was not an easy decision to make. Bohr’s model would have to show a reason for the discontinuity. How could Bohr explain it?
He explained it very simply. He postulated that an atom would only be allowed the privilege of emitting light when an electron jumped discontinuously from one orbit to another. It would be forbidden from doing so otherwise. Like Planck and Einstein before him, Bohr was setting out on a bold path. In fact, he was encouraged by their example. He felt that somehow Planck’s h factor had to be involved in the process. He knew that h had been used by both Planck and Einstein to point to the discontinuous movement of light energy in solid matter. Perhaps it could also be used inside of an atom. But how? Bohr found out.
This new secret was actually no mystery to anyone familiar with physics. It had to do with something that physicists call units. A unit is a measure of a physical quantity. Any unit can also be composd of other units. Take the common example of a monetary unit. An American dollar is a unit of money, and it is composed of other units as well. For example, a dollar is ten units called dimes, or one hundred units called pennies. Similarly, it is also one-tenth of a unit called a ten dollar bill.
Planck’s constant h also was a unit. And it too could be made up of other units. It was a unit of energy-time, something that physicists call action, and it was a unit of momentum-distance, just as a dollar is also ten dime units. But Bohr had noticed that h could be viewed as a unit of angular momentum and that observation had a direct bearing on his atomic model.
Angular momentum is a familiar experience for children. It results whenever a moving object passes a fixed point in space. If the moving object joins or connects with the point it is moving past, the object begins to whirl in a circle. Angular momentum can be thought of as momentum moving in a circle. When children run towards a tether ball hanging from a pole, they often leap and swing in a circle about the pole. By hanging on to the tether the children exhibit their angular momentum “about” the pole. Angular momentum is the product of ordinary or linear momentum and the radius or distance from the object to the reference point. Since Bohr’s electron was traveling in an orbit about the nucleus, it too was tethered, held to that nucleus by an invisible tether of electrical attraction between the electron and the atomic nucleus. Thus the electron had angular momentum. Could Planck’s constant h be used as a unit of the electron’s angular momentum?
To grasp the significance of this question, imagine that you have a ball attached to a string. Holding the loose end of the string in your hand, whirl the ball around over your head, cowboy style, as if you were about to rope a calf. The faster you whirl the ball, the greater the force you feel, holding on to the rope. When you whirl the ball faster, you increase its angular momentum.
Now picture an ice skater whirling in a spin. Notice that as the skater brings her arms toward her body, she whirls faster. Her arms are acting like balls attached to ropes. But, unlike the whirling balls, although she is spinning faster, her angular momentum remains the same. This is because the distance from her spin axis to her arms has decreased to compensate for her increased speed of rotation.
If you now picture the tiny electron whirling around in its orbit, you will realize that for a given force holding it to its circular path and for a given fixed amount of angular momentum, the speed of the electron is determined. The radius of the orbit is also determined. Everything depends on the delicate balance provided by the amount of angular momentum the electron is allowed to possess.
Bohr tried out a calculation imagining a circular orbit for the electron with one unit of angular momentum. He calculated the size of the orbit requiring that the electron have one unit of h. The orbit was the correct size; it filled out the atom. He then tried a new orbit with two units of h. It proved to be a new orbit with a larger diameter, four times the original orbit. When Bohr calculated an orbit for the electron with three units of h, the orbit grew in size to nine times the original orbit. Bohr had discovered a new model for the atom.
In this model there were only certain allowed orbits. By restricting the electron to these special or “quantized” orbits, as they were later called, Bohr successfully predicted the correct size for the atom. Each orbit grew in size as the electron increased its angular momentum. But the scale was correct.
This wasn’t the only discovery. Bohr also discovered why the electron wasn’t radiating as it whirled in its orbit. In other words he found a reason for the atom’s stability. By allowing the electron to have only whole units of h and not any other amounts of angular momentum, Bohr discovered the rule that kept the electron in a stable orbit. Only electrons with whole amounts of angular momentum (i.e., integer multiples of Planck’s Constant: 1h, 2h, 3h, etc.) would be allowed the privilege of orbiting peacefully inside of the atom. These quantized orbits were known as Bohr orbits. The integers of 1, 2, 3, and so forth were called the quantum numbers of the orbits. A quantum model of the atom had appeared.
Bohr and his atom:
Each circle represents an orbit for a planetary electron. The orbital diameters are in the ratio l:4:9. In orbit one, the electron has one unit of h, in orbit two (diameter 4) it has two units of h, and in orbit three (diameter 9) it has three units of h.
The only thing still needed was the “rule” that allowed the electron to radiate light energy. Again, there was no physical reason for the quantization rule that held the electron in a stable orbit. Bohr had made it up. So Bohr again postulated that the electron would radiate light whenever it changed from one orbit to another. He calculated the energy of the electron in each of its possible orbits. By comparing the difference in energies between the orbits and using Planck’s E = hf formula, Bohr successfully predicted the frequencies of the light observed whenever an electron made an orbital “jump.”
In January of 1913, a former classmate of Bohr’s showed him a paper written by a Swiss schoolteacher named Johann Balmer. Balmer had observed light coming from hydrogen gas in 1880. Instead of a continuous spread of colors, the light from hydrogen showed missing colors when it was passed through a prism and analyzed. The spectrum it produced appeared as a horizontal strip containing several vertical lines, like teeth in a comb. Only some teeth were missing. Ordinarily, the light we see—for example, sunlight or light from an incandescent bulb—does not break down into such a spectrum. Instead, sunlight or light from any hot solid or liquid shows a continuous spread of colors like a rainbow. But Balmer’s incomplete spectrum had been produced by hydrogen atoms in a gas. Bohr read Balmer’s paper on atomic light and became very excited. Not only could he calculate the energy of the electron in each atomic orbit, but he could also calculate the energy that the electron radiated away when it changed orbits.
Balmer’s hydrogen spectrum had missing “teeth” because the energy given out by the jumping electron was so well prescribed. Since there were only certain orbits for the electron, there had to be only certain frequencies for the light. The frequency of the light depended on the difference in energies of the electron involved in the quantum jump from one orbit to another. Balmer’s atomic light was explained.
All well and good. However, Bohr’s successful prediction was based on a very disturbing picture. The electron making the light was not oscillating or orbiting the nucleus to make the light. In fact, it wasn’t doing anything that anyone could really imagine. To make the light, it had to jump. It leaped like a desperate superman from one orbit to another inside the atom. It was not allowed to move in between orbits. Bohr tried to calculate that and failed. The best picture he could come up with was that of a quantum jump, a leap from one place to another without passing in between. As unreasonable as this picture was, it replaced any completely classical mechanical picture of that process.
What Balmer saw : Light from hydrogen atoms breaks into a spectrum of colors.
Yet Newton’s mechanics were not to be completely abandoned. Some features of the classical picture were not abandoned at all. First of all, the idea of an orbit and a planetary atom were still classical pictures based upon a continuous movement of the electron. What was not classical was the refusal of the electron to radiate when it was in a Bohr orbit. This was entirely unreasonable for a very important reason: all accelerating electrons have been observed to radiate energy. Either the Bohr-orbiting electrons were accelerating or Newton’s second law was being repealed.
According to Newton, there was a force acting on the electron. That force pulled the electron into a circular orbit, changing the momentum of the electron. Therefore, there had to be an acceleration of the electron. And it also followed that, because the electron was a particle of electricity, the electron had to give out energy whenever it accelerated. Bohr’s picture did not seem to correspond with this observed fact.
What looks continuous is really discontinuous.
But Bohr was not to be dissuaded. He noticed that his rule of allowed and forbidden radiation was dependent on the size of the electron’s quantum jump. A jump from the second orbit to the first was extremely tiny, but a sizable change on the scale of the first orbital diameter. It was, therefore, a relatively enormous jump. On the other hand, a jump from orbit 10,000 to orbit 9,999 was very large when compared to the first orbital diameter, but an extremely small change in orbit on the scale of the 10,000th orbital diameter. Thus, it was a relatively tiny jump. When Bohr calculated the radiation from a change in orbits between large atomic diameter orbits, the result he found was in agreement with classically predicted results. In other words, the smaller the relative change, the more classical and continuous the result seemed to be.
Bohr had determined another exciting feature of quantum mechanics. It applied just where it was necessary. Wherever the world appeared to be continuous, the quantum “rules” corresponded with classical rules. This was called the Principle of Correspondence. Bohr felt very encouraged. He believed that he was on to one of God’s secrets. He knew why the world appeared continuous even though it was fundamentally a discontinuous and quantum jumping world. It was all a question of relative scale. To Bohr, discontinuity was a fundamental truth.
