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November 19, November 9, It is possible to describe the basic posits of quantum theory compactly. However these posits are very likely to appear arbitrary and even a little bewildering on first acquaintance. What is needed is some understanding of why those posits were chosen and what problems they are intended to solve.
The best way to arrive at this understanding is to review the historical developments in the course of the first quarter of the twentieth century that led to quantum theory. For in that historical development one can see a naturally growing sequence of problems and solutions that eventually issues in the modern theory.
It is interesting to reflect on why this historical approach is the one now chosen, but it was not used for the exposition here of special and general relativity.
The reason is that the basic phenomena to which those theories apply are already somewhat familiar; or it takes very little to introduce those phenomena. We all can imagine things moving very fast, for example, and it takes only a little more to imagine that somehow we cannot accelerated even fast moving things through the speed of light. Matters are different with quantum theory. The phenomena that control the theory are generally unfamiliar. Those outside physics rarely have a conception of Ehrenfest's "catastrophe in the ultraviolet" for heat radiation; or the odd dependency of the photoelectric effect on frequency; or why the discreteness of the lines of an emission spectrum is classically worrisome.
The historical approach familiarizes us with these classically puzzling phenomena. Then, when the resolution emerges as quantum theory, its design and role are immediately apparent. One was particles , localized lumps of stuff that flew about like little bullets. The best investigated of the fundamental particles was the electron. Thomson had found in that the cathode rays found in cathode ray tubes--the precursor of old fashioned glass TV tubes--were deflected by electric and magnetic fields just as if they were tiny little lumps of electrically charged matter.
Atoms, a bound collection of various particles, were also particulate in character. The most celebrated interference effect arises in the two slit experiment. Waves of light depicted as parallel wavefronts moving up the screen strike a barrier with two holes in it. Secondary waves radiate out from the two slits and interfere with each other, forming the characteristic cross hatching pattern of interference. These are the same patterns seen on the surface of a calm pond in the ripples cast off by two pebbles dropped in the water.
The essential thing in these interference experiments is the way the waves combine. The patterns arise because the waves can add up two ways. In constructive interference , the phases of the waves are such that they add to form a combined wave of greater amplitude.
The figure shows the greatest possible effect of constructive interference. All the parts of the two waves line up to interfere constructively everywhere. In destructive interference , the phases are such that the waves subtract to cancel out.
The figure shows the greatest possible effect of destructive interference. All parts of the two waves line up in such a way as to interfere destructively everywhere. In ordinary cases of interference, such as the two slit experiments, both destructive and constructive interference happen in different parts of the region where the waves intersect. That leads to the complicated interference patterns seen.
Discreteness of atomic spectra and more. Thermal properties of heat radiation and more. The information about how much energy came in the various parts of the spectrum of colors could be plotted on a graph. Here is one of the graphs plotted by Otto Lummer und Ernst Pringsheim, the leading experimenters from the time. The big peak in the curve corresponds to the location in the spectrum of most of the energy of the radiation and thus determines its color.
What Lummer and Pringsheim are showing here is that data they have found is not fitted well by the curves of the best current theories. The traditional picture inherited from the great achievements of nineteenth century physics was that light is a propagating wave.
What Einstein now urged was that high frequency light sometimes behaved as if it were made up of spatially localized bundles of energy. Planck's formula gave the amount of energy in each bundle. So once again light was said to consist of a shower of corpuscles , each corpuscle now with energy equal to h x frequency of light. Einstein's core argument was ingenious.
He looked at the observed properties of high frequency light and noticed they were governed in certain aspects by exactly the same laws that govern ordinary gases. By reverse engineering those gas laws, Einstein could show that they depended essentially on gases consisting of very many spatially localized little localized lumps of matter, their molecules. He supposed that it was no accident that light and gases obeyed the same laws; they did, he urged, because the light really was made of little localized units--called "quanta"--of energy.
The best known part of Einstein's paper on the light quantum was an observation made towards the end of the paper. Einstein had been following experiments on the so-called " photoelectric effect.
According to the wave theory of light, the intensity of the light ought to determine if the light can generate these "photoelectrons. If the light was of low frequency , its individual quanta would be of low energy, so no one quanta would be energetic enough to knock electrons out of the cathode.
Increasing the intensity of the light did nothing more than increasing the number of light quanta showering on the cathode, all them too weak in energy to liberate a photoelectron. If the light was of high frequency , then each light quantum was individually energetic enough to liberate a single photoelectron. The intensity of the light did not matter. Low intensity meant that there were not many light quanta incident on the cathode.
But since only one light quantum is needed to liberate just one photoelectron, the effect would be there for high frequency light, no matter how weak the intensity of the light. Here is the emission spectrum of hydrogen gas. The light emitted by excited hydrogen has been spread out into its component frequencies by passing it through a prism or diffraction grating. The light then darkens a photographic emulsion in different places according to its frequency.
The series of lines shown is the so-called "Balmer series" that appears in the visible and near visible frequencies of light. Wavelengths are shown in units of Angstroms. Prentice-Hall, In the Rutherford model, exciting a gas by passing high voltage electricity through it would energize the electrons, which could then move farther away from the attractive pull of the nucleus.
When they fell back towards the nucleus, the energy they gained would be lost as light energy; that emitted light forms the emission spectrum. The first difficulty was that, as they fell back to the nucleus, they would pass through a continuous range of orbital frequencies and thus emit a continuous range of frequencies of light. There was no way to limit the emitted light to just a few special frequencies. The second difficulty was more serious. Nothing stops the emission of energy by the electrons through this process of light emission.
They would continue to do it until they crashed into the nucleus. According to classical electrodynamics, this would happen very quickly. It was not clear that Rutherford's model allowed matter made of atoms to exist at all. Bohr solved both problems with a proposal of breath-taking audacity.
Classical electrodynamics was quite clear: an electron orbiting the nucleus is accelerating and therefore must radiate energy. It would be like a little radio transmitter, broadcasting electromagnetic waves. In the process, it must lose energy, fall deeper into the attractive pull of the nucleus and eventually crash into it nucleus. Bohr simply posited that this was not true. Rather, he asserted that there are stable orbits arrayed around the nucleus in which an electron could orbit indefinitely without losing any energy.
Next, Bohr supposed that electrons can jump up and down between these allowed orbits. If an electron is to jump up , away from the nucleus to a higher energy orbit, it needs to gain energy.
That enables it overcome the pull of the atom's positively charged nucleus and climb away from it. The electron gets that extra energy from light falling on the atom.
The energy of the light is transferred to the electron, which can then jump up to a higher energy orbit. The incident light must deliver exactly the right amount of energy to make up the difference between the energy of the orbit left behind and the one to which the electron jumps.
In addition Bohr assumed that the amount of energy drawn from the exciting light conforms with Planck's formula: the energy is just h times its frequency. The outcome is that light only of a very specific frequency can excite the jumps between two specific orbits.
For some specified jump, the frequency of the light has to be tuned precisely so that h times the frequency exactly matches the energy needed to complete the jump. Bohr's theory also allows for the reverse process. Once an electron has jumped up to a higher energy orbit, it will not stay there. It will jump back down to a lower energy orbit. In the process, it will re-emit the energy it gained in jumping up in the form of light of a definite frequency. Once again, the energy of the light emitted will conform to Planck's formula and be equal to h times its frequency.
As a result, when an electron jumps down between two orbits, it emits light of a definite frequency that is characteristic of exactly that jump. When Bohr did that, he found a very simple way to summarize just which of the orbits were allowed. We have seen that the ordinary linear momentum of a body is just its mass times its velocity. Angular momentum is an analogous quantity that plays an important role in the dynamics of rotating or orbiting systems. For a small mass like a classical electron orbiting a nucleus, it is defined as the electron's mass x radius of orbit x angular speed of electron.
In the years immediately following, that simple condition was expanded into a broader condition that a quantity known as "action" came only in whole multiples for physical systems that returned periodically to the same initial condition.
As a result the term " quantum of action " entered the physicists' vocabulary. This sidebar should contain a brief sentence that gives you a useful idea of the physical quantity "action. It probably doesn't help too much if I tell you that the trajectories of bodies obeying classical physical laws can be picked out as those that render extremal the action added up along the trajectories. Did that help? I didn't think it would. To begin, imagine an ordinary, classical particle confined to a box.
It bounces back and forth between the walls. Classical physics allows it to move at any speed. As a result it can have a continuous range of different energies. Now imagine instead that we are confining a wave to the same box. The stable waves that can persist within the box are so called " standing waves. When a string is plucked or bowed, the base note results from a standing wave whose half-wavelength is the length of the string. There are overtones also formed that give the richness of the sound.
The essential condition is that a wave can form as long as it has nodes--the points of no displacement--at either end of the string. The matter waves that can form within the box have the same structure as these tones and overtones.
We use the double width since standing waves have nodes at each half wavelength. Each of these waves turns out to have a different energy that depends on the wavelength of the standing wave.
Thus only very few definite energies are permitted for the waves trapped in the box; the many intermediate energies between them are not allowed. First, the new theory introduced an element of probability that was unknown in classical physics.
There are many processes for whose outcomes the theory can only give probabilities. Will this radioactive atom decay now or later? The best the theory can offer are probabilities. This circumstance proved deeply troubling to many thinkers of the era, including Einstein.
They found it repugnant to think that the fundamental laws of the universe might be probabilistic and described the difficulty as a breakdown of "causality. Second, the new quantum theory worked very well for small particles. However it was far less clear how it should be applied to macroscopic bodies. Tables, chairs, houses and elephants do not obviously manifest a combination of wave and particle-like properties.
Yet the theory said that they must. We will see in the next chapter how that problem continues to vex us today.
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