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Physics

Chapter 27: 1-6 Early Quantum Mechanics

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Web Lecture

Light Waves: Interference and Diffraction

Introduction

Quantum theory was not originally developed for the sake of interpreting photoelectric phenomena. It was solely a theory as to the mechanism of absorption and emission of electromagnetic waves by resonators of atomic and subatomic dimensions. It has nothing whatever to say about the energy of an escaping electron or about the conditions under which such an electron would make its escape, and up to this day the form of the theory developed by its author has not been able to account satisfactorily for the photoelectric facts presented here with. We are confronted, however, by the astonishing situation that these facts were correctly and exactly predicted nine years ago by a form of quantum theory which is not been pretty generally abandoned.

R. A. Millikan —The opening paragraph of A Direct Photoelectric Determination of Plank's "h" (1916)

Outline

Web Lecture

by Paul Bruno, SOLA/RCA student 1999/2000. Used by permission.

The year: 1905. The man: Albert Einstein. In one year, Albert Einstein shook up the entire scientific community with the publication of three different papers in one issue of a prestigious German scientific journal, called Annalen der Physik. The first was relatively mundane; it covered the Brownian motion of molecules, and gave formulae to quantitatively describe this motion. The next two, however, were anything but ordinary; they were groundbreaking, earth-shattering, amazing, radical, revolutionary. The first of these two papers covered the special theory of relativity, which went against the grain of common sense and did away with many of the preconceived notions about time, space, and mass of physicists of that day. The second of the papers did away with many physicists' concrete notions of the nature of light. This paper was on the photoelectric effect of light, one of the main topics of Chapter 27, our current chapter, which this lecture discusses. But,before we are able to go over that phenomenon and it implications, we must comprehend and understand the groundwork for this theory, laid by scientists like Thomson and Planck. So, to do this, we go back in time to the year 1897 and the laboratory of physicist J. J. Thomson.

The cathode ray tube (CRT, like that used in our television sets and computer monitors) had already been invented by someone else before Thomson. But Thomson was the first to actually put the CRT to good use, calculating the charge to mass ratio of the of the "cathode rays" (e/m = 1.76 * 1011 C/kg), and he was the first to theorize that these extremely small, negatively charged particles were component parts of atoms. Hence, he was attributed with the discovery of the electron. Later on, R. A. Millikan made the first accurate measurement of the of the charge of the electron (e = 1.6 * 10-19 C), and, because e/m had already been calculated, the mass of the electron could then be determined (which was found to be m = 9.1 * 10-31 kg). Now, let's go forward in time three years (1900) to the den and workplace of Max Planck.

When objects are heated to high temperature, they all emit electromagnetic radiation (EM waves) of many wavelengths. We already know this from Chapter 14, Section 9, in which an equation was presented for the calculation of the amount energy carried by those EM waves, when given the emitting temperature, surface area, and emissivity of the radiating object. Around the year 1900, much study was done on the spectra of light and EM waves of other wavelengths emitted by idealized blackbody objects. Before we go on, it's important that we clarify terms here. A blackbody object is just an object with an emissivity of 1, an object that absorbs all the radiation it comes into contact with. We talk about blackbodies and blackbody radiation here just out of simplicity: it's the easiest to deal with, and, as the book says, closely approximates the radiation of real objects. However, physicists of that day ran into some problems when they examined the spectra of radiation given off by these blackbody objects. According to the classical ideas and theories of James Clerk Maxwell (inventor of the electromagnetic theory), the oscillating electric charges of the atoms of the radiating objects were the culprits producing the electromagnetic waves, and the different wavelengths of EM radiation was due to each of the electric charges having different frequencies of oscillation from one another. This took care of the fact that heated bodies emit continuous spectra of EM waves. However, when the actual spectra emitted by blackbody objects was studied in the very beginning of the twentieth century, the data did not conform to what was predicted by Maxwell's equations. Two theories to explain this phenomenon based on the classical ideas and equations of Maxwell were submitted by W. Wien and Lord Rutherford. However, both of these equations did not "fit" the data well. However, the famous German scientist, Max Planck, devised an equation that correctly reproduced the intensity/wavelength curve. However, this equation forced him to make the assumption that the energy of oscillation of each of the charges of the radiating material was "quantized"; it could only have certain, discreet energy values instead of being able to have any energy value. More specifically, the charges, depending on the frequency at which they were oscillating, they could only have an energy level equal to an integral multiple of the frequency of the oscillation multiplied by Planck's constant (named after the scientist who first discovered it and calculated its value): E = nhf, where E is the energy of the oscillation, n is a non-zero, positive integer, h is Planck's constant, and f is the frequency of the oscillation of the charge. This theory is formally called Planck's quantum hypothesis.

However, here it is interesting to note that, while we today now recognize that this hypothesis of Planck's was a radical discovery, when Planck first proposed it, it was not thought of as revolutionary or groundbreaking in any way. Most scientists (including Planck himself!) still sought to explain this quantization by classical laws and formulae. However, in a few short years, others, including and especially Einstein, recognized the theory for what it really was. So, we journey forward in time again, this time to the year 1905, to the workplace and secret study of Albert Einstein, located in Bern, Switzerland.

Growing up as an old boy, and as a young man, Albert Einstein always had a problem: his study habits. In our vernacular today, Albert Einstein was very adept at playing "hooky." During his four years at a very prestigious technical college in Switzerland, Albert almost never went to class. This becomes important because, instead of having a job at a good college as a professor of physics (which requires a professor's recommendation; Albert was in class barely enough to know his professor's names), he had to take a job at the Swiss patent office in Bern, Switzerland, reviewing patents for new electrical inventions. During the day at his place of work, Albert would expediently review all of all the patents and get rid of his daily workload quickly. After this, he would have the rest of the day free to spend in the laboratory, contemplating new theories of others and theories of his own, scribbling on spare pieces of paper long, intricate formulas, and performing his "gedanken" experiments. The material in his three 1905 papers come from the work and thinking he did while at his desk, stacks of patents lying all around. And it is there where discovered the particle nature of light and came up with his third paper on the photon theory of light and the photoelectric effect.

In this revolutionary paper, Einstein recognized Planck's quantum hypothesis for revolutionary discovery that it really was, and extended it with ideas of his own. Going back to the oscillating charges of the atoms in a radiating body, Einstein theorized that for light to be emitted, these oscillating charges must lose energy in the process for energy to be conserved. However, we know from Planck's quantum hypothesis that the energy of oscillation can't be just any old amount; if E = nhf is the energy of oscillation of a particular charge before emitting light, then its energy afterward must be E = (n - a)hf, where n, h, and f are the same as they were in the equation for the quantum hypothesis of Planck, and a is another positive, nonzero integer that is less than n. As it seems from Einstein's arguments, light must be emitted in small packets, or quanta. If the energy of the oscillation of the charge drops by only one energy "level," a = 1, the equation above for the energy afterwards becomes E = (n - 1)hf. Then, from conservation of energy, we know that the quanta (or packet) of light just emitted has an energy E = hf. Also, all light comes from radiating sources, whether they be "natural," like that emitted by the sun and the distant stars, or "artificial," like the light bulb filament in the 100 W bulb nested in the pastel blue lamp in your family room. All this seems to suggest that light is emitted as particles (yes, particles! titled photons), and not waves, as most had previously thought. Einstein thought that most physicists were going to be skeptical (to say the least) of his new theory, and they were. But Einstein was prepared. In his paper, Einstein suggested a test (which he never actually carried out himself) of his new, radical, unbelievable, ridiculous theory: "A quantitative measurement of the photoelectric effect," in the words of the book.

The apparatus for this experiment consists of a battery, and ammeter, and a photocell, all connected in circuit. A photocell is basically an oddly shaped capacitor; one of the plates resembles a satellite dish, and the other is in the shape of a very short car radio antenna. The two electrodes, as they are called in a photocell, are separated by a distance and are contained in an evacuated glass tube. When the circuit and photocell are kept in the dark, not illuminated, the ammeter indicates no steady current flowing through the circuit. However, when a light of ample frequency is shone on the larger satellite-like plate, electrons are ejected from the surface of the larger plate, cross the gap between the two "plates," and reach the receptor on the other end. This phenomenon is called the photoelectric effect. This fact can be verified by seeing that the ammeter measures a current when the light is shone.

Now, this fact, in and of itself does not contradict the wave theory of light. Light, being an electromagnetic wave, is a wave of alternately oscillating electric and magnetic fields. The field when it strikes the metal exerts a force on the electrons of the metal, and pulls them free. Differences only appear when further data is collected, such as the maximum kinetic energy of the electrons. This piece of information can be collected with a slight modification to our apparatus described above. If we spin the photocell around and reconnect it, the large plate will now be positive and the small bulbous receptor negative. If light is again shone on the larger plate, some electrons some of the fastest electrons with the greatest KE will still leave the large plate and reach the small electrode, in spite of the fact that the force of the electric field due to the electrons at the small electrode resists this movement. Again, this fact can be measured by our ammeter, which will now indicate a current running the opposite direction. (Because of the direction of the current, this actually acts to charge the battery.) This is true if we keep our source of emf, our battery, at a low voltage. However, if we increase the voltage, there will be a voltage at which the current ceases to flow through the wires and the ammeter reads 0. This voltage (labeled V0) is called the stopping voltage. What happens is that the electrons at this voltage have just enough kinetic energy to just reach the small electrode, but not enough to penetrate the surface and continue flowing through the circuit. So, at the point where this electron reaches this small receptor, its kinetic energy drops to zero; by conservation of energy, the KE has been entirely converted to potential energy. This is basically the same as the illustration the book gave when it was defining electric potential, with the two oppositely charged plates and the charged particle, except in reverse. The maximum KE (hereafter labeled KEmax) is given by this equation: KEmax = eV0. We get this equation from the law of conservation of energy, as discussed above. In this equation, e is the charge on an electron. Thus, the maximum kinetic energy of electrons can be determined.

The predictions made by these two theories of light differ as it concerns the maximum KE and the number of electrons emitted. The wave theory of light predicts than an increase in the intensity of the light beam hitting the large electrode will pull more electrons off of the electrode, and will increase the maximum KE of the molecules because of greater electric field strength. However, the photon theory predicts that only the amount of electrons coming off the large plate will change; this is because an increase in the intensity of the beam changes the amount of particles emitted. The KEmax stays unchanged. Furthermore, the wave theory of light predicts that a change in frequency should not change the maximum KE of the electrons, whereas the photon theory says that KEmax is directly proportional to the frequency. Furthermore, if the frequency f is lower than the "cutoff" frequency (labeled as f0), as the book likes to phrase it, no electrons will be ejected, no matter what the intensity. The reason for this is because hf0 = W0, where W0 represents the work function, the amount of energy needed to free the most loosely held electrons from the atom. The frequency of light need be large enough so that hf >= W0. If it is not, not a single electron will leave any of the atoms. Increases in intensity (amount of photons) won't help this situation.

As you can see, the differences between the "opinions" of the two theories of light are clearly and readily apparent. R. A. Millikan, the same physicist who first accurately measured the charge of an electron, performed the photoelectric effect experiments, and conclusively found that the observations he made supported the photon theory of light.

Further evidence for the validity of Einstein's theory came in 1923 when experiments on the Compton effect were carried out by its discoverer, A. H. Compton. This Compton effect could only be explained on the basis of the photon theory of light, and not the wave theory. The Compton effect is the reduction in frequency of light when reflected off the surface of a material. This is a result of the photon striking an electron at rest. The electron then speeds off from the collision. Because of the law of conservation of energy, we know, therefore, that some of the energy of the photon was taken away and given to the electron. Since the energy of the photon is given by E = hf, the frequency of the light must have decreased. (A more full mathematical analysis of this effect is given in Section 27-4.) But the wave theory of light offers no explanation to this phenomenon.

Well, it seems that we've come to that proverbial rock and hard place, at least, concerning light, that is. From the results of experiments performed on the photoelectric effect, the Compton effect, and others seem to indicate that light consists of many small particles today named photons. However, what about all the information and experiments described in Chapter 24 of this book? What about the double slit experiments? And the diffraction gratings? Or the diffraction patterns? And let us not forget the interference patterns like Newton's rings and the polarization of light! What about those?

It seems that light is a little more complex, intricate, enigmatic, and perhaps even interesting than we originally thought it to be.

This is precisely the problem that many physicists, even the prestigious and mighty ones, such as Max Planck, had problems with. Physicists today just have to come to terms with the fact that light behaves like a wave part of the time and like a stream of particles the other part. The results of all these experiments indicate that light has a dual nature. This nature is today formally called the "Wave-Particle Duality." It was not until long after 1905, when Einstein's paper on the photon theory of light was published, that physicists finally began to accept this fact.

As stated above, Max Planck, one of the most powerful and influential physicists of his day, was among those having difficulties with Einstein's photon theory and the results of the photoelectric effect. After the publication Albert's theories, Max Planck, who greatly admired the twenty-six year old German-born Jewish physicist, would have communicate with and have discussions with Albert via mail and Planck's assistant, who would travel from Germany to Switzerland to see Albert in person. This assistant of Planck's once asked the great Einstein, "How can light be both a particle and a wave, Herr Einstein? It doesn't make sense." Albert, with a grin, replied, "Don't let the dictionary get in the way of science. 'Particle,' 'wave,' these are merely words. The important thing is to discover the way that nature behaves. In some cases, light acts like a particle. In others, like a wave. I can accept the fact that nature has more than one mood."

To aid in the understanding of this principle, Niels Bohr put forth his principle of complementarity. It states that, for a given phenomenon, effect, or experiment, you can only apply the wave theory of light, or the particle theory, but not both. To explain the Compton effect, you use the photon theory of light, but you cannot apply it to the double slit diffraction patterns. Of course you can explain diffraction patterns by the wave nature of light, but that theory offers no explanation of the photoelectric effect and its outcome. But, to fully understand light, you must have an understanding and comprehension of both natures of light. Therefore, as indicated in the title of the principle, the two natures complement each other.

Now, let's not go forward in time at all (still in 1923), but travel to the home and workplace of Louis de Broglie, a famous French scientist who would further extend the idea of this "Wave-Particle Duality."

In that year, de Broglie wondered that if things originally thought to be waves (a.k.a., light) have a particle nature, then wouldn't it be possible, likely, and even logical if things originally thought of as strictly particles (such as electrons and other material objects) had a wave nature? Two scientists by the names of C. J. Davisson and L. H. Germer found that when they scattered electrons off of a crystal, the electrons formed a diffraction pattern, confirming de Broglie's suspicions.

Once again, it seems like God's universe is a little more complicated than we made it out to be.

image27_92756a

In a slight deviation from the discussion of all this exciting (or mundane, depending on your viewpoint) theory, we look at a practical application of the "Wave-Particle Duality": The electron microscope. The electron microscope takes advantage of the wave nature of light to enable it to produce images enlarged tens of thousands of times, and more. Because of its super high magnification, the electron microscope is often used in the field of medicine and medical research. It can even be used to produce three dimensional topographical images of microscopic cells, like DNA! So that you can get a good "feel" for the type of magnification these devices offer, I've placed a picture taken from an electron microscope directly to the left. There, you see a picture of a blood clot forming. The red objects in the picture are (in case you haven't figured it out yet) are blood cells, the blue are platelets, and the yellow stuff is the fibrin clot. The picture comes from a scanning electron microscope (SEM). The magnification of that image is 10,980 X!!! For many more interesting electron microscope pictures, head over to http://www.pbrc.hawaii.edu/~kunkel/gallery/. There you'll find 300+ interesting images taken from electron microscopes. (Btw, if you note the faint white lines on each of the images, that's just a watermark of some sort, and is not actually a part of the image.)

Now, hereafter, we discuss the different, progressing theories given for the structure of the atom in order to explain the results of certain experiments and equations; some that have already been discussed, and some that haven't yet. So, to start this discussion of atomic theory, we once again go back in time (Boy, we've been doing a lot of time travel today, haven't we?), now to the year 1911, to the laboratory of Ernest Rutherford and his companions and fellow scientists in England.

Back in 1897, the discovery of the electron was made; atoms, comprising the structure of material objects, were now thought to have a structure themselves. Thomson's model for the structure of the atom looked a lot like a popular dessert of his day in England, a plum-pudding. This model consisted of a "pudding" of positive material of some sort, and "plums" or electrons put in the pudding. However, in this year, 1911, Ernest Rutherford performed experiments whose outcome seemed to contradict Thomson's theory. In this experiment, Rutherford shot alpha particles (positively charged, heavy, relative to the mass of an electron, particles) from a rock containing radium at a thin metal sheet surrounded by a viewing screen. (A diagram for the set up of this experiment can be found on page 840, Fig. 27-23a.) According to Thomson, because the positive charge is so far spread apart, and because of the minute mass of the electron, the alpha particle should not have encountered anything to strongly repel or deflect it. Therefore, all of the alpha particles should have arrived at around the same spot on the viewing screen, having gone pretty much straight through. However, when the experiment was carried out, only most of them found this spot on the viewing screen. Others deflected wildly to the left and right of that spot, at significant angles to the normal of the foil, and some even "bounced" backwards off the thin metal sheet and hit places on the viewing screen that were on the same side from which they came, almost straight back. This seemed to indicate that instead of there being one large positive "pudding" as previously thought by Thomson, the positive charge in atoms are located at the center of that atom, concentrated in one small volume, around which the even smaller electrons orbit, according to Rutherford. Now we turn back the clocks one more time, 1885, when experiments and research was being done on hydrogen line spectrum.

Line spectra is the phenomenon when gases emit light when they are heated to high temperatures or when a large voltage passes across them, as discussed in Chapter 24. Moreover, the certain gases only radiate certain wavelengths; the line spectrum for a certain gas serves as its fingerprint. Much attention was paid to the line spectrum emitted by hydrogen. Most of the other elements emitted spectra with little or no regularity in the wavelengths given off, but the line spectrum of the element hydrogen gives off a regular pattern of lines. In this year, J. J. Balmer gave an equation for the wavelengths of a portion of the hydrogen line spectrum (which is named after him, called the Balmer series). Later on, similar, related equations derived by two other scientists, Lyman and Paschen, were for determining the wavelengths of the other portions of lines, similarly called the Lyman and Paschen series.

Later on, just like with Thomson's plum-pudding model of the atom before, problems were found with the Rutherford model of the atom. It had no explanation for the spectra just described. Moreover, because the electrons were orbiting around the nucleus, they were accelerating, so they should emit light, being accelerating charges. But, of course, we know that an atom, in its steady state, does not emit light of any kind. And there were many more "holes" found in Rutherford's theory. These holes in the atomic theory of Rutherford were filled by the great Danish physicist Niels Bohr by making modifications to Rutherford's theory.

Niels Bohr integrated the results from Planck's quantum theory Einstein's photon theory into his modifications of Rutherford's so-called "planetary" model of the atom. First the assumption was made that even though electrons were accelerating in its orbit around the nucleus, no light was emitted. This is consistent with the Einstein's photon theory of light. Next he postulated that the electrons of an atom cannot lose or gain energy at any rate (i.e., continuously), but would have to gain or lose energy in discreet amounts, in accord with Planck's quantum hypothesis. According to Bohr, an electron can absorb and reemit light by making quantized "jumps," from one energy level to another. He theorized that, because of these two premises, electrons can only occupy certain orbits of certain energy levels, and that light is emitted when an electron moves from an orbit of higher energy to an orbit of lower energy. (Here well not go into a deep, mathematical analysis of the Bohr model, as an excellent one is presented in Section 27-10.) Bohr derived equations relating amount of energy given off when a photon of light is emitted to the before and after orbit energies of the electron that emitted the photon. Bohr then found that his equations were in accord with the equations given by Balmer for the hydrogen line spectra if he made yet another assumption, that the angular momentum of the electrons orbiting the nucleus is also quantized. This assumption had no firm theoretical footing until sometime later de Broglie showed that it did, on the wave nature of matter.

Bohr's atomic theory was a success in a few areas. First, it gave an explanation of phenomenon of line spectra for gases, and even accurately described the specific line spectrum of hydrogen. However, things start to break down for Bohr's theory, too, when you begin to discuss the line spectra of other multi-electron atoms and ions, and other such problems. This leads us to the topic of our next chapter; the new theory that resolved these problems and more: quantum mechanics.

Optional

The history of the development of a theory of atomic structure is also covered in the Natural Science class. You can start with the Curies and Radiation, or jump straight into Quantum Mechanics.

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