where ν0 is the critical frequency of light below which the photoelectric effect does not occur. This picture explains at once the two laws derived from experiment. If the frequency of the incident light is kept constant, the energy content of each quantum remains the same, and the increase of light intensity results only in the corresponding increase of the number of light quanta. Thus more photoelectrons are ejected, each of them with the same energy as before. The formula giving K as the function of ν explains the empirical graphs shown in Fig. 7b, predicting that the slope of the line should be the same for all metals having a numerical value equal to h. This consequence of Einstein’s picture of photoelectric effect stands in complete agreement with experiment and leaves no doubt of the reality of light quanta.
THE COMPTON EFFECT
An important experiment proving the reality of light quanta was performed in 1923 by an American physicist, Arthur Compton, who wanted to study a collision of light quantum with an electron moving freely through space. The ideal situation would be to observe such collisions by sending a beam of light through an electron beam. Unfortunately, the number of electrons in even the most intense electron beams available is so small that one would have to wait for centuries for a single collision. Compton solved the difficulty by using X-rays, the quanta of which carry very large amounts of energy because of the very high frequency involved. As compared with the energy carried by each X-ray quantum, the energy with which electrons are bound in the atoms of light elements can be disregarded and one can regard them (the electrons) as being unbound and quite free. Considering a free collision between light quantum and an electron in the same way as one considers a collision between two elastic balls, one would expect that the energy, and hence the frequency, of scattered X-rays would decrease with the increasing scattering angle. Compton’s experiments (Fig. 8) stood in complete agreement with this theoretical prediction, and with the formula derived on the basis of conservation of energy and mechanical momentum in the collision of two elastic spheres. This agreement gave additional confirmation of the existence of light quanta.
FIG. 8 Compton scattering of X-rays. Notice that after the collision the wavelength of X-ray quantum increases because of loss of energy to the electron.
CHAPTER IV
L. DE BROGLIE AND PILOT WAVES
Louis Victor, Duc de Broglie, born in Dieppe in 1892, who became the Prince de Broglie on the death of an elder brother, had a rather unusual scientific career. As a student at the Sorbonne he decided to devote his life to medieval history, but the onset of World War I induced him to enlist in the French Army. Being an educated man, he got a position in one of the field radio-communication units, a novelty at that time, and turned his interest from Gothic cathedrals to electromagnetic waves. In 1925 he presented a doctoral thesis which contained such revolutionary ideas concerning a modification of the Bohr original theory of atomic structure that most physicists were rather skeptical; some wit, in fact, dubbed de Broglie’s theory “la ComéAdie Française.”
Having worked with radio waves during the war, and being a connoisseur of chamber music, de Broglie chose to look at an atom as some kind of musical instrument which, depending on the way it is constructed, can emit a certain basic tone and a sequence of overtones. Since by that time Bohr’s electronic orbits were fairly well established as characterizing different quantum states of an atom, he chose them as a basic pattern for his wave scheme. He imagined that each electron moving along a given orbit is accompanied by some mysterious pilot waves (now known as de Broglie waves) spreading out all along the orbit. The first quantum orbit carried only one wave, the second two waves, the third three, etc. Thus the length of the first wave must be equal to the length 2π r 1 of the first quantum orbit, the length of the second wave must be equal to one-half of the length of the second orbit, , etc. In general, the nth quantum orbit carries n waves with the length 2πrn each.
As we have seen in Chapter II, the radius of the nth orbit in Bohr’s atom is
From the equality of the centrifugal force due to the orbital motion, and the electrostatic attraction beween the charged particles, we obtain:
or
Substituting this value of e2 into the original formula, we get
or
Extracting the square root from both sides of this equation we finally obtain:
Thus, if the length λ of the wave accompanying an electron is equal to Planck’s constant h divided by the mechanical momentum mν of the particle, then
and de Broglie could satisfy his desire to introduce waves of such a nature that 1, 2, 3, etc., of them would fit exactly into the 1st, 2nd, 3rd of Bohr’s quantum orbits (Fig. 19). The result given is mathematically equivalent to Bohr’s original quantum condition and brings in nothing physically new-nothing, that is, but an idea: the motion of the electrons along Bohr’s quantum orbits is accompanied by mysterious waves of the lengths determined by the mass and the velocity of the moving particles. If these waves represented some kind of physical reality, they should also accompany particles moving freely through space, in which case their existence or non-existence could be checked by direct experiment. In fact, if the motion of electrons is always guided by de Broglie waves, a beam of electrons under proper conditions should show diffraction phenomena similar to those characteristic of beams of light. Electron beams accelerated by electron tensions of several kilovolts (which are commonly used in laboratory experiments) should, according to de Broglie’s formula, be accompanied by pilot waves of about 10−8 cm wavelength, which is comparable to the wavelength of ordinary X-rays. This wavelength is too short to show a diffraction in ordinary optical gratings and should be studied with the technique of standard X-ray spectroscopy. In this method the incident beam is reflected from the surface of a crystal, and the neighboring crystalline layers, located about 10−8 cm apart, have the function of the more widely separated lines in opitical diffraction gratings (Fig. 20). This experiment was carried out simultaneously and independently by Sir George Thomson (son of Sir J. J. Thomson) in England, and G. Davisson and L. H. Germer in the United States, who used a crystal arrangement similar to that of Bragg and Bragg, but substituted for the beam of X-rays a beam of electrons moving at a given velocity. In the experiments a characteristic diffraction pattern appeared on the screen (or photographic plate) that was placed in the way of the reflected beam, and the diffraction bands widened or narrowed when the velocity of incident electrons was increased or decreased. The measured wavelength coincided exactly in all cases with that given by the de Broglie formula. Thus the de Broglie waves became an indisputable physical reality, although nobody understood what they were.
FIG. 19 De Broglie waves fitted to quantum orbits in Bohr’s atom model.
FIG. 20 An incident wave, be it a short electromagnetic wave (X-ray) or a de Broglie wave associated with a beam of fast electrons, produces wavelets as it passes through the successive layers of a crystal lattice. Depending on the angle of incidence, dark and light interference fringes appear. (P is the phase plane.)
Later on a German physicist, Otto Stern, proved the existence of the diffraction phenomena in the case of atomic beams. Since atoms are thousands of times more massive than electrons, their de Broglie waves were expected to be correspondingly shorter for the same velocity. To make atomic de Broglie waves of a length comparable with the distances between the crystalline layers (about 10−8 cm), Stern decided to use the thermal motion of atoms, since he could regulate the velocity simply by changing the temperature of the gas. The source consisted of a ceramic cylinder heated by an electric wire wound around it. At one end of the closed cylinder was a tiny hole through which the atoms escaped at their thermal velocity into a much larger evacuated container, and in their flight through space they hit a crystal placed in their way. The atoms reflected in different directions stuck to metal plates cooled by liquid air, and the number of atoms on the different plates was counted by a complicated method of chemical microanalysis. Plotting the n
umber of atoms scattered in different directions against the scattering angle, Stern obtained again a perfect diffraction pattern corresponding exactly to the wavelength calculated from de Broglie’s formula. And the bands became wider or thinner when the temperature of the cylinder was changed.
When in the late twenties I was working at Cambridge University with Rutherford, I decided to spend Christmas vacation in Paris (where I had never been before) and wrote to de Broglie, saying that I would like very much to meet him and to discuss some problems of the Quantum Theory. He answered that the University would be closed but that he would be glad to see me in his home. He lived in a magnificent family mansion in the fashionable Parisian suburb Neuillysur-Seine. The door was opened by an impressive-looking butler.
“Je veux voir Professeur de Broglie.”
“Vous voulez dire, Monsieur le Duc de Broglie,” retorted the butler.
“O.K., le Duc de Broglie,” said I, and was let into the house.
De Broglie, wearing a silk house coat, met me in his sumptuously furnished study, and we started talking physics. He did not speak any English; my French was rather poor. But somehow, partly by using my broken French and partly by writing formulas on paper, I managed to convey to him what I wanted to say and to understand his comments. Less than a year later, de Broglie came to London to deliver a lecture at the Royal Society, and I was, of course, in the audience. He delivered a brilliant lecture, in perfect English, with only a slight French accent. Then I understood another of his principles: when foreigners come to France they must speak French.
A number of years later when I was planning a trip to Europe and de Broglie asked me to deliver a special lecture in the institute of Henri Poincaré, of which he was a director, I decided to come well prepared. I planned to write the lecture down in my (still) poor French on board the liner crossing the Atlantic, have somebody in Paris correct the text, and use it as notes at the lecture. But, as everybody knows, all good resolutions collapse on an ocean voyage offering many distractions, and I had to face the audience in the Sorbonne completely unprepared. The lecture went through somewhat stumblingly, but my French held, and everybody understood what I had to say. After the lecture I told de Broglie that I was sorry that I did not carry out my original plan of having the corrected French notes. “Mon Dieu! ” he exclaimed, “it is lucky that you didn’t.
De Broglie told me about a lecture delivered by the noted British physicist R. H. Fowler. It is well known that since English is the best language in the world, the English are of the opinion that all foreigners should learn it, thus freeing themselves from the need to learn anyone else’s language. Since the lecture in the Sorbonne had to be in French, Fowler had prepared the complete English text of his lecture, and he sent it well in advance to de Broglie, who had personally translated it into French. Thus Fowler lectured in French, using the typewritten French text. De Broglie said that after the lecture a group of students came to him, “Monsieur le Professeur,” they said, “we are greatly puzzled. We expected that Professor Fowler would lecture in English, and we all know enough English to be able to understand. But he did not speak English but some other language and we cannot figure out what language it was.” “Et par fois !” added de Broglie, “I had to tell them that Professor Fowler was lecturing in French!”
SCHRÖDINGER’S WAVE EQUATION
Creating the revolutionary idea that the motion of atomic particles is guided by some mysterious pilot waves, de Broglie was too slow to develop a strict mathematical theory of this phenomenon, and, in 1926, about a year after his publication, there appeared an article by an Austrian physicist, Erwin Schrödinger, who wrote a general equation for de Broglie waves and proved its validity for all kinds of electron motion. While de Broglie’s model of the atom resembled more an unusual stringed instrument, or rather a set of vibrating concentrical metal rings of different diameters, Schrödinger’s model was a closer analogy to wind instruments; in his atom the vibrations occur throughout the entire space surrounding the atomic nucleus.
Consider a flat metal disc something like a cymbal fastened in the center (Fig. 21a). If one strikes it, it will begin to vibrate with its rim moving periodically up and down (Fig. 21b). There exist also more complicated kinds of vibrations (overtones) like the pattern shown in Fig. 21c where the center of the plate and all the points located along the circle between the center and circumference (marked by heavy line in the figure) are at rest, so that, when the material bulges up within that circle the material outside the circle moves down, and vice versa. The motionless points and lines of a vibrating elastic surface are called the nodal points and lines; one can extend Fig. 21c by drawing higher overtones which correspond to two or more nodal circles around the central nodal point.
FIG. 21 Various vibration modes of an elastic disc fastened in the center: (a) state of rest; (b) nodal point in the center; (c) one circular nodal line; (d) one radial nodal line; (e) two radial nodal lines; (f ) three radial and two circular nodal lines.
Besides such “radial” vibrations there also exist the “azimuthal vibrations” in which the nodal lines are straight lines passing through the center as shown in Fig. 21d and e, where arrows indicate whether the membrane lifts or sinks in respect to the equilibrium horizontal position. Of course, the radial and azimuthal vibrations can exist simultaneously in a given membrane. The resulting complex state of motion should be described by two integers nr and nφ, giving the numbers of radial and azimuthal nodal lines.
Next in complexity are the three-dimensional vibrations such as, for example, the sound waves in the air filling a rigid metal sphere. In this case it becomes necessary to introduce the third kind of nodal lines and also the third integer n1 giving their number.
This kind of vibrations was studied in theoretical acoustics many years ago, and, in particular, Hermann von Helmholtz in the last century made detailed studies of the vibrations of air enclosed in rigid metal spheres (Helmholtz resonators). He drilled a little hole in the sphere, to let in sound from the outside, and used a siren which emitted a pure tone, the pitch of which could be changed continuously by changing the rotation speed of the siren’s disc. When the frequency of the siren’s sound coincided with one of the possible vibrations of air in the sphere, resonance was observed. These experiments stood in perfect agreement with the mathematical solutions of the wave equation for sound, which is too complicated to be discussed in this book.
The equation written by Schrödinger for de Broglie’s waves is very similar to the well-known wave equations for the propagation of sound and light (that is, electromagnetic) waves, except that for a few years there remained the mystery of just what was vibrating. We will return to this question in the next chapter.
When an electron moves around a proton in a hydrogen atom the situation is somewhat similar to the vibration of gas within a rigid spherical enclosure. But whereas for Helmholtz vibrators there is a rigid wall preventing the gas from expanding beyond it, the atomic electron is subject to electric attraction of the central nucleus which slows down the motion when the electron travels farther and farther from the center, and stops it when it goes beyond the limit permitted by its kinetic energy. The situation in both cases is shown graphically in Fig. 22. In the figure on the left the “potential hole” (that is, the lowering of potential energy in the neighborhood of a certain point) resembles a cylindrical well; the figure on the right looks more like a funnel-shaped hole in the ground. The horizontal lines represent the quantized energy levels, the lowest of them corresponding to the lowest energy the particle can have. Comparing Fig. 22b with Fig. 12 of Chapter II, we find that the levels of the hydrogen atom calculated on the basis of Schrödinger’s equation are identical with those obtained from Bohr’s old theory of quantum orbits. But the physical aspect is quite different. Instead of sharp circular and elliptic orbits along which point-shaped electrons run, we have now a full-bodied atom represented by multishaped vibrations of something which in the early years of wave
mechanics was called, for lack of a better name, a ψ -function (Greek letter psi ).
FIG. 22 Quantum energy levels in a rectangular potential hole (a) and in a funnel-shaped potential hole (b).
It must be remarked here that the rectangular well potential distribution shown in Fig. 22a turned out to be very useful for a description of proton and neutron motion within the atomic nucleus, and later was used successfully by Maria Goeppert Mayer and, independently, by Hans Jensen for an understanding of the energy levels within the atomic nuclei and the origin of γ-ray spectra of radioactive nuclear species.
The frequencies of different ψ -vibration modes do not correspond to the frequencies of the light wave emitted by the atom, but to the energy values of the different quantum states divided by h. Thus, the emission of a spectral line necessitated the excitation of two vibration modes, say ψm and ψn , with the resulting composite frequency
which is the same as Bohr’s expression for the frequency of light quantum resulting from the transition of the atomic electron from the energy level Em to the lower energy level En .
APPLYING WAVE MECHANICS
Apart from giving a more rational foundation to Bohr’s original idea of quantum orbits, and removing some discrepancies, wave mechanics could explain some phenomena well beyond the reach of the old Quantum Theory. As was mentioned in Chapter II, the author of the present book and, independently, a team consisting of Ronald Gurney and Edward Condon successfully applied Schrödinger’s wave equation to the explanation of the emission of α-particles by radioactive elements, and their penetration into the nuclei of other lighter elements with the resulting transformation of elements. To understand this rather complicated phenomenon, we will compare an atomic nucleus to a fortress surrounded on all sides by high walls; in nuclear physics the analogy of the fortress walls is known as a potential barrier . Due to the fact that both the atomic nucleus and the α-particle carry a positive electric charge, there exists a strong repulsive Coulomb forceoy acting on the α-particle approaching a nucleus. Under the action of that force an α-particle shot at the nucleus may be stopped and thrown back before it comes into direct contact with the nucleus. On the other hand, α-particles that are inside the various nuclei as constituent parts of them are prevented from escaping by very strong attractive nuclear forces (analogous to the cohesion forces in ordinary liquids), but these nuclear forces act only when the particles are closely packed, being in direct contact with one another. The combination of these two forces forms a potential barrier preventing the inside particles from getting out and the outside particles from getting in, unless their kinetic energy is high enough to climb over the top of the potential barrier.
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