Chapter V: The Bohr Theory of the Hydrogen Spectrum (2)

To get an idea of some of the difficulties inherent in the attempt to make concrete pictures of the nature of the processes, let us again consider the analogy between the Bohr atom of hydrogen and a special kind of musical instrument in which sounds are produced by the fall of a small sphere between discs at various heights (see p. 120). It will be most natural here to think of the sounds as developed by the sphere when it hits the lower disc, and to think of the tones of higher pitch as given by the harder blows, corresponding to the larger energy (determinative of the pitch) released by the fall. We can, however, by no means transfer such a picture to the atomic model. For in the latter we cannot think of the stationary state as a material thing which the electron can hit, and it is also unreasonable to imagine that the radiation is not emitted until the moment when the transition is over and the electron has arrived in its new stationary state. We must, on the contrary, assume that the emission of radiation takes place during the _whole_ transition, whether the latter consumes a shorter or longer time. If it were the case that a transition always took place between two successive stationary states, it would then be possible to use the musical instrument to illustrate the matter. Let us denote the discs from the lowest one up with the numbers 1, 2, 3, ... corresponding to the stationary states 1, 2, 3, ... and for the moment consider a fall from disc 6 to disc 5. We can now imagine that the space between the two discs is in some way tuned for a definite note. Thus we might place between the discs a series of sheets of paper having such intervals between them that the sphere in its fall strikes their edges at equal intervals of time, _e.g._, ¹/₁₀₀ second. The disturbance then set up will produce a sound with the frequency 100 vibrations per second. If the distance between the discs 5 and 4 is double that between 6 and 5, the sphere in the fall from 5 to 4 will lose double the energy lost in the descent from 6 to 5, and will therefore emit a note of double frequency. The sheets of paper in the space between 5 and 4 must then be packed more tightly than between 6 and 5. And so the space between any two discs may thus be said to have its own particular classification or “tuning.” In analogy with this we might think of the space about a hydrogen nucleus divided by the stationary states into sections each with its own “tuning.” But apart from the intrinsic peculiarity of such an arrangement and the particular difficulties it will meet in trying to explain the more complicated phenomena to be mentioned later, the one fact that the electron in a transition from one stationary state to another can jump over one or more intervening stationary orbits, makes such a representation impossible. If the sphere in the given example could fall from disc 6 to disc 4, it should during the whole descent emit a note of higher pitch than in the descent from 6 to 5. But this could not possibly take place, if the space from 6 to 5, which must be traversed _en route_ to 4, is tuned for a lower note. The same consideration applies to the hydrogen atom. Naturally it is not impossible to continue the effort to illustrate the matter in some concrete manner (one might, for example, imagine separate channels each with its own particular tuning between the same two discs). But in all these attempts the situation must become more and more complicated rather than more simple.

On the whole it is very difficult to understand how a hydrogen atom, where the electron makes a transition from orbit 6 to orbit 4, can during the entire transition emit a radiation with a frequency different from that when the electron goes from orbit 6 to orbit 5. Although it seems as if the two electrons in making the transition are at first under identical conditions, still, nevertheless, the one which is going to orbit 4 emits from the first a radiation different from that emitted by the one going to orbit 5. Even from the very beginning the electron seems to arrange its conduct according to the goal of its motion and also according to future events. But such a gift is wont to be the privilege of thinking beings that can anticipate certain future occurrences. The inanimate objects of physics should observe causal laws in a more direct manner, _i.e._, allow their conduct to be determined by their previous states and the contemporaneous influences on them.

There is a difficulty of a similar nature in the fact that from the same stationary orbit the electron sometimes starts for a single jump, another time for a double jump, and so on. From certain considerations it is often possible to propound laws for the probability of the different jumps, so that for a great quantity of atoms it is possible to calculate the strengths (intensities) of the corresponding spectral lines. But we can no more give the reason why one given electron at a given time determines to make a double jump while another decides to make a single jump or not to jump at all, than we can say why a certain radium nucleus among many explodes at a given moment (cf. p. 102). This similarity between the occurrence of radiation processes on the Bohr theory and of the radioactive processes has especially been emphasized by Einstein.

It must, by no means, be said that the causal laws do not hold for the atomic processes, but the hints given here indicate how difficult it will be to reach an understanding—in the usual sense—of these processes and consequently of the processes of physics in general. There is much that might indicate that, on the whole, it is impossible to obtain a consistent picture of atomic processes in space and time with the help of the motions of the nuclei and the electrons and the variations in the state of the ether, and with the application of such fundamental conceptions of physics as mass, electric charge and energy.

Even if this were the case, it does not follow that a comprehensive description in time and space of the physical processes is impossible in principle; but the hope of attaining such a description must perhaps be allied to the representation of “physical individuals” or material particles of an even lower order of magnitude than the smallest particles now known—electrons and hydrogen nuclei—and to ideas of more fundamental nature than those now known; we are here outside our present sphere of experience.

From all the above remarks it would be very easy to get the impression that the Bohr theory, while it gives us a glimpse into depths previously unsuspected, at the same time leads us into a fog, where it is impossible to find the way. This is very far from being the case. On the contrary, it has thrown new light on a host of physical phenomena of different kinds so that they now appear in a coherence previously unattainable. That this light is not deceptive follows from the fact that the theory, which has been gradually developed by Bohr and many other investigators, has made it possible to predict and to account for many phenomena with remarkable accuracy and in complete agreement with experimental observation. The fundamental concepts are, on the one hand, the stationary states, where the usual laws of mechanics can be applied (although only within certain limits), and, on the other, the “quantum rule” for transitions between the states. But at the very beginning it has been necessary in many respects to grope in the dark, guided in part by the experimental results and in part by various assumptions, often very arbitrary.

For Bohr himself, a most important guide has been the so-called _correspondence principle_, which expresses the previously mentioned connection with the classical electrodynamics. It is difficult to explain in what it consists, because it cannot be expressed in exact quantitative laws, and it is, on this account, also difficult to apply. In Bohr’s hands it has been extraordinarily fruitful in the most varied fields; while other more definite and more easily applicable rules of guidance have indeed given important results in individual cases, they have shown their limitations by failing in other cases. We can here merely indicate what the correspondence principle is.

As has been said (cf. p. 130), it has been found that in the limiting region (sufficiently low frequencies) where the Bohr theory and the classical electrodynamics are merged in their outward features, a series of frequencies ν₁, ν₂, ν₃ for monochromatic radiation, emitted by different atoms in the single jumps, double jumps, etc., of the electrons, are equal to the frequencies ω, 2ω, 3ω ... which, according to the laws of electrodynamics, are contained in each of these atoms respectively as fundamental and the first, second ... overtones in the motion of the electron. Farther away from this region the two sets of frequencies are no longer equally large, but it is easy to understand, from the foregoing, the meaning of the statement that, for example, the radiation of a triple jump with the frequency ν₃ “corresponds” to the second overtone 3ω in the revolution of the electron. It is this correspondence which Bohr traces back to the regions where there is even a great difference in two successive orbits and where the frequency produced by a transition between these orbits is very different from the frequencies of revolution in the two orbits or their overtones. He expresses himself as follows: “The probability for the occurrence of single, double, triple jumps, etc., is conditioned by the presence in the motion of the atom of the different constituent harmonic vibrations having the frequency of the fundamental, first overtone, second overtone, etc., respectively.”

In order to understand how this “correspondence,” apparently so indefinite, can be used to derive important results, we shall give an illustration. Let us assume that the mechanical theory for the revolution of an electron in the hydrogen atom had led to the result that the orbits of the electrons always had to be circles. According to the laws of electrodynamics, the motion of the electron would in this case never give any overtones, and, according to the correspondence principle, there could not appear among the frequencies emitted by hydrogen any which would correspond to the overtones, _i.e._, there would not be any double jumps, triple jumps, etc., produced, but the only transitions would be those between successive stationary orbits. The investigation of the spectrum shows, however, that multiple jumps occur as well as single jumps, and this fact may be taken as evidence that the orbits in the hydrogen atom are not usually circles. Let us next assume that, instead, we had obtained the result that the orbits of the electrons are always ellipses of a certain quite definite eccentricity, corresponding to certain definite ratios in intensity between the overtones and the fundamentals; that, for instance, the intensity of the classical radiation due to the first overtone is in all states of motion always one-half that due to the fundamental, the intensity due to the second overtone always one-third that due to the fundamental, etc. Then the radiation actually emitted should, according to the correspondence principle, be such that the intensities of the lines corresponding to the double and triple jumps, which start from a given stationary state, are respectively one-half and one-third of the intensity corresponding to a single jump from the same state.

By these examples we can obtain an idea of how the correspondence principle may in certain cases account for various facts, as to what spectral lines cannot be expected to appear at all, although they would be given by a particular transition, and concerning the distribution of intensities in those which really appear. The illustration given above, however, has really nothing much to do with actual problems, and objections may be raised to the rough way in which the illustration has been handled. The correspondence principle has its particular province in more complicated electron motions than those which appear in the unperturbed hydrogen atom—motions which, unlike the simple elliptical motion, are not composed of a series of harmonic oscillations (ω, 2ω, 3ω ...) but may be considered as compounded of oscillations whose frequencies have other ratios. The correspondence principle has, in such cases, given rise to important discoveries and predictions which agree completely with the observations.

We have dwelt thus long upon the difficult correspondence principle, because it is one of Bohr’s deepest thoughts and chief guides. It has made possible a more consistent presentation of the whole theory, and it bids fair to remain the keystone of its future development. But from these general considerations we shall now proceed to more special phases of the problem and examine one of the first great triumphs in which the theory showed its ability to lead the way where previously there had been no path.

The False Hydrogen Spectrum.

In 1897 the American astronomer, Pickering, discovered in the spectrum of a star, in addition to the usual lines given by the Balmer series, a series of lines each of which lay about midway between two lines of the Balmer series; the frequencies of these lines could be represented by a formula which was very similar to the Balmer formula; it was necessary merely to substitute _n_ = 3½, 4½, 5½, etc., in the formula on p. 57 instead of _n_ = 3, 4, 5, etc. It was later discovered that in many stars there was a line corresponding to _n″_ = ³/₂ or _n′_ = 2 in the usual Balmer-Ritz formula (p. 59). It was considered that these must be hydrogen lines, and that the spectral formula for this element should properly be written

1 1
ν = K ---------- - ---------
( _n″_/2 )² ( _n′_/2 )²

where _n″_ and _n′_ can assume integral values. This was done since it was not to be believed that the spectral properties of chemically different elements could be so similar. This view was very much strengthened when Fowler, in 1912, discovered the Pickering lines in the light from a vacuum tube containing a mixture of hydrogen and helium. It could not quite be understood, however, why the new lines did not in general appear in the hydrogen spectrum.

According to the Bohr theory for the hydrogen spectrum it was impossible—except by giving up the agreement (cf. p. 129) with electrodynamics in the region of high orbit numbers—to attribute to the hydrogen atom the emission of lines corresponding to a formula where the whole numbers were halved. The formula given above might, however, also be written as

( 1 1 )
ν = 4K(------ - ------).
( _n″_² _n′_² )

If the earlier calculations had been carried out a little more generally, _i.e._, if instead of equating the nuclear charge with 1 elementary electric quantum _e_, as in hydrogen, it had been equated with N_e_ where N is an integer, then the frequency might have been written as

( 1 1 )
ν = N²K(------ - ------).
( _n″_² _n′_² )

This formula is evidently the same as that just given when N equals 2. Now we know that helium has the atomic number and nuclear charge 2 (cf. p. 90); a normal neutral helium atom has two electrons and it is, therefore, very different from a hydrogen atom. If, however, a helium atom has lost one electron and therefore has become a positive ion with one charge, it is a system like the hydrogen atom with only one single electron moving about the nucleus. It differs in its “outer” characteristics from the hydrogen atom only in having a nuclear charge twice as great, _i.e._ its spectral formula must be given with N = 2, or N² = 4. The formula for the supposed hydrogen lines would consequently fit the case of a helium atom which has lost an electron. Bohr was aware of this, and he therefore suggested that the lines in question were due, not to hydrogen, but to helium.

At first all the authorities in the field of spectroscopy were against this view; but most of the doubt was dispelled when Evans showed that the lines could be produced in a vacuum tube where there was only helium with not a trace of hydrogen.

In a letter to _Nature_ in September 1913, Fowler objected to the Bohr theory on the ground that the disputed line-formula did not exactly correspond to the formula with 4K, but that there was a slight disagreement. Bohr’s answer was immediate. He called attention to the fact that—since temporarily he had sought only a first approximation—in his calculations he had taken the mass of the nucleus to be infinite in comparison to the mass of the electron, so that the nucleus could be considered exactly at the focus of the ellipse described by the electron. In reality, he said, it must be assumed that nucleus and electron move about their common centre of gravity, just as in the motion in the solar system it must be assumed that not the centre of the sun, but the centre of gravity of the entire system remains fixed. This motion of the nucleus leads to the introduction of a factor M/(M + _m_) in the expression for the constant K given on p. 129, where M is the mass of the nucleus and _m_ that of the electron, which in hydrogen is ¹⁄₁₈₃₅ that of the nucleus. In helium, M is four times as large as in hydrogen, so that the given factor here has a slightly different value. The difference in the values for K for the hydrogen and for the helium spectrum which was found by Fowler, is 0·04 per cent., which agrees exactly with the theoretical difference.

Bohr thus turned Fowler’s objection into a strong argument in favour of the theory.

The Introduction of more than one Quantum Number.

During the first years after 1913, Bohr was practically alone in working out his theory, at that time still assailed by many, and in showing its application to many problems. In 1916, however, the theorists in other countries, led by the well-known Munich professor, Sommerfeld, began to associate themselves with the Bohr theory, and their investigations gave rise to much essential progress. We shall here mention some of the most important contributions.

In the theory for the hydrogen spectrum propounded above, it was assumed that we had to do with a single series of stationary orbits, each characterized by its quantum number. But as shown by theoretical investigations each of the stationary orbits must, when more detail is asked for, also be indicated by an additional quantum number.

This is closely connected with the fact that the motion of the electron is not quite so simple as previously assumed. We have assumed that the electron moves about the nucleus just as a planet moves about the sun (according to Kepler’s Laws), in an ellipse with the sun at one focus, since the electron is influenced by an attraction inversely proportional to the square of the distance, just as the planets are attracted by the sun according to Newton’s Law. We must, however, remember that we are here concerned with the electric attraction which at a given distance is determined, not by mass, but by the electric charges in question. If the latter remain unchanged, while the mass of the electron varies, the motion will be changed, because the same force has less effect upon a greater mass. According to the Einstein principle of relativity, the mass of the electrons, in accordance with ideas expounded long ago by J. J. Thomson, will not be constant, but to a certain extent depend upon the velocity, which will vary from place to place, when the orbit is an ellipse. As a result of this, the motion becomes a central motion of more general nature than a Kepler ellipse. Since the influence of the change of mass is very small, the orbit can still be considered as an approximate Kepler ellipse; but the major axis will slowly rotate in the plane of the orbit. In reality, the orbit will not therefore be closed, but will have the character which is shown in Fig. 26; this, however, corresponds to a much more rapid rotation of the major axis than that which actually takes place in the hydrogen atom, where—even in case of the swiftest rotation—the electron will revolve about 40,000 times round the nucleus at the same time as the major axis turns round once.

If the electron moves in a fixed Kepler ellipse, the energy content of the atom will be determined by the major axis of the ellipse only. If these axes for the stationary states with quantum numbers 1, 2, 3 ... are respectively denoted by 2_a_₁, 2_a_₂, 2_a_₃ ... the frequency, for instance, in the transition from No. 3 to No. 2—since it is determined by the loss of energy—will be the same whether the orbits are circles or ellipses. If, on the other hand, the electron moves in an ellipse which itself rotates slowly, the energy content, as can be shown mathematically, will depend not only upon the major axis of the ellipse, but also to a slight degree upon its eccentricity, or, in other words, on its minor axis. Then in the transition 3-2 we shall get different energy losses and consequently different frequencies, according as the ellipse is more or less elongated. If it were the case that the eccentricity of the ellipses for a certain quantum number could take arbitrary values, then in the transition between two numbers we could get frequencies which may take any value within a certain small interval, _i.e._, a mass of hydrogen with its great quantity of atoms would give diffuse spectral lines, _i.e._ lines which are broadened over a small continuous spectral interval. This is, however, not the case; but long before the appearance of the Bohr theory it had been discovered that the hydrogen lines, which we hitherto have considered as single, possess what is called a _fine structure_. With a spectroscopic apparatus of high resolving power each line can be separated into two lying very close to each other. This fine structure can now be explained by the fact that in a stationary state with quantum number 3 and major axis of the orbit 2_a_₃, for instance, the eccentricity of the orbit has neither one single definite value nor all possible values, but, on the contrary, it has several discrete values of definite magnitude, to which there correspond slightly different but definite values of the energy content of the atom. It is now possible to designate the series of stationary orbits, which have the major axis 2_a_₃ with the principal quantum number 3, with subscripts giving the _auxiliary quantum number_ for stationary orbits corresponding to the different eccentricities, so that the series is known as 3₁, 3₂, 3₃. Instead of a single line corresponding to the transition 3-2, there are then obtained several spectral lines lying closely together and corresponding to transitions such as 3₃-2₂, 3₂-2₁, etc. By theoretical considerations, requiring considerable mathematical qualifications, but of essentially the same formal nature as those Bohr had originally applied to the determination of the stationary orbits in hydrogen, Sommerfeld was led to certain formal quantum rules which permit the fixing of the stationary states of the hydrogen atom corresponding to such a double set of quantum numbers. The results he obtained as regards the fine structure of the hydrogen lines agree with observation inside the limit of experimental error.

Although Sommerfeld’s methods have also been very fruitful when applied to the spectra of other elements, they were still of a purely formal and rather arbitrary nature; it is, therefore, of great importance that the Leiden professor, Ehrenfest, and Bohr succeeded later in handling the problem from a more fundamental point of view, Bohr making use of the correspondence principle previously mentioned. It should be said here, by way of suggestion, that Bohr used the fact that the motions of the electrons are not simple periodic but “multiple periodic.” We see this most simply if we think of the revolution of the electrons in the elliptical orbit as representing one period, and the rotation for the major axis of the ellipse as representing a second period.

Fig. 27 shows a number of the possible stationary orbits in the hydrogen atom according to Sommerfeld’s theory; for the sake of simplicity the orbits are drawn as completely closed ellipses. If we examine, for instance, the orbits with principal quantum number 4, we have here three more or less elongated ellipses, 4₁, 4₂, 4₃, and the circle 4₄; in all of them the major axis has the same length, and the length of the major axis is to that of the minor axis as the principal quantum number is to the auxiliary quantum number (for the circle 4 : 4 = 1). On the whole, to a principal quantum number _n_ there correspond the auxiliary quantum numbers 1, 2 ... _n_, and the orbit for which the auxiliary quantum number equals the principal quantum number is a circle. We see that in the more complicated hydrogen atom model there is possibility for a much greater number of different transitions than in the simple model (Fig. 25, p. 119). Some of the transitions are indicated by arrows. Since the energy content of the atom is almost the same for orbits with the same principal quantum number and different auxiliary quantum numbers, three transitions like 3₃-2₂, 3₂-2₁ and 3₁-2₂ will give about the same frequency, and therefore spectral lines which lie very close together. In a transition like 4₄-4₃ the emitted energy quantum _h_ν, and also ν, will be so extremely small that the corresponding line will be too far out in the infra-red for any possibility of observing it.

It must be pointed out that the above considerations only hold if the hydrogen atoms, strictly speaking, are undisturbed. Thus, very small external forces, which may be due to the neighbourhood of other atoms, etc., will be sufficient to cause changes in the eccentricity of the stationary orbits. In such a case the above definition of the auxiliary quantum number becomes obviously illusory, and the original character of the fine structure disappears. This is in agreement with the experiments, since the Sommerfeld fine structure can be found only when the conditions in the discharge tube are especially quiet and favourable.

Influence of Magnetic and Electric Fields on the Hydrogen Lines.

As previously mentioned (p. 76), the spectral lines are split into three components when the atoms emitting lights are exposed to magnetic forces. The agreement found here between observation and the Lorentz electron theory was considered as strong evidence of the correctness of the latter. According to the Bohr theory, the picture upon which this explanation rested must be abandoned entirely; but fortunately it has been shown that the Bohr theory leads to the same results; and, moreover, Bohr, with the assistance of the correspondence principle, has been able to set forth the more fundamental reason for this agreement.

The German scientist, Stark, showed, in 1912, that hydrogen lines are also split by electric fields of force. In Fig. 28 it is shown how very complicated this phenomenon is; here the classical electron theory could not at all explain what happened. This phenomenon could also be accounted for by the extended Bohr theory (with the introduction of more than one quantum number), as it was shown independently by Epstein and by Schwarzschild in 1916; further, the correspondence principle has again shown its superiority, since it makes possible an approximate determination of the different intensities of the different lines. A calculation carried out by H. A. Kramers has shown that the theory gives a remarkably good agreement with the experiments.

Not until we think of the extraordinary accuracy of the measurements which are obtained by spectrum analysis, can we thoroughly appreciate the importance of the quantitative agreement between theory and observation in the hydrogen spectrum that has just been mentioned. Moreover, we must remember how completely helpless we previously were in the strange puzzles offered even by the simplest of all spectra, that of hydrogen.