# The simple theory of the series spectrum of hydrogen

by Niels BohrThe frequencies of the lines of the series spectrum of hydrogen may, if we look apart from the fine structure of the single lines revealed by instruments of high dispersive power, be represented by the formula

```
ν = K
1
n002
−
1
n02
, (35)
```

# where K is a constant, and n 0 and n 00 a set of two entire numbers, different for the different lines of the spectrum. According to the general principles of the quantum theory of line spectra discussed in the first section of Part I, we shall therefore expect that this spectrum is emitted by a system which possesses a series of stationary states in which the numerical value of the energy in the n th state, omitting an arbitrary constant, with a high degree of approximation is given by En = Kh n2 , (36) where h is Planck’s constant which enters in the fundamental relation (1). Now according to Rutherford’s theory of atomic structure, a neutral hydrogen atom must be expected to consist 71 of an electron and a positive nucleus of a mass very large compared with that of the electron, which move under the influence of a mutual attraction inversely proportional to the square of the distance apart. Assuming that the motion in the stationary states may be determined by ordinary mechanics, and neglecting for the moment the small modifications claimed by the theory of relativity, we find that each of the particles will describe an elliptical orbit with their common centre of gravity at one of the foci, and from the well known laws for a Keplerian motion we have that the frequency of revolution ω and the major axis 2α of the relative orbit of the particles, quite independent of the degree of eccentricity of this orbit, are given by ω = r 2W3 (M + m) π 2N2e 4Mm , 2α = Ne2 W , (37) where W is the work necessary to remove the electron to infinite distance from the nucleus, while Ne and M are the charge and the mass of the nucleus, and −e and m the charge and the mass of the electron. As explained in Part I, there will in general be no simple connection between the motion of a system in the stationary states and the spectrum emitted during transitions between these states; such a connection, however, must be expected to exist in the limit where the motions in successive stationary states differ comparatively little from each other. In the present case this connection claims in the first place that the frequency of revolution tends to zero for increasing n. 72 According to (36) and (37) we may therefore put the value of W in the n th stationary state equal to Wn = Kh n2 . (38) Moreover, since (35) can be written in the form ν = (n 0 − n 00)K n 0 + n 00 n02n002 , it is seen to be a necessary condition that the frequency of revolution for large values of n is asymptotically given by ωn ∼ 2K n3 , (39) if we wish that the frequency of the radiation emitted during a transition between two stationary states, for which the numbers n 0 and n 00 are large compared with their difference n 0 − n 00, shall tend to coincide with one of the frequencies of the spectrum which on ordinary electrodynamics would be emitted from the system in these states. But from (37) and (38) it will be seen that (39) claims the fulfilment of the relation K = 2π 2N2 e 4Mm h 3 (M + m)

# 2π 2N2 e 4m h 3 (1 + m/M) . (40) As shown in previous papers, this relation is actually found to be fulfilled within the limit of experimental errors 73 if we put N = 1 and for e, m, and h introduce the values deduced from measurements on other phenomena; a result which may be considered as affording a strong support for the validity of the general principles discussed in Part I, as well as for the reality of the atomic model under consideration. Further it was found that, if in formula (35) for the hydrogen spectrum the constant K is replaced by a constant which is four times larger, this formula represents to a high degree of approximation the frequencies of the lines of a spectrum emitted by helium, when this gas is subject to a condensed discharge. This was to be expected on Rutherford’s theory, according to which a neutral helium atom contains two electrons and a nucleus of a charge twice that of the nucleus of the hydrogen atom. A helium atom from which one electron is removed will thus form a dynamical system perfectly similar to a neutral hydrogen atom, and may therefore be expected to emit a spectrum represented by (35) if in (40) we put N = 2. Moreover a closer comparison of the helium spectrum under consideration with the hydrogen spectrum has shown that the value of the constant K in the former spectrum was not exactly four times as large as that in the latter, but that the ratio between these constants within the limit of experimental errors agreed with the value to be expected from (40), when regard is taken to the different masses of the nuclei of the atoms of hydrogen and helium corresponding to the different atomic weights of these elements.1 ) 1 ) For the literature on this subject the reader is referred to the 74 Introducing the expression for K given by (40) in the formulæ (37) and (38), we find for the values of W, ω and 2α in the stationary states Wn = 1 n2 2π 2N2 e 4Mm h 2 (M + m) , ωn = 1 n3 4π 2N2 e 4Mm h 3 (M + m) , 2αn = n 2 h 2 (M + m) 2π 2Ne2Mm . (41) Now for a mechanical system as that under consideration, for which every motion is periodic independent of the initial conditions, we have that the value of the total energy will be completely determined by the value of the quantity I, defined by equation (5) in Part I. As mentioned this follows directly from relation (8), which shows at the same time that for a system for which every motion is periodic the frequency will be completely determined by I or by the energy only. For the value of I in the stationary states of the hydrogen atom we get by means of (8) from (37) and (41), since in this case I will obviously become zero when W becomes infinite, I = Z ∞ Wn dW ω

# s π 2N2e 4Mm 2(M + m) Z ∞ Wn W−3/2 dW

s 2π 2N2e 4Mm Wn(M + m) = nh. papers cited in the introduction. 75 This result will be seen to be consistent with condition (24) which, as mentioned in Part I, presents itself as a direct generalisation to periodic systems of several degrees of freedom of condition (10) which determines the stationary states of a system of one degree of freedom, and which again on Ehrenfest’s principle of the mechanical transformability of the stationary states forms a rational generalisation of Planck’s fundamental formula (9) for the possible values of the energy of a linear harmonic vibrator. In this connection it will be observed, that the relation discussed above between the hydrogen spectrum and the motion of the atom in the limit of small frequencies is completely analogous to the general relation, discussed in § 2 in Part I, between the spectrum which on the quantum theory would be emitted by a system of one degree of freedom, the stationary states of which are determined by (10), and the motion of the system in these states. It will at the same time be noted that, in case of hydrogen, this relation implies that the motion of the particles in the stationary states of the atom will not in general be simply harmonic, or in other words that the orbit of the electron will not in general be circular. In fact if the motion of the particles were simply harmonic, as the motion of a Planck’s vibrator, we should expect on the considerations in Part I that no transition between two stationary states of the atom would be possible for which n 0 and n 00 differ by more than one unit; but this would obviously be inconsistent with the observations, since for instance the lines of the ordinary Balmer series, accord- 76 ing to the theory, correspond to transitions for which n 00 = 2 while n 0 takes the values 3, 4, 5, . . . . In connection with this consideration it may be remarked that, adopting a terminology well known from acoustics, we may from the point of view of the quantum theory regard the higher members of the Balmer series (n 0 = 4, 5, . . . ) as the “harmonics” of the first member (n 0 = 3), although of course the frequencies of the former lines are by no means entire multipla of the frequency of the latter line. While in the above way it was possible to obtain a simple interpretation of certain main features of the hydrogen spectrum, it was not found possible in this way to account in detail for such phenomena in which the deviation of the motion of the particles from a simple Keplerian motion plays an essential part. This is the case in the problem of the fine structure of the hydrogen lines, which is due to the effect of the small variation of the mass of the electron with its velocity, as well as in the problems of the characteristic effects of external electric and magnetic fields on the hydrogen lines. As mentioned in the introduction, a progress of fundamental importance in the treatment of such problems was made by Sommerfeld, who obtained a convincing explanation of the fine structure of the hydrogen lines by means of his theory of the stationary states of central systems, in which the single condition I = nh was replaced by the two conditions (16); and the theory was further developed by Epstein and Schwarzschild, who on this line established the general theory, based on the conditions (22), 77 of the stationary states of a conditionally periodic system for which the equations of motion may be solved by means of separation of variables in the Hamilton-Jacobi partial differential equation. If the hydrogen atom is exposed to a homogeneous electric or to a homogeneous magnetic field, the atom forms a system of this class, and, as shown by Epstein and Schwarzschild as regards the Stark effect and by Sommerfeld and Debye as regards the Zeeman effect, the theory under consideration leads to values for the total energy of the atom in the stationary states, which together with relation (1) lead again to values for the frequencies of the radiations emitted during the transitions between these states, which are in agreement with the measured frequencies of the components into which the hydrogen lines are split up in the presence of the fields. As pointed out in Part I, it is possible moreover to throw light on the question of the intensities and polarisations of these components on the basis of the necessary formal relation between the quantum theory of line spectra and the ordinary theory of radiation in the limit where the motions in successive stationary states differ very little from each other. In the following sections the mentioned problems will be discussed in detail. As regards the fixation of the stationary states we shall not, however, follow the same procedure as used by the authors just mentioned, which rests upon the immediate application of the conditions (22), but it will be shown how the conditions which fix the stationary states of the perturbed atom may be obtained by a direct examination of the small deviations of the motion 78 of the electron from a simple Keplerian motion. In this way it seems possible to obtain a more direct illustration of the principles discussed in Part I; and we shall see moreover that the treatment in question may be used also in cases where the method of separation of variables cannot be applied. In Part III the problem of the series spectra of other elements will be treated from a similar point of view. As pointed out by the writer in an earlier paper, a simple explanation of the pronounced analogy between these spectra and the hydrogen spectrum is offered by the fact, that the atomic systems, involved in the emission of the spectra under consideration, in a certain sense may be regarded as a perturbed hydrogen atom. On the other hand, a clue to the interpretation of the characteristic difference between the hydrogen spectrum and the spectra of other elements was first obtained by Sommerfeld’s theory of the stationary states of central systems referred to above. As shown by Sommerfeld, it is possible on this theory to account in general outlines for the well known laws governing the frequencies of the series spectra of the elements; and, as it will be shown in Part III, it is also possible, on the basis of the formal relation between the quantum theory and the ordinary theory of radiation, in this way to obtain a simple interpretation of the laws governing the remarkable differences in the intensities with which the various series of lines appear, which on the combination principle would constitute the complete spectra under consideration. As regards the detailed discussion of these spectra, however, it is necessary to bear in mind that 79 the part played by the inner electrons in the atoms of the elements in question forms a far more intricate problem than the perturbing effect of a fixed external field on the hydrogen atom. For the treatment of this problem the theory of conditionally periodic systems based on the conditions (22) does not seem to suffice, while, as it will be shown in Part III, it appears that the method of perturbations exposed in the following lends itself naturally also to this case.