The Dreams That Stuff is Made of

by Stephen Hawking

  (57)

  whence comes

  (58)

  (applying abbreviation (46) for the principal quantum number). In the approximation we are aiming at we may expand with respect to the small quantities εk and get

  (59)

  Further, in the calculation of these small quantities, we may use the approximate value (45) for A in (55). We thus obtain, noticing the two D values, by (42),

  (60)

  Addition gives, after an easy reduction,

  (61)

  If we substitute this, and the values of A, B1, and B2 from (42) in (59), we get, after reduction,

  (62)

  This is our provisional conclusion; it is the well-known formula of Epstein for the term values in the Stark effect of the hydrogen spectrum.

  k1 and k2 correspond fully to the parabolic quantum numbers; they are capable of taking the value zero. Also the integer n, which has evidently to do with the equatorial quantum number, may from (40) take the value zero. However, from (46) the sum of these three numbers must still be increased by unity in order to yield the principal quantum number. Thus (n + 1) and not n corresponds to the equatorial quantum number. The value zero for the latter is thus automatically excluded by wave mechanics, just as by Heisenberg’s mechanics.eu There is simply no proper function, i. e. no state of vibration, which corresponds to such a meridional orbit. This important and gratifying circumstance was already brought to light in Part I. in counting the constants, and also afterwards in § 2 of Part I. in connection with the azimuthal quantum number, through the nonexistence of states of vibration corresponding to pendulum orbits; its full meaning, however, only fully dawned on me through the remarks of the two authors just quoted.

  For later application, let us note the system of proper functions of equation (32) or (32ʹ) in “zero approximation”, which belongs to the proper values (62). It is obtained from statement (37), from conclusions (39) and (49), and from consideration of transformations (47) and (48) and of the approximate value (45) of A. For brevity, let us call a0 the “radius of the first hydrogen orbit”. Then we get

  (63)

  The proper functions (not yet normalised!) then read

  (64)

  They belong to the proper values (62), where l has the meaning (46). To each non-negative integral trio of values n, k1, k2 belong (on account of the double symbol ) two proper functions or one, according as n > 0 or n = 0.

  § 4. ATTEMPT TO CALCULATE THE INTENSITIES AND POLARISATIONS OF THE STARK EFFECT PATTERNS

  I have lately shownev that from the proper functions we can calculate by differentiation and quadrature the elements of the matrices, which are allied in Heisenberg’s mechanics to functions of the generalised position- and momentum-co-ordinates. For example, for the (r r ʹ)th element of the matrix, which according to Heisenberg belongs to the generalised co-ordinate q itself, we find

  (65)

  Here, for our case, the separate indices each deputise for a trio of indices n, k1, k2 , and further, x represents the three co-ordinates r, θ , φ. ρ(x ) is the density function; in our case the quantity (34). (We may compare the self-adjoint equation (32ʹ) with the general form (2)). The “denominator” (. . . )– in (65) must be put in because our system (64) of functions is not yet normalised.

  According to Heisenberg,ew now, if q means a rectangular Cartesian co-ordinate, then the square of the matrix element (65) is to be a measure of the “probability of transition” from the rth state to the rʹth, or, more accurately, a measure of the intensity of that part of the radiation, bound up with this transition, which is polarised in the q-direction. Starting from this, I have shown in the above paper that if we make certain simple assumptions as to the electrodynamical meaning of ψ, the “mechanical field scalar”, then the matrix element in question is susceptible of a very simple physical interpretation in wave mechanics, namely, actually: component of the amplitude of the periodically oscillating electric moment of the atom. The word component is to be taken in a double sense: (1) component in the q-direction, i.e. in the spatial direction in question, and (2) only the part of this spatial component which changes in a time-sinusoidal manner with exactly the frequency of the emitted light, | Er − Erʹ| / h. (It is a question then of a kind of Fourier analysis: not in harmonic frequencies, but in the actual frequencies of emission.) However, the idea of wave mechanics is not that of a sudden transition from one state of vibration to another, but according to it, the partial moment concerned—as I will briefly name it—arises from the simultaneous existence of the two proper vibrations, and lasts just as long as both are excited together.

  Moreover, the above assertion that the qrrʹ’s are proportional to the partial moments is more accurately phrased thus. The ratio of, e.g., qrrʹ to qrr” is equal to the ratio of the partial moments which arise when the proper function ψr and the proper functions ψrʹ and ψr” are stimulated, the first with any strength whatever and the last two with strengths equal to one another—i.e. corresponding to normalisation. To calculate the ratio of the intensities, the q-quotient must first be squared and then multiplied by the ratio of the fourth powers of the emission frequencies. The latter, however, has no part in the intensity ratio of the Stark effect components, for there we only compare intensities of lines which have practically the same frequency.

  The known selection and polarisation rules for Stark effect components can be obtained, almost without calculation, from the integrals in the numerator of (65) and from the form of the proper functions in (64). They follow from the vanishing or non-vanishing of the integral with respect to φ. We obtain the components whose electric vector vibrates parallel to the field, i.e. to the z-direction, by replacing the q in (65) by z from (33). The expression for z, i.e. ( λ1 − λ2), does not contain the azimuth φ. Thus we see at once from (64) that a non-vanishing result after integration with respect to φ can only arise if we combine proper functions whose n’s are equal, and thus whose equatorial quantum numbers are equal, being in fact equal to n + 1. For the components which vibrate perpendicular to the field, we must put q equal to x or equal to y (cf. equation (33)). Here cos φ or sin φ enters, and we see almost as easily as before, that the n-values of the two combined proper functions must differ exactly by unity, if the integration with respect to φ is to yield a non-vanishing result. Hence the known selection and polarisation rules are proved. Further, it should be recalled again that we do not require to exclude any n-value after additional reflection, as was necessary in the older theory in order to agree with experience. Our n is smaller by 1 than the equatorial quantum number, and right from the beginning cannot take negative values (quite the same state of affairs exists, we know, in Heisenberg’s theory).ex

  The numerical evaluation of the integrals with respect to λ1 and λ2 which appear in (65) is exceptionally tedious, especially for those of the numerator. The same apparatus for calculating comes into play as served already in the evaluation of (52), only the matter is somewhat more detailed because the two (generalised) Laguerre polynomials, whose product is to be integrated, have not the same argument. By good luck, in the Balmer lines, which interest us principally, one of the two polynomials , namely that relating to the doubly quantised state, is either a constant or is a linear function of its argument. The method of calculation is described more fully in the mathematical appendix. The following tables and diagrams give the results for the first four Balmer lines, in comparison with the known measurements and estimates of intensity, made by Starkey for a field strength of about 100,000 volts per centimetre. The first column indicates the state of polarisation, the second gives the combination of the terms in the usual manner of description, i.e. in our symbols: of the two trios of numbers (k1, k2, n + 1) the first trio refers to the higher quantised state and the second to the doubly quantised state. The third column, with the heading Δ, gives the term decomposition in multiples of 3h2 F / 8π2 me, (see equation (62)). The next column gives the intensities observed by Stark, and 0 there signif
ies not observed. The question mark was put by Stark at such lines as clash either with irrelevant lines or with possible “ghosts” and thus cannot be guaranteed. On account of the unequal weakening of the two states of polarisation in the spectrograph, according to Stark his results for the || and for the ⊥ components of vibration are not directly comparable with one another. Finally, the last column gives the results of our calculation in relative numbers, which are comparable for the collective components (|| and ⊥) of one line, e.g. of Hα, but not for those of Hα with Hβ, etc. These relative numbers are reduced to their smallest integral values, i.e. the numbers in each of the four tables are prime to each other.

  INTENSITIES IN THE STARK EFFECT OF THE BALMER LINES

  TABLE 1

  Hα

  TABLE 2

  Hβ

  TABLE 3

  Hγ

  TABLE 4

  Hδ

  FIG. 1 Hα ||-components.

  FIG. 2 Hα ⊥-components.

  In the diagrams it is to be noticed that, on account of the huge differences in the theoretical intensities, some theoretical intensities cannot be truly represented to scale, as they are much too small. These are indicated by small circles.

  A consideration of the diagrams shows that the agreement is tolerably good for almost all the strong components, and taken all over it is somewhat better than for the values deduced from correspondence considerations.ez Thus, for example, is removed one of the most serious contradictions which arose, in that the correspondence principle

  FIG. 3 Hβ || -components.

  FIG. 4 Hβ ⊥-components.

  FIG. 5 Hγ ||-components.

  gave the ratio of the intensities of the two strong ⊥-components of Hβ, for Δ = 4 and 6, inversely and indeed very much out, in fact as almost 1 : 2, while experiment requires about 5 : 4. A similar thing occurs with the mean (Δ = 0) ⊥-components of Hγ, which decidedly preponderate experimentally, but are given as far too weak by the correspondence principle. In our diagrams also, it is admitted that such “reciprocities” between the intensity ratios of intense components demanded by theory and by experiment are not entirely wanting. The theoretically most intense ||-component (Δ = 3) of Hα is furthest out; by experiment, it should lie between its neighbours in intensity. And the two strongest ||-components of Hβ and two ⊥-components (Δ = 10, 13) of Hγ are given “reciprocally” by the theory. Of course, in both cases the intensity ratios, both experimentally and theoretically, are pretty near unity.

  FIG. 6 Hγ ⊥-components.

  Passing now to the weaker components, we notice first that the contradiction which exists for some weak observed components of Hβ to the selection and polarisation rules, of course still remains in the new theory, since the latter gives these rules in conformity with the older theory. However, components which are extremely weak theoretically are for the most part unobserved, or the observations are questionable. The strength ratios of weaker components to one another or to stronger ones are almost never given even approximately correctly; cf. especially Hγ and Hδ. Such serious mistakes in the experimental determination of the blackening are of course out of the question.

  Considering all this, we might feel inclined to be very sceptical of the thesis that the integrals (65) or their squares are measures of intensity. I am far from wishing to represent this thesis as irrefutable. There are still many alterations conceivable, and these may, perhaps, be necessitated by internal reasons when the theory is further extended.

  FIG. 7 Hδ || -components.

  FIG. 8 Hδ ⊥-components.

  Yet the following should be remembered. The whole calculation has been performed with the unperturbed proper functions, or more precisely, with the zero approximation to the perturbed ones (cf. above § 2). It, therefore, represents an approximation for a vanishing field strength! However, just for the weak or almost vanishing components we should expect theoretically a fairly powerful growth with increasing field strength, for the following reason. According to the view of wave mechanics, as explained at the beginning of this section, the integrals (65) represent the amplitudes of the electrical partial moments, which are produced by the distribution of charges which flow round about the nucleus within the atom’s domain. When for a line component we get as a zero approximation very weak or even vanishing intensity, this is not caused in any way by the fact that to the simultaneous existence of the two proper vibrations corresponds only an insignificant motion of electricity, or even none at all. The vibrating mass of electricity—if this vague expression is allowed—may be represented as the same in all components, on the ground of normalisation. Rather is the reason for the low line intensity to be found in a high degree of symmetry in the motion of the electricity, through which only a small, or even no, dipole moment arises (on the contrary, e.g., only a four-pole moment). Therefore it is to be expected that the vanishing of a line component in presence of perturbations of any kind is a relatively unstable condition, since the symmetry is probably destroyed by the perturbation. And thus it may be expected that weak or vanishing components gain quickly in intensity with increasing field strength.

  This has now actually been observed, and the intensity ratios, indeed, alter quite considerably with field strength, for strengths of about 10,000 gauss and upwards; and, if I understand aright, in the wayfa shown by the present general discussion. Certain information on the question whether this really explains these discrepancies could of course only be got from a continuation of the calculation to the next approximation, but this is very troublesome and complicated.

  The present considerations are of course nothing but the “translation” into the language of the new theory of very well-known considerations which Bohrfb has brought forward in connection with calculation of line intensities by means of the principle of correspondence.

  The theoretical intensities given in the tables satisfy a fundamental requirement, which is set up not only by intuition but also by experiment,fc viz., the sum of the intensities of the ||-components is equal to that of the ⊥-components. (Before adding, undisplaced components must be halved—as a compensation for the duplication of all the others, which occur on both sides.) This makes a very welcome “control” for the arithmetic.

  It is also of interest to compare the total intensities of the four lines by using the four “sums” given in the tables. For this purpose I take back from my numerical calculations the four factors, which were omitted in order to represent the intensity ratios within each of the four line groups by the smallest integers possible, and multiply by them. Further, I multiply each of these four products by the fourth power of the appropriate emission frequency. Thus I obtain the following four numbers:

  I give these numbers with still greater reserve than the former ones because I am not sure, theoretically, about the fourth power of the frequency. Investigationsfd which I have lately published seem to call, perhaps, for the sixth. The above method of calculation corresponds exactly to the assumptions of Born, Jordan, and Heisenberg.fe Fig. 9 represents the results diagrammatically.

  FIG. 9 Total Intensities.

  Actual measured intensities of emission lines, which are known to depend greatly on the conditions of excitation, naturally cannot here be used in a comparison with experience. From his researchesff on dispersion and magneto-rotation in the neighbourhood of Hα and Hβ, R. Ladenburg has, with F. Reiche,fg calculated the value 4.5 (limits 3 and 6) for the ratio of the so-called “electronic numbers” of these two lines. If I assume that the above numbers may be taken as proportional to Ladenburg’sfh expression,

  then they may be reduced to (relative) “electronic numbers” by division by , i.e. by

  Hence we obtain the four numbers,1.281, 0.2386, 0.08975, 0.04418.

  The ratio of the first to the second is 5.37, which agrees sufficiently with Ladenburg’s value.

  § 5. TREATMENT OF THE STARK EFFECT BY THE METHOD WHICH CORRESPONDS TO THAT OF BOHR

  Mainly to give an example of the
general theory of § 2, I wish to outline that treatment of the proper value problem of equation (32), which must have been adopted, if we had not noticed that the perturbed equation is also exactly “separable” in parabolic co-ordinates. We therefore now keep to the polar co-ordinates r, θ, φ, and thus replace z by r cos θ. We also introduce a new variable η for r by the transformation

  (66)

  (which is closely akin to transformation (48) for the parabolic co-ordinate ξ). For one of the unperturbed proper values (45), we get from (66)

  (66ʹ)

  where a0 is the same constant as in (63). (“Radius of the innermost hydrogen orbit.”) If we introduce this and the unperturbed value (45) into the equation (32), which is to be treated, then we obtain

  (67)