" the Universal Meaning of the Quantum of Action", by Jun Ishiwara

Home , Hugo Tetrode, Jun Ishiwara

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A more fundamental definition of the scalar quantity h will unfold differently in any event. From the same perspective the Sommerfeld Rule seems generally valid because it identifies h with the invariant h of mechanics that is the action size (up to the factor of 2π ). Sommerfeld has used it to explain the photoelectric effect; an application thereof to the statistics of the radiation unfortunately will not produce the desired result. 4 After futile attempts now only one path remains along which for me to proceed. I formulate the following basic assumption: 108 "An elementary structure of the matter or a system of immense number of elementary structures exist in a stationary periodic motion or in a statistical equilibrium. The state is completely determined by coordinates p1, 5 p2,..., pf and the associated momentum coordinates q1, q2, ..., qf . Then the movements in nature will always take place in such a way that a decomposition of each state plane piqi in those elementary areas of equal probability is allowed, whose mean value in a certain point of the state space:

j 1 h = q dp (1) j i i i=1 X Z equals a universal constant.” 6 The application of the last sentence to the statistics of black-body radiation on the one hand and the gas of molecules on the other hand already provided an experimental proof of the universality of h. The value of h calculated by O. Sackur [Sackur 1912a, Sackur 1912b] and H. Tetrode [Tetrode 1912a, Tetrode 1912b] from the chemical constants of argon and mercury vapor is in excellent agreement with the value that is well-known from radiation 7. It should now be examined whether the statement is really applicable also to the process of individual electron movements, especially to the case of the Bohr atom model and the photoelectric problem. Before I concentrate on each part individually, I would like to outline a general observation.

4 In the 1911 paper, Sommerfeld emphasizes the use of the expression “quanta of action”, instead of “quanta of energy”. Here, he compares the classical (with the Rayleigh’s formula) and quantum (with the Planck’s formula) for black-body theory and shows the limits where both expressions coincide. After he presents his quantization hypothesis τ h L dt = Z0 2π and analyses the photoelectric effect, Sommerfeld tries to expand it to relativistic systems. At the end, there is a discussion with Johannes Stark (1874-1957), Einstein, Leo K?nigsberger (1837-1921) and Heinrich Rubens (1865-1922). The second paper cited here, the work with his student Peter Debye (1884-1966), "Theory of the photoelectric effect from the standpoint of the quantum of action", is a more extensive and detailed version of the theory for photoelectric effect exposed on the first one. The theory of Debye-Sommerfeld for photoelectric effect was not equivalent to the Einstein’s one - see section III of [our analysis paper] for comments about this point. 5 Ishiwara writes against the popular convention, where the q-letter indicates the position and the p-letter indicates the momentum. 6 The condition of Ishiwara is different from the condition proposed by Bohr [Waerden 2007], Sommerfeld [Sommerfeld 1916a, Sommerfeld 1916b, Sommerfeld 1923] and Wilson [Wilson 1915],

qi dpi = ni h Z where, for j degrees of freedom, we have j equations, with i varying from 1 to j - i. e., there are no more summation over i, and no division by j [Abiko 2015]. In (6), we follow the convention where the q-letter indicates the position and the p-letter indicates the momentum. See a detailed discussion in section III of [Pelogia 2017]. 7 In this paragraph, Ishiwara refers to two papers of Otto Sackur (1880-1914), "The importance of the elementary quantum of action for the theory of gases and the calculation of chemical constants", that was published in the book "In Honour W. Nernst to his 25-year-old Philosophical Doctoral Jubilee", and "The universal significance of the so-called ’elementary quantum of action’ "; and to two papers of Hugo Tetrode (1895-1931), "The chemical constant of the gases and the elementary quantum of action" and "Correction to my work : ’ The chemical constant of the gases and the elementary quantum of action ’". They talk about the chemical constant, a constant which occurred in the expression for the absolute entropy of a gaseous substance, that was used by them to the quantization of translational motion of atomic particles on monoatomic gases and to analyze the deviation of the classical equation at low temperatures. [Mehra 1982, Mehra 2001] At this time, the expression ’value of radiation’ would have referred not necessarily to h itself, but to other constants used at that time and that could be related to Planck’s one. For example, in his original work about photoelectric effect [Arons 1965], Einstein did not used h, but R (the universal gas constant), α and β (the constants that appeared on the original expression of Planck for the black-body spectra [Feldens 2010, Planck 2000a, Planck 2000b, Studart 2000]). The papers of Sackur and Tetrode, cited by Ishiwara, are illustrative about it. On the other side, Robert Andrew Millikan (1868-1953), in his studies about the photoelectric effect, was making measurements of h itself - and his results were cited by Sommerfeld [Debye 1913, Sommerfeld 1916a] in the name of Millikan. Sometimes, the name cited is James Remus Wright (1883? - 1937), a member of Millikan’s team who obtained his Ph. D. in 1911 (his thesis "The positive potential of aluminum as a function of the wave-length of the incident light" [Wright 2017] is only his paper published in Physical Review [Wright 1911]) and, after that he held the position of Chief of Department at the University of Philippines from 1911 to 1919, working with new oil refining processes at the Standard Oil Company from 1919 until two years before his death [Cattel 1921, Wright 1937].

2 Imagine an electron; its position being expressed by the radial vector r that is drawn from the point of origin. By marking the diﬀerentiation with respect to the (Minkowski) proper time τ 8by a top marked point, the momentum quantity of the electron is given by m0r˙ with m0 denoting the rest mass. The quantum theorem stipulates here that the integral 1 h = (m0r,˙ dr) (1’) j Z has a universal value in an elementary region with equal probabilities. Assuming j equals to 1, 2 or 3, jθ depending 109 on the movement being one-, two- or three-dimensional. Now, the area of equal probability is likely to be bounded by two speciﬁc stationary movements. By extending the integral (1’) over the whole area, which is bounded by one of said stationary movements, it must equal an integer multiple of h; thus

njh = m0r˙ dr (2) Z when n is an integer. By looking at r as a function of the proper time τ, one can rewrite the integral:

2 m0r,˙ dr = m0r˙ dτ Z Z By putting 9

1 2 T = m0r (3) 2 (2) becomes in

θ njh =2 T dτ (4) 0 Z where θ is the period of the considered movement. It is further more

θ θ 2 θ m0r˙ dτ = m0r r˙ 0 m0 (r r¨) dτ 0 | | − 0 Z Z The ﬁrst term on the right vanishes because of the periodicity; however, the second is equal to twice the time integral of the Clausius Virial V because

m0 (¨r r) = (Kr)= 2V (5) − is to be set. In the central force whose magnitude is inversely proportional to the square of the distance V equals the potential of the force . Therefore relation (4) may also be written as: ℜ θ njh =2 (T + V ) dτ (6) 0 Z From the above it is clear that the classic action principle:

2 δ (T V ) dτ = δ m0c dτ =0 − Z Z still remains valid here, independent of the quantum theorem. Therefore it seems to me that the Sommerfeld 110 hypothesis is losing its basic principle.

8Here, the reference to "proper time" is significant in the concerns of Ishiwara with relativity[Einstein 1982, Hu 2007]. Indeed, Ishiwara dedicated most of his time to Einstein’s relativity, and he was a scholar and commentator on the theory, gaining reputation both in Japan and China, where he was known as “the only expert of relativity studies in Japan (. . . ) [who] had a unique understanding of the theory of relativity”. The first paper by Ishiwara on the theory was published in 1909 and was the first in Japan on this topic. [Hu 2007, Sigeko 2000] 9In the equation (3) there is a typographical error, appearing r2 instead r˙2, but it causes no effects in further developments, because the correct expression for kinetic energy is used.

3 2. The Bohr model of the atom

The following text will explain the application of the quantum theorem to stationary motion of electrons inside the atom. For the sake of simplicity we focus on the atomic nucleus and an orbiting electron and leave the eﬀect of the other electrons aside. Following Coulomb’s law the electron carries out the central movement around the nucleus of O (a); so it describes as a focal point an ellipse with greater semiaxis a and the eccentricity ε. With O being the point of origin and the elliptical plane being the xy plane, the path is expressed by equation10

2 2 y (x + εa) + = a2 (7) 1 ε2 − The law of areas then can be expressed as

m0 (xy˙ yx˙)= f (8) − f means the constant angular momentum. Diﬀerentiating (7) with respect to the proper time, we obtain y (x + εa)x ˙ + y˙ =0 (9) 1 ε2 − Elimination of y and y˙. from (7), (8) and (9) yields

2 2 2 2 2 2 2 2 2 m0 1 ε a 1 ε εax x˙ = f a (x + εa) − − − − 11 n o which through the introduction of the variable

′ x = x + a can be rewritten as: 2 2 2 ′ 2 ′2 2 2 ′2 m0 1 ε a (a εx ) x˙ = f a x − − − After taking the square root

2 ′2 ′ f√a x m0x˙ = − (10) ∓a√1 ε2 (a εx′) − − a I neglect the terms proportional to the square of the velocity against that of the speed of light c. 10The equation (7) describes an ellipse with his left focus (F = εa) on the origin O of the system of coordinates, with greater semiaxis over the x-axis, as indicated on the Fig. 1. y

OC x

FF Figure 1: Ellipse described by the equation (7), with center on the C point and origin O of the system of coordinates over the left focus. 11 In the equation deﬁning the new variable there is a typographic error, the expression truly used by Ishiwara is x′ = x + εa. This change of variable is equivalent to set the origin O of the system of coordinates on the center C of the ellipse.

4 By eliminating x˙ one easily ﬁnds 12: 111

fx′ m0y˙ = (11) a (a εx′) − The formula (2) thus writes itself, since j =2 to set,

′ ′ 2nh = m0x˙ dx + m0y˙ dy Z Z or because of the symmetry of the ﬁgures with respect to the x′-axis

a a ′ ′ dy ′ nh = m0x˙ dx + m0y˙ ′ dx − − dx Z a Z a These integrals are calculated according to (7), (10) and (11) as follows 13:

a 2 ′2 2 ′ f √a x 1 ε x ′ nh = 2 − ′ −′ 2 ′2 dx a√1 ε − − a εx − (a εx ) √a x Z a ( − ) − a ′ − − f a + εx ′ πf = 2 2 ′2 dx = 2 −a√1 ε −a √a x √1 ε − Z − − Thus we conclude h f = n 1 ε2 (12) π − p h √ 2 The angular momentum of the electron motion therefore equals an integral multiple of π 1 ε . In the case where the elliptical orbit turns into a circle, it obtains only − h f = n (12’) π Thus it is shown not only that the Bohr hypothesis generally fails, but also that it still is not quite correct even for the circular motion. We will now examine how this modiﬁcation of the basic assumption has further consequences. The period of the electron motion is given by 14

2 a 4am0√1 ε a εx´ θ = dτ = − −2 ′2 dx´ − f − √a x Z Z a 2 2 − 2πa m0√1 ε = − f

Designating the number of vibrations per unit of time with ν, it follows thereafter 112

12 To arrive at (11), we must ﬁnd the expression for y˙

1 ε2 (x + εa)2 y˙2 = − x˙ 2 a2 (x + εa)2 − 2 ′ 2 ′2 (1−ε )x (1−ε )x 13 There is another typographical error on the following equation, instead , the second integrand is . (a−εx′)√a2−x′2 (a−εx′)√a2−x′2 14 The period θ of the movement by a central conservative force is given by [Fetter 2003] m ab θ = 2π 0 f where b = a√1 ε2 is the minor semi-axis of the ellipsis, then we have in a direct way the result. However, Ishiwara adopts the − ′ dx′ time integration, with τ = τ (x ,y). It is easy to see that dτ = 2 x˙ ′ and, with (10), we can obtain the period. However, there is a typographical error in the intermediate expression, the correct form is

2 a 2a m0√1 ε a εx´ θ = dτ = − − ′ dx´ Z − f Z−a √a2 x 2 −

5 f 2 =2πa m0ν (13) √1 ε2 − and with (12)

nh 2 =2πa m0ν (14) π On the other hand, if the electric charge of the revolving electron is e and for the atomic nucleus is e′, we have, as is well-known 15, −

f 2 a 1 ε2 = (15) − m0ee´ From (12) and (15) it follows that

n2h2 a = 2 (16) π m0ee´ Finally, by eliminating a and solving for ν one obtains from (14) and (16)

2 2 2 π m0e e´ ν = (17) 2n3h3 The average kinetic energy of the system according to (4) with the value j =2, amounts to:

T = nhν (18) or by inserting ν from (17)

2 2 2 π m0e e´ T = (19) 2n2h2 Assuming diﬀerent integers for n, one gets a series of values that correspond to states of diﬀerent probabilities. Yet only during the transition between two such states can the electron emit a radiation energy 16. The vibration frequency ν1 of the emitted radiation is to be determined herein so that the energy output is exactly equal to hν1. According to (19) 17

2 2 2 π m0e e´ 1 1 ν1 = (20) 2h3 n2 − n2 1 2 To identify this law with Balmer’s theory, one now has to assume that the central charge of the hydrogen atom consists of two electrical elementary quanta. For if one sets 18,

e´ =2e then (20) turns into 113

2 4 2π m0e 1 1 ν1 = (21) h3 n2 − n2 1 2 which is just what is found in Bohr’s theory.

15 The equation (15) can be obtained considering the conservation of energy under Coulombian potential. 16 According to the atomic model of Bohr. [Bohr 1913a, Bohr 1913b, Bohr 1913c, Bohr 1963, Parente 2013] 17 We can see here that the equation (20), if applied to Hydrogen atom (e´ = e), will furnish a frequency of transition π2m e4 1 1 ν = 0 1 3 2 2 2h n1 − n2 diﬀerent from the real one - compare with equation (4) of [Bohr 1913a] and (21) 2π2m e4 1 1 ν = 0 1 3 2 2 h n1 − n2 which was obtained by Sommerfeld with his formula (6) (see footnote 5). 18 Obviously, this is a hypothesis contrary to the very deﬁnition of a chemical element and it is not necessary if we use the Bohr- Sommerfeld-Wilson version, equation (6). For a more detailed analysis of the original Sommerfeld calculation, see [Castro 2016].

6 The fact that the hydrogen atom in actual fact in the neutral state contains two electrons is in my opinion still significantly supported by the following circumstances: 1. The adoption of two rotating electrons can completely explain the large value of the magnetic susceptibility of liquid hydrogen according to the Langevin theory of diamagnetism. 19 2 1 dr 2. If one takes addictional quantities of the order of c2 dt into consideration, one obtains a more rigorous formula for ν1, which is similar to that derived by Allen [Allen 1915a], and fits the precise spectral measurements 20 better [Allen 1915b]. The numerical figures regarding this I intend to communicate in a forthcoming work.

3. The photoelectric phenomenon

Our fundamental law also seems to show its special power in regards to the explanation of photoelectric phenomenon. We ﬁrst imagine the photoelectric process after Sommerfeld [Debye 1913] 21, as follows: An electron is resonant with incident radiation, so a large amount of the energy is accumulated in it, but only until it reaches a certain value, which could then satisfy the equation (4), if the incident radiation would stop at just the same moment and the electron would still continue its stationary motion inside the atom. If one imagines the simplest case of perfect resonance, one can write for the linear oscillation of the electron along the x-axis (b): eC x¨ +4π2ν2x = cos(2πντ) (22) m0 where C is the amplitude of the incident electric ﬁeld, ν the number of vibrations per unit of time. 114 With the initial conditions:

x = 0 andx ˙ = 0 for t =0 one gets the solution 22: eC x = τ sin (2πντ) (23) 4πm0ν and thus the kinetic and the potential energy of the oscillating electron results in:

2 2 1 2 e C 2 T = m0x˙ = 2 2 [sin (2πντ)+2πντ cos(2πντ)] (24) 2 32π m0ν 2 2 2 2 2 e C 2 U =2π m0ν x = [τ sin (2πντ)] (25) 8m0 The electron now separates itself from the atom after an accumulation time θ1 has elapsed. Now, the time θ1 is determined according to our hypothesis as follows:

19 A contemporary reference about the diamagnetism is [Essen 2012] and, for Langevin’s theory in particular, see the profound analysis of Navarro and Olivella [Navarro 1992]. 20 Several points can be analyzed here. The relativistic treatment, that Ishiwara says he wants to share in a forthcoming work, was conducted soon after the publication of this paper, by Sommerfeld [Sommerfeld 1916b, Sommerfeld 2014], and allowed to explain the fine structure of spectral lines. Ishiwara cites the works of Herbert Stanley Allen (1873 - 1954), who suggests, to obtain more precise spectral formulas, following on from the previous work of other researchers, to treat the atom as consisting of a magnetic core, electrically charged and surrounded by one or more electrons, with the quantization of the electronic orbital angular momentum. In the first paper, however, the conclusion is, "that the magnetic forces set up by the atom are not in themselves sufficient to account for more than a small fraction of the effect that would be necessary to give the observed distribution of lines in spectral series" [Allen 1915a]. In the second paper, published in the same volume as the first one, Allen supposes that the magnetic moment of the nucleus has different values for the steady states of motion but, one more time, the model just "appears that it is possible to account for the series spectrum of hydrogen" [Allen 1915b]. 21 Once more, is important to note that Ishiwara follows the theory of the photoelectric effect by Sommerfeld, not Einstein’s one. See section III of [Pelogia 2017]. b Neglecting the damping. 22 The simple form of the solution (23) is due to: 1) the absence of the damping term on (22), as emphasized in the footnote [b], 2) the form of initial conditions and 3) the fact that the frequency of the incident wave is the same as the natural frequency of the electron (resonance case).

7 We assume in our thoughts that the electron, from the time τ = θ1 onwards continues with stationary vibrations. Then the quantum law (4) should apply. Since j =1 in our example, it reads 23

θ nh =2 T dτ (26) 0 Z where θ means the period of the considered vibrations. This means instead of (22) we have

x¨ +4π2ν2x =0 which results in:

x = a sin (2πντ + δ) It therefore is calculated:

1 2 2 2 2 2 T = m0x˙ =2π m0ν a cos (2πντ + δ) 2 and therefore

θ 2 2 T dτ = π m0νa 0 Z With this value one obtains from (26) 115

2 nh a = 2 (27) 2π m0ν the same relationship as in (14). On the other side, the amplitude a is the one which the resonant oscillation (23) assumes at the time τ = θ1. Hence 24 eC a = θ1 (28) 4πm0ν Comparing (27) and (28), the result is

√8m0νnh θ1 = (29) eC It is remarkable that this value of the accumulation time completely agrees with the value calculated from the Sommerfeld theory, although the basic assumption of the latter theory is different from ours. 25 The difference between the two theories shows, of course, in other relationships, i.e., if one asks for the energy of the electron at the end of the accumulation time. At time τ = θ1 the kinetic energy is according to (24) e2C2 2 2 2 2 2 T 2 2 πνθ1 πνθ1 πνθ1 π ν θ πνθ1 = 32π m0ν sin (2 )+2 sin (4 )+4 1 cos (2 ) . Because of the large value of νθ1 [Debye 1913] the last term in the brackets considerably exceeds the preceding ones, and one can write with sufficient approximation:

2 2 e C 2 T = [θ1 cos(2πνθ1)] (30) 8m0 Furthermore according to (25)

2 2 e C 2 U = [θ1 sin (2πνθ1)] (31) 8m0 The photoelectrically liberated electron therefore has the energy:

23 Here, as Ishiwara considers j = 1, there will be no influences of his incorrect condition (1) - see footnote 5. 24 If x (θ1) is the largest value of (23), we must have sin (2πνθ1) = 1. 25 Here, where Ishiwara talks about Sommerfeld’s theory, he is talking about the Eq. (4), presented by the first time at the first Solvay Congress [Jammer 1966, Straumann 2011], not to the “definitive” version,

qi dpi = ni h Z

8 2 2 e C 2 T + U = θ1 8m0 or as a result of (29)

T + U = nhν (32) 116 This is consistent with Einstein’s law. 26 For the liberation of the electron should happen in reality at maximum T , where T reaches an integral multiple of hν (c).

In Sommerfeld’s theory the value of the kinetic energy at the relevant point of time is also an integral multiple of hν; after all, this applies with approximation

T = U (33) Should the free electron after Einstein’s law ﬂy with this kinetic energy T , then the following question can be answered only with diﬃculty by Sommerfeld: 27 Where does the potential energy U remain? Herein I also see a reason to prefer our theory over the latter one. Institute of Physics of Sendai University.

References

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