Abstract
Niels Bohr’s model of the hydrogen atom is widely cited as an example of an inconsistent scientific theory because of its reliance on classical electrodynamics (CED) together with assumptions about interactions between matter and electromagnetic radiation that could not be reconciled with CED. This view of Bohr’s model is controversial, but we believe a recently proposed approach to reasoning with inconsistent commitments offers a promising formal reading of how Bohr’s model worked. In this paper we present this new way of reasoning with inconsistent commitments and compare it with other approaches before applying it to Bohr’s model and offering some suggestions for how it might be extended to account for subsequent developments in old quantum theory (OQT).
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Notes
That is, classically or intuitionistically or indeed with any other logic we choose.
Forcing, a weakly aggregative logic due to P.K. Scotch and R.E. Jennings, preserves a generalization of consistency called level: the level of Σ, l(Σ) is the least n for which there is a consistent n-covering of Σ. For a level n, we can capture all the aggregation that follows from forcing using the rule 2/n+1: where \(2/n+1(\alpha _{0},...\alpha _{n}) = {\bigvee }_{0\leq i \neq j \leq n} (\alpha _{i} \wedge \alpha _{j})\), \({\Gamma } \vdash \alpha _{0},...{\Gamma } \vdash \alpha _{n} / {\Gamma } \vdash 2/n+1(\alpha _{0},...\alpha _{n})\).Apostoli and Brown (1995) This is the strongest formal principle of aggregation we can apply to a set of level n without trivialization.
When speaking of Bohr’s account of the hydrogen atom we will use the word ‘model’, reserving ‘theory’ for a set of sentences closed under a consequence relation.
It is tempting to allow every sentence consistent with \({\sigma _{i}^{k}}\) to permeate from \({\sigma _{j}^{k}}\) before closing under \(\vdash \) to form \(\sigma _{i}^{k+1}\). But with just two elements σ 1 and σ 2 such that \(\sigma _{1} \cup \sigma _{2} \vdash \bot \), closing both under \(\vdash \) and then allowing every sentence consistent with each to permeate in from the other is disastrous if the underlying logic is classical: for any sentence α consistent with \(\sigma _{1}^{k-1}\), consider the sentence \(\phi \rightarrow \alpha \), where ϕ follows from \(\sigma _{1}^{k-1}\) and ¬ϕ follows from \(\sigma _{2}^{k-1}\). This is trouble– \(\phi \rightarrow \alpha \) is a consequence of (and so contained in) \(\sigma _{2}^{k-1}\), and it’s consistent with \(\sigma _{1}^{k-1}\). So every such sentence permeates into σ k−1 before we close under \(\vdash \) to form \({{\Sigma }_{1}^{k}}\). But we already have ϕ in \({\sigma _{1}^{k}}\), so when we close again we find \({\sigma _{1}^{k}}\) includes every α consistent with \(\sigma _{1}^{k-1}\). But if \(Cl(\sigma _{1}^{k-1})\) is not maximal consistent, then for some β∈L, both β and ¬β are consistent with \(\sigma _{1}^{k-1}\). So unless \(Cl(\sigma _{1}^{k-1})\) is maximal consistent, \({\sigma _{1}^{k}}\) is trivial.
A general plan for eliminating inconsistency in scientific theories was proposed in Norton (1987), where Norton emphasized the separation between quantum theory and CED in the course of showing that Planck’s derivation of the black-body radiation law can be obtained using a sub-theory of CED that is consistent with quantum rules limiting the energy states of resonators and the radiation field. But in the same paper, Norton also invokes a weakly aggregative approach to Planck’s original theory, remarking “...one could not derive any proposition within the theory because of the tacit introduction of a nonclassical device, the two domains of calculation with inarticulated restrictions on the exchange of results between them.”Norton (1987, p. 348) C&P provides a systematic way of specifying such restrictions.
While some paraconsistent logics do aim at this goal, C&P does not.
The question of whether Bohr’s account was logically inconsistent is difficult to answer directly: From a purely logical point of view, Bohr’s description of the atom combined with CED implied that his atom could not have a stable ground state. Since Bohr’s model included a stable ground state, the sentences used in the course of applying his theory to account for the hydrogen spectra were inconsistent. However, Bohr’s personal views could still have been consistent: for example, he might have accepted CED instrumentally for purposes of observing light on macroscopic scales, while regarding it as unreliable on the atomic scale. But such interpretive questions about personal beliefs are not our concern here. Applying CED to the stationary states was obviously disastrous– the atom would rapidly collapse, radiating at increasing frequencies along the way. But this consequence is not logically inconsistent, and while rapidly collapsing hydrogen atoms seem consistent enough in themselves, they are clearly inconsistent with observation. To stay consistent with observation, Bohr had to avoid applying CED to his stationary states. But classical electrodynamics was the only available way to model the radiation emitted by his atoms. Bohr apparently dealt with this tension simply by assuming ‘for now’ that no radiation occurs while the atom is in a stationary state– an uneasy kind of stipulation. Our C&P structure allows us to retain CED for purposes of interpreting spectral data while systematically avoiding the disastrous collapse of the stationary states by confining CED and Bohr’s account of the stationary states in separate cells of our proposed C&P structure.
Bohr’s initial treatment approximated by treating the ratio of the proton’s mass to the electron’s as infinite. After Fowler claimed that Bohr’s calculation of the Pickering lines fell outside the bounds of experimental data, Bohr wrote a letter to Nature in which this assumption was dropped. The corrected calculation gave improved agreement with the Pickering lines and predicted several as yet unobserved lines (see Pais (1991, p. 149)) and Mehra and Rechenberg (1982, p. 192)
The Appendix provides a standard account of Bohr’s original model of the hydrogen atom, specifying the stationary states, deriving the energy differences between them and calculating the frequencies of the resulting radiation (or the radiation absorbed) using Planck’s rule. The equations used in these calculations can be read as a detailed list of the contents of σ Q .
As an example of such treatments of instruments and their interaction with light, the location of brightness maxima for various wavelengths λ of plane wave light diffracted from a grating are determined by calculating the difference of path lengths from each line of the grating to each point on the illuminated surface: maxima appear when the difference of path lengths to the surface equals λ.
Of course if we include a rich mathematics in each cell of our structure, the possibility of proving consistency can’t be ruled out, and this in turn would imply inconsistency; but that uncertainty runs deeper than the tensions in OQT, and deeper than we venture here.
The resolution of Pauli’s difficulty emerged from the re-interpretation of stationary states in quantum mechanics, which allowed states corresponding to these states forbidden by old quantum theory; see Vickers (2013, p. 67f).
In a conversation about OQT.
Bohr’s original expression for the quantization condition was given in terms of the electron’s kinetic energy, \(W = \tau h \frac {\omega } {2}\), where ω is the angular frequency of the electron’s orbit and τ is the quantum number Bohr (1913a, p. 5). Since W=π ω L this is equivalent to our 1.
Note that in Bohr (1913a) Bohr omits the Z in 3., using instead e for the charge of the electron and E for the charge of the nucleus, a for the radius of the orbit and τ for the quantum number. Most contemporary texts use r for the radius and n for the quantum number; here we follow contemporary usage.
Note that Bohr used ‘W’ for energy.
While the experimental data of the time did not provide precise values for e/m and h, Bohr applied the available values to 11, obtaining a value of 3.1×1015, which compared well to the spectrally determined value of 3.29×1015 Bohr (1913a, p. 9). Programmatically, Bohr’s proposal provided a general account of the relation between quantized energy levels of atomic systems and frequencies of their spectral emissions, drawing a direct link between physical models of atoms and spectroscopic observations for the first time. A further significant empirical advance was Bohr’s explanation for the absence of certain hydrogen lines in terrestrial observations, as due to the high ambient pressure at the earth’s surface and the size of electron orbits at higher energy levels: “According to the theory the necessary condition for the appearance of a great number of lines is therefore a very small density of the gas”Bohr (1913a, p. 9).
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Appendix: Bohr’s Hydrogen Atom
Appendix: Bohr’s Hydrogen Atom
In this appendix we follow through a standard contemporary treatment of the main features of Bohr’s account of the hydrogen atom with our proposed C&P structure in mind. Like Bohr’s, this account is focused on the properties of stationary states and inferring from them the energy and frequency of light emitted or absorbed in transitions between states (Eisberg 1961, p. 115f). We begin with Bohr’s key postulates:
- 1.:
-
\(L = \frac {nh}{2\pi } = n\hbar \) Footnote 14
- 2.:
-
E i −E f =h ν a b , or \(\nu = \frac {E_{i} - E_{f}}{h}\)
1 picks out the quantized orbits of the electron from the continuum of circular orbits allowed by classical mechanics, while 2 applies the Planck/Einstein relation between the energy and frequency of a quantum of light to the light emitted in a transition between stationary states. The most radical element in Bohr’s approach was his simple postulation that these states are stable. In the end, a new electrodynamics would be required to provide a satisfactory account of these states, but Bohr’s approach was simply to set that problem aside and attempt to characterize the states and the consequences of transitions between them as best he could without a new electrodynamics.
The next step applies the quantization rule, 1, to the classical models of the stationary states:
- 3.:
-
\(\frac {Ze^{2}}{r^{2}} = \frac {mv^{2}}{r}\)
- 4.:
-
\(L = mvr = \frac {nh}{2\pi } = n\hbar \) Footnote 15
Combining 3 and 4 leads to helpful results regarding the radius and velocity of the orbiting electron in the stationary states:
- 5.:
-
\(r = \frac {n^{2} \hbar ^{2}}{mZe^{2}}, n = 1,2,3...\)
- 6.:
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\(v = \frac {n \hbar m Z e^{2}}{m n^{2} h^{2}} = \frac {Ze^{2}}{ n \hbar }, n = 1,2,3...\)
Given experimental values for m, e and \(\hbar \) 5 predicts r 1=5.3x10−9 c m, compatible with evidence suggesting the order of magnitude of atomic radii was about 10−8 c m. Further, v 1 (the highest velocity for an orbiting electron) is 2.2x108 c m/s e c, which is less than 1% of c, indicating that special relativity need not be brought into the calculations.
Next, the total energy of the stationary states is obtained and applied to give the key results, Bohr’s calculation of the Rydberg constant and the general law of the hydrogen spectrum.
- 7.:
-
\(V = {\int }_{\infty }^{r} \frac {Ze^{2}}{r^{2}}dr = - \frac {Ze^{2}}{2r} \)
- 8.:
-
\(T = \frac {1}{2} mv^{2} = \frac {Ze^{2}}{2r} \), giving
- 9.:
-
\(E = \frac {Ze^{2}}{2r} = -T \)
Where V is the potential energy of the electron (with the 0 of potential energy defined as the state in which the electron is at rest, infinitely far from the nucleus), T is the kinetic energy and E the total energy.Footnote 16
Applying 2 to the energy levels given by 9 allows us to derive Bohr’s formula for the hydrogen spectrum and to determine the Rydberg constant for a nuclear charge Z:
- 10.:
-
\(\nu = \frac {mZ^{2}e^{4}}{4\pi \hbar ^{3}} \left (\frac {1}{{n_{f}^{2}}} - \frac {1}{{n_{i}^{2}}}\right ) \)
Re-writing 10 to follow Rydberg’s formula for the hydrogen spectrum, written in terms of wave number, k=1/λ=ν/c, we get an expression for the Rydberg constant for hydrogen:
- 11.:
-
\(R_{H} = \frac {me^{4}}{4\pi c \hbar ^{3}}\)
Up to this point the principles of CED have played no role in the derivation. From the C&P point of view, the entire argument has been conducted within σ Q . However, identifying this value for the Rydberg constant with the quantity known empirically from spectral observations requires a link to CED, in which frequencies, wavelengths and spacings of spectroscopic gratings interact in ways that, at the time, only CED captured. So the application of this theory to data on the hydrogen spectrum implicitly invokes the principles contained in σ C .Footnote 17
Relying on CED in this way does not require inconsistent beliefs: some, including Bohr, hoped that quantum constraints did not apply to the radiation field, but only to its interaction with matter. Other physicists accepted that CED would need to be replaced by a quantum theory of electrodynamics. But at the time no other theory provided an account of the interactions between light and instruments that underlie the observational practice of spectroscopy: for the purpose of reasoning about spectra, CED was indispensable.
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Brown, M.B., Priest, G. Chunk and permeate II: Bohr’s hydrogen atom. Euro Jnl Phil Sci 5, 297–314 (2015). https://doi.org/10.1007/s13194-014-0104-7
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DOI: https://doi.org/10.1007/s13194-014-0104-7