Theoretical and Experimental Values for the Rydberg Constant Do Not Match

In many areas of physics and chemistry, the Rydberg constant is a fundamental physical constant that plays an important role. It comes into play as an indispensable physical constant in basic equations for describing natural phenomena. The Rydberg constant appears in the formula for calculating the wavelengths in the line spectrum emitted from the hydrogen atom. However, this Rydberg wavelength formula is a nonrelativistic formula derived at the level of classical quantum theory. In this paper, the Rydberg formula is rewritten as a wavelength formula taking into account the theory of relativity. When this is done, we come to an unexpected conclusion. What we try to determine by measuring spectra wavelengths is not actually the value of the Rydberg constant R ∞ but the value , n m R of Formula (18). R ∞ came into common use in the world of nonrelativistic classical quantum theory. If the theory of relativity is taken into account, R ∞ can no longer be regarded as a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.


Introduction
In many areas of physics and chemistry, the Rydberg constant is a fundamental physical constant (abbreviated below as "physical constant") that plays an important role. It comes into play as an indispensable physical constant in basic equations for describing natural phenomena. Around the end of the 19 th century, Balmer, Rydberg, and others discovered that the following relation holds be- This R ∞ is the Rydberg constant. Its value is fixed, and does not depend on the spectra series or atom. The physical status of this empirical formula, which was obtained experimentally, was indicated by Bohr. In Bohr's theory of the hydrogen atom, R ∞ is given by the following formula. Here, it is shown that R ∞ is an approximate value obtained by assuming that the mass of the proton is infinite. To obtain a value with higher precision, the mass of the electron must be replaced with the reduced mass of the electron and proton. This results in a correction of about 1/2000. The correction due to the theory of Dirac, which treats the hydrogen atom relativistically, is on the order of ( ) 2~1 137 α α . Furthermore, the correction of quantum electrodynamics (QED) is on the order of 3 α .
A crucial point is that all of these corrections vary in proportion to R ∞ . Therefore, even when these corrections are made, it is thought that the value of R ∞ can be found with high precision.
In the classical quantum theory of Bohr, the energy levels of the hydrogen atom are given by the following formula [1].
Here, BO E refers to the total mechanical energy predicted by Bohr. Also, α is the following fine-structure constant.
The photonic energy emitted during a transition between energy levels The Rydberg formula can be derived from Formula (5) Formula (6) is derived from Formula (3), and thus is not a wavelength formu-K. Suto la taking into account the theory of relativity.
Incidentally, the author has already derived the following energy levels of the hydrogen atom taking into account the theory of relativity (Appendix A).
Here, However, the term "relativistic" used here does not mean based on the special theory of relativity (STR). It means that the expression takes into account the fact that the mass of the electron varies due to velocity. According to the STR, the electron's mass increases when its velocity increases. However, inside the hydrogen atom, the mass of the electron decreases when the velocity of the electron increases. Attention must be paid to the fact that, inside the hydrogen atom, the relativistic mass of the electron n m is smaller than the rest mass e m .
The following figures the energy levels of the hydrogen atom derived by Bohr, and the energy levels derived by the author (Figure 1).
Comparing Formulas (8b) and (3b), it is evident that Formula (3) is an approximation of Formula (7). That is, Now, the author has already shown that the following wavelength formula can be derived from Formula (7) Formula (10) is a wavelength formula taking the theory of relativity into account. In this paper, Formula (10) is rewritten in a form similar to Formula (6). It is also checked what happens to the part corresponding to R ∞ in Formula (6) in the newly derived formula.

The Relativistic Wavelength Formula Obtained by Rewriting Formula (10)
If the Taylor expansion of Formula (10) is taken, the following formula is obtained.
The following relationship is used here.
It is evident from this that the previously-known Formula (6) is an approximation of Formula (10).
Next, Formulas (6) and (10) are further compared. If, in Formula (10), the values of the Compton wavelength C λ and fine structure constant α are determined, it is possible to calculate the spectra wavelengths. Of course, Formula (6) too must predict the spectra wavelengths. However, at present, the spectra wavelengths are first measured, and then the value of R ∞ is determined based on those values. Formula (6) is not for calculating wavelengths, but for determining the Rydberg constant. Next, the following table summarizes the wavelengths calculated using the Rydberg Formula (6) and Formula (10) for wavelengths derived by the author ( Table 1).
The discussion thus far has already been presented in another paper (reference [2]). The discussion presented for the first time in this paper begins here.
Formula (10) is first rewritten as the following formula similar to Formula (6).
First, Formula (10) can be written as follows taking Formula (12) into account.

(
) ( ) Here, the following , n m R is defined.
It is thus evident that the value we try to determine by measuring spectra wavelengths is not R ∞ in Formula (6)  is not a physical constant, as is evident from Formula (18).
Next, Table 2 summarizes the values of Ordinarily, we determine the value of the Rydberg constant in Formula (6) by measuring the spectra wavelengths. If this value matches with the theoretical value (Formula (2)), then the validity of Bohr's model of the atom is confirmed.
However, the value we try to determine through experiment is the value of Originally, Formula (1) is given as follows. However, writing out the right side of Formula (2) is bothersome, and for reasons of convenience, this expression was replaced with the single symbol R ∞ . (However, this is not a description of the actual history.) Therefore, R ∞ is not a physical constant on a par with c or e. It is also not the case that we discovered a physical constant R ∞ . R ∞ came into common use in the world of nonrelativistic classical quantum theory. If the theory of relativity is taken into account, R ∞ can no longer be regarded as a physical constant.

Conclusions
The formula for wavelengths in classical quantum theory is the following.
In contrast, the author has previously derived the following formula, more precise than Formula (22), by taking into account the theory of relativity.
( ) ( ) is not a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.