Relativistic Correction of the Rydberg Formula

The relationship E = −K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.


Introduction
In the classical quantum theory of Bohr, the energy levels of the hydrogen atom are given by the following formula [1] [2].
Here, BO E refers to the total mechanical energy predicted by Bohr. Also, α is the following fine-structure constant.
Bohr thought the following quantum condition was necessary to find the energy levels of the hydrogen atom. e 2 2 .
n n m v r n ⋅ π = π  (3) The energy of the hydrogen atom is also given by the following formula.
If E in Equation (1b) is substituted into Equation (4), then the following formula can be derived as the orbital radius of the electron.
The photonic energy emitted during a transition between energy levels ( ) Here, R ∞ is the Rydberg constant, which is defined by the following equation.
The Rydberg formula can be derived from Equation (6)

Relationship Enfolded in Bohr's Quantum Condition
This section to Section 4 are excerpts from another paper, but this material is repeated because it is needed here. The Planck constant h can be written as fol- Here, C λ is the Compton wavelength of the electron.
When Equation (9) is used, the fine-structure constant α can be expressed as follows.
Also, the classical electron radius e r is defined as follows.
From this, the following relationship can be derived [4].

The Relation between Kinetic Energy and Momentum Derived from the STR Relationship
The energy-momentum relationship in the special theory of relativity (STR) holds in an isolated system in free space. Here, if 0 m is rest mass and m relativistic mass, the relationship can be written as follows.
( ) ( ) What is the relationship between relativistic kinetic energy and momentum if this relationship holds?
Incidentally, Sommerfeld once defined kinetic energy as the difference between the relativistic energy 2 mc and rest mass energy 2 0 m c of an object [5].
That is, Sommerfeld believed that Equation (18), which can be derived from Equation (17), can also be applied to the electron in a hydrogen atom.
First, it is clear that the following formula holds [4].
Expanding the left side of this equation yields the following.   The following formula is obtained from this.
Here, re K is relativistic kinetic energy and re p relativistic momentum. The "re" in re K and re p stands for "relativistic".
Equation (23) is the formula for relativistic kinetic energy. Classical (non-relativistic) kinetic energy, in contrast, is defined as follows.
In classical theory, mass does not depend on velocity. That is, Equation (23) and Equation (24)

Energy-Momentum Relationship of the Electron Derived with Another Method
The author has previously derived the following relationships applicable to the electron constituting a hydrogen atom [6]. These energy relationships can be illustrated as follows (Figure 1). In this paper, Equation (25) will be derived more simply by using a method different from that used previously. The logic of Equations (19) to (23) is borrowed to accomplish that purpose. Now, it is clear that the following equation holds.
Next, the relativistic kinetic energy of the electron can be defined as follows by referring to Equation (23).
Hence, the energy levels of a hydrogen atom re,n E are:

Relativistic Energy of a Hydrogen Atom Derived from Equation (16)
When both sides of Equation (16) Next, the numerator and denominator of Equation (39) are multiplied by: When this is done, Equation (39) is as follows.  Next, Table 1 summarizes the energies of a hydrogen atom obtained from Equation (1) and Equation (34) [7].
The following values of CODATA were used when calculating energies. The results derived from section 3 to 5 are summarized here in Table 2.
In deriving the energy levels of a hydrogen atom, Sommerfeld began from Einstein's energy-momentum relationship. However, that is a mistake. The Einstein relation that holds in an isolated system in free space is not applicable in the space inside a hydrogen atom where there is potential energy. The author derived, for the first time, Equation (25) that is applicable to an electron in a hydrogen atom.

Discussion
In the sections up to the previous section, the groundwork was laid for finding a formula for the wavelength of a photon emitted from a hydrogen atom.
The differences in energy between different energy levels in the hydrogen atom can be found with the following formula.