The Quantum Condition That Should Have Been Assumed by Bohr When Deriving the Energy Levels of a Hydrogen Atom

Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. This famous quantum condition was not derived logically, but it beautifully explained the energy levels of the hydrogen atom. Therefore, Bohr’s quantum condition was accepted by physicists. However, the energy levels predicted by the eventually completed quantum mechanics do not match perfectly with the predictions of Bohr. For this reason, it cannot be said that Bohr’s quantum condition is a perfectly correct assumption. Since the mass of an electron which moves inside a hydrogen atom varies, Bohr’s quantum condition must be revised. However, the newly derived relativistic quantum condition is too complex to be assumed at the beginning. The velocity of an electron in a hydrogen atom is known as the Bohr velocity. This velocity can be derived from the formula for energy levels derived by Bohr. The velocity v of an electron including the principal quantum number n is given by αc/n. This paper elucidates the fact that this formula is built into Bohr’s quantum condition. It is also concluded in this paper that it is precisely this velocity formula that is the quantum condition that should have been assumed in the first place by Bohr. From Bohr’s quantum condition, it is impossible to derive the relativistic energy levels of a hydrogen atom, but they can be derived from the new quantum condition. This paper proposes raising the status of the previously-known Bohr velocity formula.


Introduction
N. Bohr was the first to derive the energy levels of an electron forming a hydrogen atom (this will be abbreviated below as energy levels of the hydrogen atom).
This Introduction reviews the history up to derivation of the energy levels of the hydrogen atom with the assistance of the writings of Dr. H. Ezawa in Japanese.
In 1884, J. J. Balmer noticed that the wavelengths λ of the spectral lines emitted from a hydrogen atom could be described with the following formula.
After that, W. Ritz transformed this formula as follows.
From Formula (4), Bohr predicted the following relationship.
The energy of the hydrogen atom is discontinuous. Bohr thought that when the electron transitions from a state with energy E n to a state with energy E m , the electron emits a photon with energy hν . He also obtained the following formula for energy levels.
At the time, the value of cR was known through experiment, but details concerning R were not known. Thus Bohr decided to derive the energy levels of the hydrogen atom using another method.
First, Bohr considered the case where the electron moves at constant speed around the atomic nucleus (proton). If r is taken to be the radius of a circular orbit, and v is taken to be the speed of the electron, then the following Newtonian equation of motion holds. (7) This equation indicates the equality of the centrifugal force acting on the electron (left side) and the Coulomb attraction received by the electron from the atomic nucleus. Here, the electron mass was set to m e , and the charge was set to -e.
Also, since the energy of the electron can be expressed by the sum of the ki- in the case of a circular orbit, and thus the energy can be written as follows (the discussion here concerns a circular orbit as a special form of an elliptical orbit).
Here, if both sides of Formula (10) are squared, Incidentally, the angular momentum L when an electron moves in a circle can be expressed as mvr. Here, if the number n is affixed to the energy E and angular momentum L, then Formula (12) becomes as follows.
The energy in Formula (6)  It was found that L, for which a unique value was not known, could be expressed with the following equation.
Bohr substituted in the not very precise numeric values for physical quantities that were known at the time and conjectured L n to be as follows.   Here, BO,n E signifies the energy levels derived by Bohr. Also, α is the following fine-structure constant. Formula (17) is not a logically derived formula. It is a formula derived by assuming the quantum condition in Formula (16). At the time, L. de Broglie noticed that light, thought to be a wave in the classical theory, exhibits particle characteristics. He also predicted that the electron, thought to be a particle, would exhibit wave characteristics. He also assumed that when the wavelength λ of the wave accompanying an electron in circular motion satisfies the following relationship, that electron is state.
The following relationship holds between the momentum p and wavelength λ of the electron. According to de Broglie, Bohr's quantum condition was able to acquire a substantive meaning, and thus it came to be that the energy levels of the hydrogen atom in Formula (17), found by assuming Formula (16), were believed to be correct.
Also, if E n in Formula (17) is substituted into Formula (10), then the following formula can be derived as the orbital radius of the electron. Here, BO,n r is the orbital radius of the electron predicted by Bohr's theory. Also, B a is the orbital radius when n = 1, i.e., the Bohr radius. The content of this paper thus far simply reiterates the information in another paper by the author [2]. However, this was deemed necessary for the discussion in subsequent sections. The content of this introduction is a shared understanding of physicists. Journal of Applied Mathematics and Physics was accepted because it enabled the energy levels (17a) of a hydrogen atom to be derived correctly. However, the value of Formula (17a) does not match perfectly with the value predicted by the completed theory of quantum mechanics. This is likely because Bohr did not take account of the theory of relativity.
When the theory of relativity is taken into account, Bohr's quantum condition (21) and de Broglie's hypothesis (19) must be revised.

Relationship Enfolded in Bohr's Quantum Condition
Bohr thought the following quantum condition was necessary to find the energy levels of the hydrogen atom.
In Bohr's theory, the energy levels of the hydrogen atom are treated non-relativistically, and thus here the momentum of the electron is taken to be e m v .
Also, the Planck constant h can be written as follows [3]. When Formula (24) is used, the fine-structure constant α can be expressed as follows. Also, the classical electron radius e r is defined as follows.
If Formula (22) is written using e r and α, the result is as follows.    This shows that the electron mass which appears in Bohr's quantum condition is rest mass.

Bohr's Energy Levels (17)
When both sides of Formula (31) are squared, and then multiplied by e 2 m ,

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 The following formula is obtained from this. Formula (41) is the formula for relativistic kinetic energy [2]. Classical (nonrelativistic) kinetic energy, in contrast, is defined as follows. Next, the relativistic kinetic energy of an electron in a hydrogen atom is defined as follows by referring to Formula (41).  here is just the principal quantum number. Therefore, re,n E is not a formula that predicts all the relativistic energy levels of the hydrogen atom.) However, the term "relativistic" used here does not mean based on the 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 .
In this way, two formulas have been obtained for the relativistic kinetic energy of the electron in a hydrogen atom (Formulas (43), and (45)).
The following Figure 1 illustrates the energy levels BO,n E derived by Bohr and the energy levels re,n E derived in this paper.
Incidentally, the following equation can be derived from Formulas (43) and (45). Formula (47) is the energy-momentum relationship applicable to the electron in a hydrogen atom (the author calls this "Suto's energy-momentum relationship").
The author already derived this relationship (47) using another method [5] [6]. The difference between Einstein's relationships (35) and (47) arises due to the presence/absence of potential energy.
The relation between n m and e m is as follows due to Formula (49).

Bohr's Quantum Condition and De Broglie's Hypothesis Derived from a Relativistic Standpoint
In the discussion in the previous section, it was possible to find re,n p and re,n r for relativistically deriving Bohr's quantum condition (23) and de Broglie's hypothesis (19).

Bohr's Quantum Condition
If the values of Formulas (52) and (56) Here, we use the fact that Formula (27) can be written as follows.
Therefore, the right side of Formula (60) is as follows.
Here, we take into account the fact that the following relationship holds.