A Surprising Physical Quantity Involved in the Phase Velocity and Energy Levels of the Electron in a Hydrogen Atom

The phase velocity phase v of a material wave is given by the following equation: phase .  If this formula is rewritten using the Planck-Einstein relation E hν  and the de Broglie formula / p h λ  , it becomes: phase / . v E p  Next, the values for E and p , obtained from a relation applicable to the electron in a hydrogen atom derived by the author, are substituted into this equation. When that is done, multiple formulas relating to the phase velocity of the electron wave are derived. These formulas contain still-unknown ultra-low energy levels of the hydrogen atom and electron orbital radii n r  corresponding to those energy levels. n r  also appear in the formula for energy levels of the hydrogen atom derived in this paper. This serves as grounds for the existence of ultra-low energy levels in addition to the already-known energy levels of the hydrogen atom.


Introduction
According to Maxwell's electromagnetism, the following relationship holds between the momentum p and energy E of light. .


(1) Also, Einstein asserted, based on consideration of the photoelectric effect, that light has a particle nature, although it had previously been regarded as a wave.
If a photon as a single particle is assumed to have a frequency ν, Einstein concluded it has the following energy. .


(2) Here, h is the Planck constant.Also Formula (2) can be written as follows using the angular frequency ω . , ω is defined as follows.
. 2 Incidentally, the quantum condition of Bohr, which provided a good explanation of the stability of the hydrogen atom, contains an integer called a quantum number (Bohr, 1913).
In classical physics, integers appear in interference and normal modes.Thus, de Broglie thought that, if light -previously thought to be a wave -has a particle nature, then perhaps the electron-thought to be a particle-has a wave nature.Thus, he applied Formula (5) to matter.
In classical physics, the following relation holds between momentum p and kinetic energy K.The phase velocity phase v and group velocity group v of a material wave are defined as follows (in the following, these may be abbreviated as p , In light of the above, the phase velocity of the wave is as follows.
Also, the group velocity of the wave is as follows.
g 2 2 de Broglie noticed that the velocity of a body is the group velocity of the material wave.He also concluded that the velocity, obtained from the product of the material wave's wavelength and frequency, is the phase velocity.
Incidentally, the author has already derived a formula for the relativistic energy levels of a hydrogen atom.The only quantum number in this formula is the principal quantum number n.The energy levels derived by the author are quantitatively nowhere near the solutions of the Dirac equation.However, Dirac derived the relativistic wave equation by assuming that Einstein's energy-momentum relationship holds even within the hydrogen atom.On the other hand, the author has already pointed out that Einstein's relation does not hold in the hydrogen atom, where potential energy exists.
Thus far, in deriving the relativistic energy levels of the hydrogen atom, the author has not considered at all the conclusions reached by de Broglie.
Therefore, this paper examines whether new findings are obtained regarding the energy levels of the hydrogen atom when the assertions of de Broglie are taken into account.
In light of these points, Section 5 of this paper derives a formula for the phase velocity of the electron in a hydrogen atom.The formulas needed to achieve that objective are confirmed in the subsequent sections.

Formulas for Kinetic Energy and Momentum Derived From the Special Theory of Relativity
According to the special theory of relativity (STR), the following relation holds between the energy and momentum of a body moving in free space (Einstein, 1961).
Here, 0 m is the rest mass of the body, and m is the relativistic mass.Now, Formula ( 13) is rewritten as follows.
Comparing Formulas ( 13) and ( 14), the relativistic momentum re p can be defined as follows.
The "re" subscript of re p stands for "relativistic."Incidentally, Einstein and Sommerfeld defined the relativistic kinetic energy re K as follows (Sommerfeld,  1923).
The following relation holds due to Formulas ( 15) and ( 17).The formula for kinetic energy of a body is given by Formula (7) in classical physics, but in the STR, this becomes Formula (18).
Incidentally, the following relationship holds between 0 m and m in the STR.p c m m 

Energy Levels of the Electron in a Hydrogen Atom
Section 3 confirms the energy-momentum relationship applicable to the electron in a hydrogen atom.
This relation has already been derived by the author with three types of methods.One of these is presented here (Suto, 2011;Suto, 2018;Suto, 2020a;Suto, 2020b).
The relativistic kinetic energy of an electron in a hydrogen atom is defined as follows by referring to Formulas (17) and ( 18).This paper defines re,n E as the relativistic energy levels of the hydrogen atom.(The quantum number used here is just the principal quantum number.Therefore, re,n E is not a formula which 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 (22), and ( 23)).
The following formula can be derived from Formulas ( 22) and ( 23).Rearranging this, the following relationship can be derived.
Formula ( 25) is the energy-momentum relationship applicable to the electron in a hydrogen atom.Now, in the past, Dirac derived the following negative solution from Formula (13).
If the same logic is applied to Formula (25), then the following formula can be derived.
However, Formula ( 27) does not incorporate the discontinuity peculiar to the micro world.Therefore, Formula ( 27) must be rewritten into a relationship where energy is discontinuous.
In order to incorporate discontinuity into Formula ( 27), the author has previously shown that the following relation can be used (Suto, 2021a).(Appendix) .
Here, α is the following fine-structure constant.Using the relation in Formula (28), Formula ( 27) can be written as follows (Suto, 2014).
From this, it is evident that Formula (34) derived by Bohr is an approximation of Formula (32).
Incidentally, the fine-structure constant can be written as follows.
Here, e r is the following classical electron radius.Also, the following formula was used here (Suto, 2020c).
Here, C λ is the Compton wavelength of the electron.
If α in Formula ( 35) is substituted into Formula (32a), The energy levels (34) of the hydrogen atom derived by Bohr include the Planck constant and fine-structure constant.However, and α are not included in Formula (38).It is very significant that a formula can be derived for energy levels which does not contain the constants and α which are important in quantum mechanics.
Incidentally, in Formula (34) for the energy levels of the hydrogen atom derived by Bohr, the energy of an electron at rest infinitely far from the proton was regarded as zero (Figure 1).
The rest mass energy of the electron is not taken into account in Bohr's theory.Thus, the author derived a formula (32) for the energy levels of the hydrogen atom, taking into account the rest mass energy of the electron (Suto, 2021b).(Figure 2) Figure 1 Figure 2 Figure 1.In Bohr's theory, the energy when the electron is at rest at a position infinitely distant from the proton (atomic nucleus) is defined to be zero.

Orbital Radius of an Electron in a Hydrogen Atom
The total mechanical energy of the hydrogen atom is given by the following formula.
Also, if the formula for potential energy is used, then re,n E can be written as follows (Suto, 2018b).
  Next, if the electron orbital radii corresponding to the energy levels in Formula (30) are taken to be, respectively, Also, Formulas ( 46) and ( 47) can be written as follows (Suto, 2017a).
  In this paper, n r  is called the orbital radius, as is customary.However, a picture of the motion of the electron cannot be drawn, even if that motion is discussed at the level of classical quantum theory.The electron in a hydrogen atom is not in orbital motion around the atomic nucleus.The domain of the ordinary hydrogen atom that we know starts from n r  The following ratio is obtained from Formulas ( 46) and ( 47).
Here, if we set The author pointed out that an electron with negative mass forming dark hydrogen atom exists near the atomic nucleus (proton) (Suto, 2017b;Suto, 2021c).

Phase Velocity of the Electron Wave in a Hydrogen Atom
The important formulas derived in Sections 2 to 4 were all derived in the past by the author.In Section 5, the phase velocity of the electron wave in the hydrogen atom is derived from the standpoint of relativity theory based on the discussion in the previous sections.
First, the electron's phase velocity p,n v is given by the following formula.Formula (54) can be written as follows using the relationship of Formulas ( 2) and ( 5) (velocity and frequency are easily confused, so caution is necessary). p, .
Due to the above, the formula for the relativistic kinetic energy of the electron corresponding to Formula (1) is as follows.
The energy of a photon is found as the product of the photon's momentum and the speed of light.The kinetic energy of an electron, in contrast, is determined by the product of the electron's momentum and its phase velocity.
In Formula (57), the velocity of the electron as a wave and its velocity as a particle are both involved in the relativistic kinetic energy of the electron.A single formula incorporates the particle/wave duality.
Incidentally, there are multiple formulas for the kinetic energy and momentum of the electron, as is also evident from Table 2. Here, the phase velocity of the electron wave is derived with three methods by appropriately combining those formulas.When that is done, Formula (58) can be written as follows.
Next, the following equation obtained from Formula (28) is used.
Formula (62) can also be written as follows.
  In the second method, the phase velocity is defined as follows.
Rearranging this equation, In the third, phase velocity is defined as follows.
Rearranging, the following is obtained.
Naturally, the phase velocities derived with the three methods all agree.Also, the following formula can be derived from Formulas ( 52) and (66).
Taking the ratio of p,n v and g,n v from Formula (60), the result is as follows.
Developing the Taylor expansion of Formula (71), Incidentally, Formula (62) can be written as follows by using Formula (61). .
Next, let us consider the kinetic energy of the electron.

Discussion
This paper considers the meaning of the following equation obtained from Formulas ( 65) and (70).Incidentally, despite the fact that energy is described with an absolute scale in Formula (30), there is a negative solution.
This problem can be solved by considering the situation in the following way.The electron has a latent negative energy of 2 e mc  (Suto, 2020b).
In the state where all of the photon energy of the electron has been discharged, the electron energy is 2 e mc  not zero.
That is, the mass specific to the electron is e m  .However, this mass is the value obtained by assuming that the electron approaches from the center of the proton ( 0 r  ) to a distance of e /4 r .
In the state ab 0 E  , the photon energy of the electron 2 e mc and the specific negative energy Here, the true photon energy tab,n E  of the electron is defined as follows.
The "tab" subscript of this energy indicates the true, absolute photon energy.The descriptor "tab" is applied because absolute energy ab,n E has already been defined.).
In other words, the relation between The following illustrates the relationship of the three types of energy levels (Figure 3).

Conclusion
A. Previously, the phase velocity of a material wave has been defined with Formulas (10) and ( 54).However, this paper derived the following formula as the wave phase velocity of the electron in a hydrogen atom.Next, the following relation holds due to Formula (101).Comparing here with Formula (59), it is evident that the following relation holds.and the energy levels of the hydrogen atom.The existence of these states has been predicted mathematically, but has not yet been experimentally verified.However, due to the discussion in this paper, it has been confirmed that ultra-low energy levels of the hydrogen atom definitely exist.

Figure 2 .
Figure 2. According to the STR, the energy of an electron at rest at a position where r  is the orbital radii of an electron in a hydrogen atom n r  and the orbital radii of an electron with a negative mass .
velocity of the electron wave when the principal quantum number is in the n wavelength and frequency of the electron wave.
denominator on the right side of Formula (82) express some kind of energy levels, and it is convenient if n r  corresponds with the energy levels of 22 e n m c m c  (the same is also true for 22 e n m c m c  ).
way, it is possible for an electron in state ab 0 E  to emit a photon and transition to negative energy levels.

E
photon energy of an electron whose energy levels are at ab,n E  (

Figure 3 .
Figure 3. Relationship of three types of energy incorporated in the formulas for the electron phase velocity,

Table 1 .
Physical quantities described in classical physics and the STR

Table 2 .
Physical quantities of an electron in a hydrogen atom based on classical physics and the STR Furthermore, the relation between positive and negative energy levels is as follows.
  A formula like the following was also derived as the formula for electron phase velocity.In this paper, the following formulas were derived as formulas for the energy levels of a hydrogen atom.