etic energy . When this is done, we obtain:
If found from Equation (7) is used, then the energy levels (2) derived by Bohr are obtained. That is,
Equation (4) can be derived from Equation (7) ,but Equation (4) cannot be derived from Equation (1).
Incidentally, the following Rydberg formula is available for calculating spectrum wavelength.
Recently, the author has shown that Equation (10) is correct within the scope of classical theory, but it becomes an approximation when the theory of relativity is taken into account . In this way, it has been shown that even the Rydberg constant in Equation (10) cannot, strictly speaking, be regarded as a fundamental physical constant.
It has also been pointed out that there are ultra-low energy levels that cannot be predicted by quantum mechanics in the hydrogen atom. The incompleteness of quantum mechanics has also become evident   .
Thus, the next section examines whether Planck constant can truly be called a universal constant.
2. Planck Constant Derived from Fundamental Physical Constant
Below is Einstein’s formula expressing the equality of energy and mass .
Meanwhile, Einstein’s relational expression which applies Planck’s quantum hypothesis to photons is as follows .
The photon’s energy E is proportional to its frequency ,and this constant of proportionality is known as the Planck constant. Equation (11) and Equation (12) are traditionally thought to be representative equation of the special theory of relativity and quantum mechanics, the roots of modern physics, and these two equations have been thought to have similar importance.
Now, if is the mass of an electron, an electron’s rest mass energy can be represented by the following equation.
If the law of conservation of momentum is taken into account, then a single photon of energy is never emitted from an electron with rest mass energy .
However, this does not mean that the existence of a photon with energy equal to is forbidden.
If is the frequency of a photon carrying an amount of energy equivalent to ,the following is true.
If we subsequently combine equals from Equation (13) and Equation (14), we obtain:
Here, the left half is the energy contained within the electron. This energy is emitted as multiple photons when it is externally emitted. The right half of this equation is the sum of all of this photon energy as represented by the frequency of a photon when converted into a single photon. Fundamentally these two types of energy have different characteristics, but from a quantitative perspective, it is possible to combine them as equals.
Incidentally, an electron’s Compton wavelength is represented by the following equation.
The Planck constant can be defined as follows from Equation (16).
Incidentally, Equation (13) can be written as follows.
is the wavelength of a photon with energy ,but when Equation (17) is taken into account, Equation (18) can be written as follows.
Comparing Equation (17) and Equation (18), it is evident that in Equation (18) matches the electron’s Compton wavelength.
Thus, if we call the Compton frequency, and rewrite it as, then Equation (18) becomes as follows.
Next, let’s consider the case where the energy of the photon is not but rather.
The following relation holds between and. (Naturally, m and are not the mass of the photon.)
At this time, the following relation holds.
From Equation (21) and Equation (22) it is evident that all photon energies can be described using the following formula.
We next substitute the following values for physical quantities in .
By doing so, becomes as follows.
Meanwhile, the Planck constant has the following value .
and h are a perfect match. The currently known values for or were not determined through experimentation. was determined through precise calculations from Rydberg constant formulas, and was obtained by substituting in the formula
Based on measured data from theoretical formulas or experiments designed to represent the fundamental laws of physics, many fundamental physical constants are being adjusted to avoid conflicts from arising between these con