ight in vacuum c and the Newtonian constant of gravitation G, also plays an important role when assembling Planck units from universal constants. From the above, the Planck constant is a constant by name, but it has come to be strongly regarded as being the smallest unit of angular momentum.
Incidentally, in deriving the equation for the energy levels of the hydrogen atom, Bohr assumed the following quantum condition including the Planck constant:
Taking this quantum condition as a departure point, Bohr derived the following energy levels .
Here, α is the following fine-structure constant.
However, Equation (2) is a formula of classical quantum theory, and does not take into account the theory of relativity. When the theory of relativity is taken into account, the following energy levels can be derived .
The “re” in stands for “relativistic”.
Equation (4) is an elementary formula that includes only the principal quantum number n.
Here, let’s examine the relationship between Equation (2) and Equation (4).
If the part in parentheses in Equation (4) is developed as a Taylor series,
From this, it is evident that Equation (2) derived by Bohr is an approximation of Equation (4) .
Incidentally, the quantum condition is not assumed when deriving Equation (4). If the quantum condition is necessary, the following quantum condition can be inferred from the derived Equation (4) .
However, it is difficult to predict and assume a complex condition like Equation (6) at the beginning. In Equation (6), the Planck constant doesn’t seem to play an essential role for values of α or higher. Also, the Planck constant does not play an important role when deriving Equation (4).
The author has shown that the following quantization procedure is more important than quantum condition (1) .
Equation (7) is built into Equation (1). Equation (7) is a mathematical procedure for shifting from the classical world dominated by continuity to the discrete quantum world.
Here, let’s multiply both sides of Equation (7) by the formula for classical kinetic 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 followin