In 1900, when deriving a formula that derived an experimental value of black-body radiation, Planck proposed the quantum hypothesis stating that the energy of a harmonic oscillator with oscillation frequency would quantize at the integral multiple of . This was the first time that the Planck constant h appeared in physics theory . Since this time, the Planck constant has been thought to be a universal constant defined in the realm of quantum theory, but the essence of this constant cannot be considered to be well understood. In this paper, using non-historic reasoning, the true essence of this constant is revealed.
Beforehand, let us verify the following points regarding fundamental physical constants and the Planck constant. Fundamental physical constants play an essential part in elementary formulas that describe natural phenomena and can be largely divided into universal constants and material constants.
Universal constants include the speed of light in vacuum c, Newtonian constant of gravitation G, and Planck constant h.
Material constants can be divided into micro material constants and macro material constants. Also, micro material constants can be divided into atomic and nuclear constants and electromagnetic constants.
Atomic and nuclear constants include the electron rest mass ,electron’s Compton wavelength ,and include such constants as the fine-structure constant α and the Rydberg constant . Electromagnetic constants include the elementary charge e.
The Boltzmann constant k and the Avogadro constant NA are examples of macro material constants.
However, the Planck constant h is thought to be a universal constant representative of quantum mechanics and not dependent on human definition, unlike micro material constants such as α or .
Because the Planck constant has an action quantity dimension, it was at first called an action quantum when quantum theory originally emerged. h appears in the inequality when Heisenberg discovered the uncertainty principle in 1927. Planck constant h, along with the speed of light 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 kin