Max Planck
Quantum Mechanics
Abstract
This research will illustrate how Planck units can be derived from first principles, uncovering their complex relationships not only with the electron
but also with gravity and thermodynamics. The investigation began due to the realization that the Reduced Planck constant is largely misunderstood
and typically treated as a scalar value. By utilizing the component parts of the Reduced Planck constant instead of a fixed value, numerous new insights emerge.
Introduction
Max Planck was a German theoretical physicist born on April 23, 1858, in Kiel, Germany. He is best known for being the originator of quantum theory, which
revolutionized our understanding of atomic and subatomic processes. Planck's work on quantum theory earned him the Nobel Prize in Physics in 1918.
One of his most significant contributions is the introduction of the Planck constant (denoted as \(h\)), which is a fundamental constant in quantum mechanics.
This constant is crucial for describing the behavior of particles at the quantum level. Planck's work laid the foundation for modern physics and
has had a profound impact on various fields, including quantum mechanics and thermodynamics.
Planck units are a set of natural units that are defined using fundamental physical constants. As mentioned they were named after the German physicist Max Planck, who first proposed them. Planck units provide a system of measurement that is based on natural physical scales, making them very useful in theoretical physics, particularly in quantum mechanics and cosmology.
Planck Length
Arguably, the most important of these units is the Planck length itself;
$$l_p = 1.616\ 314\ 0683 × 10^{-35}\text{m}$$
The Planck length is an extremely small value that, once again, seems beyond accurate calculation.
To determine the Planck length, we must begin from first principles by considering the Bohr radius, which is the assumed distance
between the nucleus of a hydrogen atom and its ground state electron, and is known with high precision its value being;
$$a_0 = 5.291\ 7721\ 0903 × 10^{-11}\text{m}$$
The simple question can be asked as to whether the Bohr radius is the smallest value that exists, clearly not, it must consist of many smaller entities
or quanta. What is expected is that each of these individual lengths that make up the Bohr radius logically should be in dimensions of the Planck length
itself. It follows that a simple calculation can be made to verify this fact.
It may seem unusual but the values that can be used are Avogadro's and Eulers numbers. Avogardro's number to ascertain the number of states or elements and
Euler's number to zero in as it were on the exact value.
Quanta
Using Avogadro's and Euler's number it follows that;
$$N_q = \frac{a_0}{2 N_A e} = 3.273\ 975\ 1593 × 10 ^{24}$$
Where:
\(N_q\) Number of quanta
\(a_0\) is the Bohr radius
\(N_A\) is Avogadro's number
\(e\) is Eulers number
The number of "quanta" \(N_q\) is the count of individual component parts, entities or quanta which make up the Bohr radius.
It is clearly a large number of units, but what does it represent? By dividing the Bohr radius by the Planck length, we indeed obtain the
number of elementary units or quanta which make up the Bohr radius. Performing this calculation yields the value \(3.273\ 975\ 1593 × 10 ^{24}\).
Having determined the number of quanta in the Bohr radius from first principles, it can be asked why this exact value. It is not a coincidence it is due
to the implicit relationship with the Boltzmann constance which is one of the base values used to calculate the planck units. This number can be validated
calculating the size of each individual quanta, by dividing the Bohr radius by the number of quanta.
The result, is exactly as expected, the value of Planck length
$$l_p = 1.616\ 314\ 0683 × 10^{-35}$$
This result clearly illustrates the close relationship between the quantum scale Planck units and the atomic scale and temperature. The calculation it not
particularly unusual except that it serves to support the idea of one cohesive matrix of correlating equations. These values being based upon the somewhat arbitrary
ontological units such as time, the orbit of the Earth around the sun, weight based upon historical norms and the length of a platinum bar in Paris.
The Reduced Planck Constant
Possibly, the most utilized value, the Reduced Planck constant, itself represents the angular momentum of the ground state electron in a hydrogen atom.
Traditionally, it is considered to express the amount of energy in a quantum, but rather it can be more accurately represented as angular momentum.
Max Planck determined its value in response to the "ultraviolet catastrophe," assuming it to be the proportionality between the frequency and
energy of electromagnetic radiation.
The value of the reduced Planck constant can also be calculated directly from the properties of the electron as follows;
$$\hbar = m_ea_0v_e = 1.054\ 571\ 8176 × 10 ^{-34}\text{Js}$$
Where:
\(\hbar\) is Plancks reduced constant
\(m_e\) is the electron mass
\(a_0\) is the Bohr radius
\(v_e\) is the electron velocity
As in prior calculations this is based only upon the properties of the ground state electron.
One of the more popularly quoted values is that of the Planck constant itself which is;
$$ h=2 \pi\hbar = 6.626\ 070\ 1499\ × 10 ^{-34} \text{Js}$$
With the principal Planck unit being established, the remaining two important values are the Planck mass and Planck time.
Planck Mass
The Planck mass can be calculated similarly to the Planck length, using the number of quanta in the Bohr radius as follows;
$$m_p = N_q m_e \alpha = 2.176\ 354\ 8377 × 10 ^{-8}\text{kg}$$
Where:
\(m_p\) is the Planck mass
\(N_q\) is the number of quanta
\(m_e\) is the Electron mass
\(v_e\) the Electron velocity
\(c\) the speed of light
Planck Time
The final Planck unit of any importance being that of the Planck time, in a similar manner as above this value can be calculated as follows;
$$t_p = \frac {a_0}{N_qc} = 5.391\ 443\ 3973 × 10 ^{-44}\text{s}$$
Where:
\(t_p\) Planck time
\(a_0\) the Bohr radius
\(N_q\) the number of quanta
\(c\) the speed of light