Introduction

In the following I shall show that a connection exists between the energy states of a hydrogen atom and the total content of energy of the Universe. It will be shown that the mechanical energy of an electron in a certain state is equal to a definite fraction of the total energy of the Universe. It will be shown that the discontinuous properties are a consequence of the quantization of space, time and mass.

In 1913 Niels Bohr showed (Niels Bohr (1885-1962): 'On the Constitution of Atoms and Molecules', Philosophical Magazine, vol. 26, p. 1 (1913)), how it was possible to quantize the energy of a H-atom, and thus give a quantitative "explanation" of the experimental data concerning the line spectra of the H-atom. To quantize the energy possibilities in an H-atom, Bohr assumed ad hoc that the angular momentum of the electron, viz. the product of its mass, velocity and distance from the proton, was equal to a natural number multiplied by Planck's Constant divided by 2*. By means of this ad hoc supposition, Bohr could deduct a quantized energy formula, i.e. the electron can only have quite definite and separate values of its energy.

However, the physical quantity we call the angular momentum is calculated by the fundamental physical quantities, distance, time and mass, which is also the case for all other physical quantities! As a consequence it is reasonable to assume that the quantization is already present at this fundamental level! I shall show that the quantization at a "higher" level is a consequence of the quantization of space, time and mass.Fundamentally we have:
The space quantum – elementary length r₀ is given by:

(1)

where h is Planck's Constant, c₀ is the velocity of light, and M₀ is the total matter/energy mass of the Universe. The time quantum – elementary time t₀ is given by:

(2)

The mass of an "elementary" particle is assumed to be a fraction of the total mass of the Universe. Every finite physical distance is a natural number – the space quantum number multiplied by elementary length.
It means that the following is valid:

(3)

Likewise every physical time interval is equal to a natural number – the time quantum number multiplied by elementary time t₀. Thus the following is valid:

(4)

All physical quantities, such as velocity, acceleration, force, work, energy etc., are defined as usual, but we must now take into consideration the quantization of space, time and mass. This fundamental 'atomization' implies that all movement is discontinuous, all movements take place in 'jumps'. Furthermore all physical processes in a definite system will characteristically consist of discontinuous changes of certain physical quantities.

Processes in a physical system can be described by the change of certain quantum numbers, belonging to definite physical quantities. All physical quantities can be expressed by the cosmically fundamental quantities: the space quantum, the time quantum and the total mass of the Universe.

Velocity and acceleration in the discontinuous movement

(5)

where is the distance travelled in the time interval . In this definition equation is the space quantum number belonging to the distance , and the time quantum number belonging to the time interval . The ratio between these two quantum numbers defines a rational velocity quantum number, denoted by . This velocity quantum number is from the interval between 0 and 1, as no velocity surpasses c₀.
If a particle changes its velocity in a certain time interval, the particle is said to accelerate. The magnitude of this acceleration is defined by the following expression:

(6)

In this definition equation, we shall call the acceleration quantum number which is seen to be a rational number, lying in the interval 0 to 1. The ratio between the velocity of light and elementary time c₀/t₀ gives a theoretical upper limit for the acceleration of a particle. This upper limit is denoted by a_max.
If – in a certain reference system – it is registered that a particle accelerates, we say that it is influenced by a force.

In the following I shall show that the mechanical energy, viz. the sum of the kinetic energy and the potential energy, of an electron, being in a certain state of movement in the hydrogen atom, is quantized, and that this quantum energy can furthermore be expressed by a certain fraction of the total energy of the Universe.

We shall make the calculations on the following idealized model of the hydrogen atom: we shall presume that the electron moves in a closed 'quantum circle' around a proton, which we shall presume is at rest in a chosen reference system. We shall presume that the H-atom does not interact with other partial systems, which of course is impossible in the real Universe. In a first approximation we shall disregard magnetic forces, caused by the movement of the electron in its orbit and own rotation (spin) of the electron and the proton.

Furthermore we shall assume that both the electron and the proton are mathematical points, which of course is not realistic. However, this mathematical idealization has proved fruitful in other contexsts, f.i. in the ideal gas model, in which it is also assumed that the particles have no extension.

To make the model a little more physical, we can presume that both the proton and the electron have an extension equal to elementary length. Coulomb's electrostatic force law, which we shall use, is valid for charges of mathematical points.

(7)

where m_e is the mass of the electron at rest, v its discontinuous velocity in a certain 'quantum orbit', k_c is the Coulomb Constant, e the electric elementary quantum, and r the average 'quantum radius'.

In order that the electron shall be able to move in a closed orbit, it must be influenced by a 'centripetal force', towards the proton. Thus we can write Newton's 2nd Law for the movement of the electron as:

(8)

The right side of equation (8) is the expression for the electric Coulomb force. By means of equation (8) we can transform equation (7) as:

(9)

Inserting the quantized velocity expression from equation (5) we get:

(10)

where the mass m_e of the electron is expressed as a whole fraction of M₀, the total mass of the Universe, thus:

(11)

where we shall call n_m the mass quantum number.
We finally transform to:

(12)

where we can call n_e the electron's cosmic energy quantum number  
From equation (12) we see the following: the quantum energy of an electron in a certain quantum orbit is equal to a whole fraction of the total energy of the Universe, which is logically reasonable. That the energy is negative shows that the electron is in a 'bound' quantum orbit, and besides, the sign is decided by the choice of origo for the electric potential energy.

As the light constant c₀ is equal to the ratio between elementary length and elementary time, we see the following from equation (12):   The quantum energy of the atom is determined by elementary length, elementary time and the total energy of the Universe. To this comes a quantum number, dependent of the used system. In equation (12) n_e is the cosmic energy quantum number of the atom.
Niels Bohr's quantum energy formula contains the mass of the electron, its electric charge, Planck's Constant and the Coulomb Constant, quantities characterized by the atom physical level, and not so fundamental in their nature as the quantities I have used in my deduction.

As the total mass in the Universe is around 1.6 · 10⁶⁰ kg (see my quantum cosmology), which is an extremely high figure, the cosmic energy quantum number is also extremely high, in the order of 10⁹⁵. Practical calculations with such high figures are difficult, and we can therefore for practical reasons again introduce the rest mass of an electron in equation (12).
My purpose of the preceding deductions has been to show the holistic connections in our Universe. I shall formulate it as follows: Everything determines everything!

Quantum Physical deduction of the Rydberg Formula.

In my treatise: Rational Quantum Physics, I have made a quantum physical treatment of the empirically found Rydberg Formula.
In this section I shall give a quantum physical deduction of the formula.
We consider two energy states of the atom, E₁ and E₂, given by:

(13)

The ratio between these two energies is:

(14)

where n_e,1/n_e,2 gives the ratio between the electron's cosmic energy quantum numbers in the two states. This is furthermore given by:

(15)

i.e. the square of the ratio between the two rational velocity quantum numbers, corresponding to the electron's two velocities.
We can now express the ratio between these two velocity quantum numbers by space quantum numbers, which can only take natural numbers, viz. 1, 2, 3, etc.
If an electron is in a higher state of energy, E₂, it will, in a quantum jump 'seek' down to the lower energy E₁. During this process, conservation of angular momentum shall be valid, i.e. the product of the mass of the electron, its velocity and its distance from the center of force shall be constant. Assuming a photon is emitted 'radially', the following must be valid:

(16)

Expressing velocities and distances by quantized quantities, we get:

(17)

where is the velocity quantum number, and n_r is the space quantum number. Using this in equation (15) we get:

(18)

Using this in equation (14) we get a connection between the quantum energies and the space quantum numbers. We can thus write:

(19)

As the lowest energy state of the atom, E₁, corresponds to a space quantum number n_r1 = 1, we can write equation (19) as follows:

(20)

where we get the higher energy states by giving n_r2 the values 2, 3, 4, etc. It should be denoted that these energies are negative, which can be seen from equation (13), and which corresponds to the bound states of the electron.

If an electron goes from f.i. a higher energy state E₂ to the lower state E₁, the difference in energy will be emitted as a photon. As these two energy states correspond to the space quantum numbers n_r1 = 1 and n_r2 = 2, we can write:

(21)

where E_f gives the energy of the emitted photon. Equation (21) thus only expresses the demand for conservation of energy.
The numerical value of E₁ is equal to the ionization energy, i.e. the positive energy necessary to transfer to the atom in order to break away an electron from the energy state E₁. In order to calculate the energy states of the atom it is sufficient to measure the ionization energy.

We can now write equation (21) in a general way, corresponding to the space quantum numbers n_r1 and n_r2. We get

(22)

In equation (22), E_ion is the measured ionization energy for a real atom. The energy of an emitted photon can now be calculated by giving the natural numbers m₁ and m₂ the values: m₁ = 1, 2, etc. and m₂ = m₁ + 1.

Equation (22) is formally identical to the formulas found by Balmer and Rydberg on a purely empiric base. They were not able to give a reason based on the fundamental physical condition of our Universe.

It should be noted that the previous deductions are based on 'point charges', and this result in not realistic geometrical conditions. In a more realistic model, it is necessary to take into account the extension of the particles, which will give another mathematical expression of the Coulomb energy. Likewise the magnetic energy conditions due to the electron's and the proton's motions in orbit and their own motions (spin) must be considered. If these energy amounts are inserted in the calculations, it will be possible to describe the hydrogen atom's so-called fine structure and hyper fine structure.

Finally I shall again underline that the previous calculations show, that the quantum aspects in our Universe are caused by the fundamental quantization of space, time and mass, and that this quantization is determined by the total and definite mass of our Universe.

2nd January 1997

  E-mail: LNi@Herlufsholm.dk Louis Nielsen, Herlufsholm.