### Completing The Early Bohr Theory;

#### New Insights Into Bohr’s Early Quantum Theory And Classical Electromagnetic

Abstract: Bohr Theory is completed to include the magnetic currents of the electron and proton, giving both their orbital and cyclotron velocities in the hydrogen atom. Using field forms similar to the Hertz potentials and the Maxwell-Lorentz equations, a system of equations is derived. The solution method looks like Bohr’s method, though there are differences; it is extended to encompass the photon in its role as binding energy. Behind the theory is a self-perpetuating cycle as the photon transverses from electron to proton and back again. The elementary quantum of action is seen as a series of coherent lossless collisions adding energy to the electron and proton during their orbits. In the principal mode  the phase cycle has four parts each of π/2 radians obeying Maxwell's equations; they appear as a resonance phenomenon where both electron and proton receive and transmit the photon. Analytic expressions derived for the particle radii and frequencies match Bohr Theory and compare well with other data.

### Introduction

In 1913 Bohr presented his early quantum theory in which he used the known value of what is today termed Planck’s constant to determine the electron’s orbital radius, termed the Bohr radius in his honour, and its frequency within the hydrogen atom. In doing so he used the experimentally confirmed ‘theory of energy radiation’ that had in the previous two decades begun to revolutionize the classical understanding of physics. Bohr’s work was the first scientific venture into the mathematical physics of what was to become quantum theory. Bohr spoke of the various experimental results that could not be solved via classical means including classical electromagnetics in introducing the new formulation:

“The way of considering a problem of this kind has (…) undergone essential alterations in recent years owing to the development of the theory of the energy radiation, and the direct affirmation of the new assumptions introduced in this theory, found by experiments on very different phenomena such as specific heats, photoelectric effect, Röntgen &c. The result of the discussion of these questions seems to be a general acknowledgment of the inadequacy of the classical electrodynamics in describing the behaviour of systems of atomic size.(…) Whatever the alteration in the laws of motion of the electrons may be, it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant, or as it often is called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed as this constant is of such dimensions and magnitude that it, together with the mass and charge of the particles, can determine a length of the order of magnitude required.”

The essential theory that Bohr presented can be written as follows:

The centripetal force between the electron and the nucleus is balanced by their coulombic attraction:

\frac{me v2}{r}=\frac{q2}{4πε0r}

where m_e = mass of electron

qe = charge of electron in Coulombs
ε 0= electric permittivity of free-space

ve = electron velocity in its orbital rotation in hydrogen atom

re = electron radius in its orbital rotation in hydrogen atom

Rearranging we get

(2)

Since the electron’s energy comes from its orbit, the energy is preserved. Furthermore according to Bohr, the angular momentum of the electron is a whole number n of the elementary quantum of action

where  = Planck’s constant. Therefore we may write

(3)

Substituting (3) into (1) we derive the radius in terms of Planck’s constant:

(4)
At this stage we are able to find the radius of the electron’s orbital rotation in hydrogen’s atom, the Bohr radius

Turning to obtain the frequency of the electron’s orbital rotation, the total energy of the electron is the sum of its kinetic and potential energies:

(5)

Substituting (2) into (5):

(6)

Substituting (3) into (6):

Remembering that electrons in outer orbits move faster than inner orbits, this implies that when an electron falls to a lower energy level, it emits a photon of energy hν; the difference in energy between the inner level Ei and the energy in the outer level, Eo

(7)

In essence this is the early Bohr Theory; it was the first time a quantum method had been used to solve an atomic problem. This solved a number of conundrums including how the electron managed to be stable in its orbit around the nucleus without spiraling into the proton. Bohr assumed the energy was somehow due to its motion within the atom. The critical assumption was what Bohr termed Planck’s ‘theory of energy radiation’ involving Planck’s constant . Bohr theorized that in Planck's theory of energy radiation the energy in an atomic system does not take place in the continuous way assumed in classical electrodynamics, rather it occurs in quantum emissions radiated out from an atomic vibrator of frequency ν in an emission equal to n, where n is an integer, and h is recognized as a universal constant.

According to Bohr’s understanding of classical physics, in particular electromagnetism, the rotation of the electron around the charged core must lead to the gradual loss of energy in the form of radiation till the collapse of the atomic system. When charged particles such as the electron and the proton move along a curved orbit they emit energy since their velocity changes over time. This was the prevailing understanding of science at that precise point in scientific history when quantum theory was first proposed. In his 1913 paper the radius of the electron’s orbit and its frequency around the nucleus were found but not its magnetic motion which is another orbit usually called a cyclotron rotation.

What has remained unquestioned after a century of quantum mechanical computations is its accurate results for the radius and frequency of the electron in the hydrogen atom and its applicability to other hydrogenic systems. Indeed Bohr Theory is still used to introduce quantum mechanics before the more complex valence shell atoms. Included in this early theory was the use of Coulomb’s electrostatic equation even though Maxwell’s time varying electromagnetic (EM) formulation had been known for about 50 years. Even in 1888 Hertz had used an EM potential formulation to determine the electric and magnetic sinusoidal far fields of a half-wave dipole antenna. The use of Coulomb’s electrostatic field form is surprising to say the least and to the author’s knowledge has remained uncommented upon for a century concerning this time invariant equation that is juxtaposed inside a time-varying situation, the motions of atoms. An explanation of this puzzling fact is found by going through the Maxwell-Lorentz equations and finding an understanding of the elementary quantum of action as an effect of a series of elastic photon-electron and photon-proton collisions. Crucially these collisions are periodic and occur as a precise integer number during each rotation; they are also naturally synchronous across each atom.

In the following report Bohr’s early quantum theory is extended to determine the proton’s orbital motion, and the cyclotron motion of both the electron and proton in the hydrogen atom. Following presentation of the extended calculations, a discussion follows on the assumptions behind the theory. The elementary quantum of action is seen as a series of coherent lossless collisions that add energy to the electron and proton during their orbits. To permit a constant addition of energy per revolution there are an integer number of these collisions per revolution. Using field forms similar to the Hertz potentials and the Maxwell-Lorentz equations, a system of equations is derived. The solution method follows Bohr’s original method extending it to encompass the photon in its role as binding energy. Behind the theory is a self-perpetuating cycle as the photon transverses from electron to proton and back again. In the principal mode of the resulting eigenvalue problem the phase cycle has four parts each of π/2 radians obeying Maxwell's equations; they appear as a resonance phenomenon where both electron and proton receive and transmit the photon. Finally the extended results are compared with the Bohr Theory and other spectroscopic data.

### Extending Bohr’s Solution Technique:

The Bohr Theory can be extended in the following way:

1. As seen the Bohr Theory finds the orbital velocity of the electron via its kinetic energy and its potential energy in the electric field acting on it. To find the electric field assume for the moment a stationary or ‘infinitely massive’ proton at the centre of the orbital motion of the electron Now the centripetal or repulsive force on electron is balanced by the Coulombic attraction between the electron and proton. Next applying the Planck theory of energy radiation the angular momentum mvr is an integer multiple of . As we have seen this gives a solution for the Bohr radius and the frequency of the electron in terms of Planck’s constant. In summary h is assumed known a priori and the electron’s orbital radius re,o and frequency νe,o are determined. (rB ) and ωe,o)

2.  In the same way the orbital velocity of the proton can be found via the electric field acting on the proton. The proton’s frequency must be synchronous with the electron, ωp,o = ωe,o. Assume the electron is stationary or ‘infinitely massive’ at the centre of motion of the proton. This gives the proton’s orbital radius and its frequency using the Bohr radius to estimate the electric field acting on the proton. In summary the electron’s radius and frequency re,o ωe,o are known a priori and therefore the proton’s orbital radius and frequency can also be determined rp,o and νp,o). Note: Since mp is ~1836 times greater than me the motion of proton rp,o due to the electric field of electron is << rB  (rp,o << re,o)

3. Find the cyclotron radius of the electron due to the magnetic field of proton acting on the electron. Again requires the proton to be ‘infinitely massive’ at the centre of the cyclotron motion of the electron. Need to determine magnetic field due to proton. Note: There is a relationship between the electric and magnetic field for electromagnetic waves E0 = cB0 where Eand B0 are the scalar amplitudes of the coupled electromagnetic waves.  The frequencies of all particle motions must be equal because the atom as a whole acts synchronously. The electrons for example in large orbits (outer shells) complete their rotations in the same time as electrons in small orbits (inner shells).

4. Finally find the cyclotron velocity of the proton due to the magnetic field of electron acting on the proton; this is the same as 3 above.

Summary: Start with rB (= re,o)>> rp,o , then rB >> re,c, finally rB>> re,c>> rp,c and spectroscopy shows the small quantum steps with Zeeman effect compared with Stark effect. So far this is a generalized solution. We will use the Maxwell-Lorentz equations to solve the problem deterministically without probabilistic solutions as in conventional quantum theory.

### Discussion

According to classical electrodynamics applications are assumed to be ‘large’ in atomic terms; this means that there are large amounts of energy, mass, and charge involved in any problem. When individual atoms are involved the problem becomes ‘quantum’, i.e the number of particles involved in the atom becomes discrete. In the new description outlined above this quantum description includes the photon(s) that acts as the binding energy by transiting between the electron and the proton. In classical problems  the Liénard–Wiechert (LW) potentials can be used to calculate the electric and magnetic fields directly between charged particles.  At the atomic level the LW potentials are incorrect because the problem is quantum and not large. Now the electric and magnetic fields can be described as quantum where the fields between particles are composed of a stream of photons. Thus the fields do not spread out over all solid angles away from the charged particles; for instance the fields are not Coulombic, although they still obey Maxwell’s equations.

Further, in both Bohr theory and the extended theory the distances in calculating the electric and magnetic fields assume time-averaged periodic motions rather than taking the distances directly between moving charged particles. This implies that there are two distinct orthogonal distances involved in measuring the two fields (forming a 'bispinor') rather than the single distance measured directly between charges as in the LW potentials. This suggests a change of metric is needed in classical electromagnetics and in quantum theory. At the same time notice the atomic motions being calculated are an equilibrium state, an eigenvalue for a wave-particle model, a standing wave, or more precisely two coupled standing waves inside the hydrogen atom. Thus far the method (with its correct solutions for  rB  and ωe,o) implies that both classical EM and current quantum theory are lacking a correct metric in terms of the number of variables in the problem and also that the fields need to be time-averaged over each period (for some as yet unconsidered reason; it is found in the following that the forces due to the fields are a series of small collisions between the photon and the atomic particles).

In the next section the Maxwell-Lorentz equations are examined using the suggested change in metric. Also the photon in its role as binding energy is examined. It is found there is a set of equations that indicate the effect of both the electric and magnetic fields is due to a series of elastic collisions between the photon and electron, and photon and proton. This collision based field implies the forces (not their amplitudes) due to the fields are 'near-field' effects and not 'far-field' effects. In this way the time averaged form for the fields is understood as physical, remembering the situation is an equilibrium state of the atom, so regular periodic motions are a valid assumption. At the same time the overall physics is that of a coupled electric and magnetic field, so the use of electrostatic forms for the fields needs further examination.