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The Harmonic Neutron Hypothesis, HNH, has demonstrated that many of the fundamental physical constants are associated with quantum integers,
*n*, within a classic integer and partial harmonic fraction system, and follow a known two-dimensional, 2D, power law geometry. These are exponents of a fundamental frequency,
*v*
_{F}, the basis of which is the annhilation frequency of the neutron,
*v*
_{n0}. Our goal to a first approximation is to derive the frequency equivalents of the Rydberg constant,
*v*
_{R}, the Bohr radius, v
_{a0}, the electron, v
_{e-}, and the reciprocal fine structure constant, 1/
*α* all from
*v*
_{n0}, π, and a small set of prime integers only. The primes used in the derivations are respectively 2, 3, 5, 7, and 11. This is possible since it is known that the number 3 is associated with
*R*, 5 with
*a*
_{0}, 7 with
*e*
^{-}, and 11 with 1/
*α*. In addition, the interrelationships of the frequency ratio equivalents of these natural units with 2 and π are known, thus allowing for the derivation of any one from the others. Also the integer and partial fractions of
*a*
_{0},
*e*
^{-}, and
*n*
^{0} define Planck time squared,
*t*
_{P}
^{2}. An accurate estimate of t
_{P}
^{2} from v
_{F} alone is also related to the integer 2 since gravity is a kinetic force. Planck time squared,
*t*
_{P}
^{2} scales the Y-axis, and
*v*
_{F} scales the X-axis. In conclusion the quantum properties of hydrogen are derived from only the natural unit physical data of the neutron, to a relative precision ranging from 2.6 × 10
^{-3} to 6.7 × 10
^{-4}. This supports the hypothesis that many of the fundamental constants are related to
*v*
_{n0}.

The primary method organizing and conceptualizing the fundamental physical constants is the Standard Model SM [

The directly observable properties of hydrogen, H, include the proton, p^{+}, electron, e^{−}; the Bohr radius, a_{0}; and the ionization energy, developed with Rydberg’s constant, R. These represent some of the most fundamental constants of mass, distance, and electromagnetic bosonic energy scaling factors in physics. As frequency equivalents these are inter-related with the fine structure constant, α, by the ubiquitous factors 2, and π [^{+}, if any two of hydrogen’s physical constants are known then the other two can be derived. They are not truly independent, but they represent a unique integrated quantum linear domain ratio system that is linked through classic Euclidean geometric scaling factors, 2 and π. Consequently these constants correlate with different physical values, for mass; distance; frequency; and for electromagnetic energy which when normalized to a single Hertzian unit are unified. These same four constants are related to Coulomb’s law; the free space constants of permittivity and permeability; Planck’s constant, h, and the speed of light, c through computational definitions of a_{0}. These entities also define the classic kinetic energy relationship of mass times velocity squared divided by two. The fine structure constant α is the ratio of a specific velocity and the speed of light, c. In Special Relativity this is called a β ratio. If the mass is that of the electron, the velocity is

If the fundamental constants represent a truly integrated unified system, all of them should be derivable from a single natural unit value in the ideal situation, or at least just a few fundamental values. This paper explores the possibility that all of the fundamental constants are inter-related in a similar fashion as these four hydrogen constants. In classic simple harmonic systems it is possible to derive any harmonic frequency provided one frequency and its single harmonic number are known. This is also a property of quantum spectra such as that developed in the Rydberg series. This type of integration of multiple physical constants into a unified system, while commonplace; though it is not typical to imagine that the whole structure of the fundamental constants represents such a system.

The goal is to logically derive to a first approximation the frequency equivalents of the electron,

The following is a limited review and explanation of the HNH. The details have been described in multiple previous publications, and will not be repeated [

The primes used in our model define a global organization of inter-relationships among the fundamental constants in a resonant harmonic system. And we postulate that higher prime numbers, higher composites of those primes, and larger partial fractions are possibly associated with higher order physical entities. The HNH model has accurately explained the global organization of the fundamental constants based purely on integer properties including why black body radiation and the elements form a consecutive integer series, why the fundamental constants cluster around the neutron in a partial fraction pattern, and why the SM constants are grouped in pairs of three entities each [

We define a collection of different integer n values. They are referred to as a consecutive integer series,

cy. Each physical state in the linear domain, for example, distance; time or frequency; and mass is defined by an integer from a consecutive series,

constants are defined by integer partial fraction exponents of

For the fundamental particles and bosons their integer-n values are logically and computationally derived based upon the relative scale between the individual constant and the neutron, similar in concept to the chemical periodic chart. We also find that the physical constants must follow power law properties within a harmonic system with the four natural unit frequency equivalent values that scale the whole harmonic system, namely the neutron,

The harmonic fractions and partial harmonic fractions not only define the possible harmonic frequency ratios, but also define the X, and Y coordinates of a unit circle. When n is associated with the harmonic fraction

nus the square of the harmonic fraction,

and

lows the second computation. Therefore, harmonic and partial fractions are defined by the scaling of a unit circle on a Cartesian plane, and are logically closely related to all sinusoidal periodic relationships.

The model has predictive power beyond standard methods by analyzing the whole system as an integrated harmonic spectrum rather than individual, isolated and independent physical values. The methods of prediction are similar to standard methods seen in atomic spectra. We use dimensional analysis methods similar to that of Rayleigh and Buckingham’s Pi Theorem, where the exponential base is the dimensionless neutron annihilation

frequency,

in kg, s equals one second, c equals the speed of light, and h equals Planck’s constant. The Buckingham Pi Theorem states that physical laws are independent of the form of the physical units. Therefore, acceptable laws of physics are homogeneous in all dimensions.

All of the physical phenomena are evaluated as frequency equivalents, and secondarily as dimensionless coupling constant ratios. The actual physical units can be stripped away then logically re-installed, after the dimensionless calculations. The system is standard physical unit-independent. Arbitrary physical units such as kilogram, kg; second, s; electron volt, eV; Joule, J; meter, m; electron charge, e; and the speed of light, c, are transformed to frequency equivalents, and purged from the calculations. In the linear domain all fundamental constants are defined solely by ratios. In the exponential domain they are uniquely associated with harmonic or

integer fraction exponents of

This fundamental ratio

scaled as

In this type of single-variable physical system, the units are all dimensionless coupling constants and completely defined by exponents or integer values of

Previous HNH predictions and derivations have been made from two (2) finite integer sets. The first set includes four natural units based on known atomic quantities as frequency equivalents. Included in this set is the neutron,

All of the data for the fundamental constants were obtained from the websites: http://physics.nist.gov/cuu/Constants/ and www.wikipedia.org. The NIST site http://physics.nist.gov/cuu/Constants/energy.html has an online physical unit converter that can be used for these types of calculations so the values used in the model are all standard unit conversions.

The floating point accuracy is based upon known experimental atomic data for hydrogen and the neutron, of approximately 5 × 10^{−8}. Subscript “k” denotes a known experimental value, and subscript “d” represents a derived value. All of the known fundamental constants are converted to frequency equivalents,^{2} (speed of light squared) then dividing by h (Planck’s constant). Distances are converted by dividing the wavelength into c. Energies in Joules are converted to Hz by dividing by h. The eV value for the neutron is 939.565378 (21) × 10^{6}. Its frequency in Hz is converted to eV by multiplying by the constant, 4.13566750 (21) × 10^{−15} eV/Hz. The eV was converted to frequency by multiplying by the constant 2.41798930 (13) × 10^{14} Hz/eV. N_{A} equals Avogadro’s number, and amu equal atomic mass unit.

Our model has two parallel domains both describing identical physical values. One domain is the frequency equivalent of any physical value. This is the linear domain of possible physical states. The other domain is a set

of exponents applied to the base

specific value, Equation (2). This is the domain of the fundamental constants. The known exponent, exp_{k}, of a

fundamental constant is the ratio of the log_{e} of its frequency equivalent,

The known or derived exponent minus the quantum fraction,

constant. The known or derived frequency equivalent of a constant, v, is calculated by raising

exponent. In a classic simple harmonic system there are no δ values since all of the possible harmonic frequencies are defined by the

This is associated with

Equation (4) demonstrates that

wk δ-line, bwk, awk; and both the slope and Y-intercept of the EM δ-line [

this paper only

Each individual fundamental constant is plotted on a 2D power law universal harmonic plane,

The X-axis is scaled by

Physical constant | Value |
---|---|

2.2718590 (01) × 10^{23} | |

53.780055612 (22) | |

bwk_{k}, y-intercept, weak force, wk line ^{−3} | 3.51638329 (18) × 10^{−3 } 3.5242141 (2) × 10^{−3} |

awk_{k}, slope, weak force, wk line awk_{d} relative error, 7.2873 × 10^{−2} | 3.00036428 (15) × 10^{−3 } 3.21900854 (16) × 10^{−3} |

bem_{k}, Y-intercept, electromagnetic, EM line ^{−2 } | −3.45168347 (17) × 10^{−3 } −3.5242141 (2) × 10^{−3} |

aem, slope, electromagnetic, EM line _{ } relative error, 2.1013 × 10^{−2} | −3.45168347 (17) × 10^{−3 } −3.5242141 (2) × 10^{−3} |

minus 1. This centers the neutron at the origin (0, 0). All of these ratio values and all points on the X-axis are rational numbers. These define discrete integer-based locations. This axis location is defined by the individual partial harmonic fractions or the sum of multiple partial harmonic fractions defining a composite constant such as Planck time squared,

The Y-axis, δ, is related to the difference between the known subscript k, or derived subscript d, the exponents and the harmonic or quantum fraction, Equation (3). The Y-axis is scaled by the

to the

stant. The degenerate exponent value is the partial fraction or

There are two fundamental lines defined by four points that scale the global δ Y-axis 2D power law plane.

These are referred to as δ-lines,

rived exponent equals the sum of the partial fraction or

The first line of the 2D plane is defined by the points of the Bohr radius,

The second δ-line is defined by the points of Planck’s constant,

Planck time,

In prior publications, the HNH has accurately derived the square of Planck time,

_{P} value of_{P} value is

applied to

We define the degeneracy of any physical constant to be related to the frequency of the neutron raised to a simple integer ratio exponent; all values of which fall on the X-axis and therefore have no δ value. An estimated degenerate value of

gravity is a kinetic-inducing force. Since 2 is a known constant the slope of this point to the h-point scales the

entire Y-axis, independent of force. This is referred to as the

The

Constant unit | n_{ie} or n_{ife} | 1 ± 1/n_{ife} or qf |
---|---|---|

2 × elemental graviton, binding kinetic energy of the electron in hydrogen, boson | −1 n_{ie} | −1 |

electromagnetic energy, h, boson | 0 n_{ie} | 0 |

neutron, elemental mass, strong force, fermion 939.565378 (21) × 10^{6} MeV/c^{2}, 2.2718590 (01) × 10^{23} Hz | 1 n_{ie} | 1 |

Rydberg constant, R, EM energy, boson, known 1.09737315 (5) × 10^{7 }m^{−1}, 3.28984196 (17) × 10^{15} Hz | −3 n_{ife} | 2/3, 1 − (1/3) |

Rydberg constant, derived, relative error, 2.5971 × 10^{−3} 1.09452317 (5) × 10^{7} × m^{−1}, 3.28129794 (23) × 10^{15} Hz | ||

Bohr radius, a_{0}, distance, known 5.2917721092 (17) × 10^{−11} m, 5.66525639 (28) × 10^{18} Hz | −5 n_{ife} | 4/5, 1 − (1/5), |

Bohr radius, a_{0}, derived, relative error, 1.9287 × 10^{−3} 5.3019982(3) × 10^{−11} m, 5.65432962 (31) × 10^{18} Hz | ||

electron, e^{−}, mass, matter, fermion, known 0.510998925(20) × 10^{6} eV/c^{2}, 1.235589964(62) × 10^{20} Hz | −7 n_{ife} | 6/7, 1 − (1/7), |

electron, e^{−}, derived, relative error, 1.2579 ×10^{−3} 0.51035615 ×10^{6} eV/c^{2}, 1.23403574 (07) × 10^{20} Hz | ||

reciprocal fine structure constant, coupling constant, α^{−1}, known | −11 n_{ife} | 1/11, 1 − (10/11) |

reciprocal fine structure constant, α^{−1}, derived, relative error, 6.7215 × 10^{−4} | ||

Planck time squared, ^{−86} s^{2} | qf | −128/35 |

Planck time squared, ^{−2} 1.91137824(09) ×10^{−86} s^{2} |

ħ degenerate value

The

The slope and Y-intercept of the line from the

The^{−3} for bwk, and 2.1013 × 10^{−2}, for bem.

Ref. [_{R} to the wk line. The X-axis value of this wk entity product must equals the sum of the partial fractions_{d}.

It is assumed that the _{d} equals_{k} is

Multiplying _{d} equals _{k} equals

The derived exponent for

ponent is

The derived exponent for

There are multiple possible ratios of

ship arbitrarily chosen for the derived

A robust model that accurately scales from classic, to quantum, to cosmic physical constants does not exist [

There have been a few publications demonstrating indirect dominance of prime numbers manifest in physical systems. One of our previous papers demonstrated that the global organization of the fundamental constants including the particles and bosons can be represented within a purely mathematical system [

Previous publications have demonstrated that starting with the four natural units of frequency equivalents of the n_{0}, e^{−}, a_{0}, and R it is possible to derive many other fundamental constants to within or beyond their known experimental ranges including the quarks, the proton,

The geometry of the 2D harmonic plane is scaled in the X-axis dimension solely by

scaled by

Also

and partial fractions, and the factor 2, Equation (5). Thus,

In the physical domain

The quote of Gilles Cohen-Tannoudji paraphrases the significance of Planck time within the global context of physical systems [

“The new interpretation of the gravitational constant, when it is associated with h and c, opens up amazing prospects: thus the Planck time and length suggest a quantum structure of space-time itself. Imagine the fascinating implication of a limit to the divisibility of space and especially the divisibility of time!”

These derivations are based on the concept that

It is possible to a first approximation derive the frequency equivalents of the electron, Bohr radius, Rydberg constant, and the reciprocal of the fine structure constant starting with the single natural physical unit of the frequency equivalent of the neutron and a few integers. These derivations support the hypothesis that the fundamental constants represent a classic unified harmonic power law integer based system.

_{0} define the wk-line with slope of awk and Y-intercept, bwk (dashed green solid line). The EM line is defined by h and R (solid green line). The points defined by (0, bwk) and (0, bem) are plotted. The known values are plotted as points the derived values are plotted as open circles demonstrating their relative error. The dot dash red arrow represents the_{d} and_{d} slope. From those derived slopes and Y-intercepts the

Y-intercept of the EM δ-line, bem, aem used for the derivations of e^{−}, a_{0}, R, and 1/α. The derived values closely approximate the known values.

I would like to thank Richard White MD, Vola Andrianarijaona Ph.D. for their support and help.

Donald WilliamChakeres,RichardVento, (2015) The Association of the Neutron, and the Quantum Properties of Hydrogen, with the Prime Numbers 2, 3, 5, 7, 11. Journal of Modern Physics,06,2145-2157. doi: 10.4236/jmp.2015.614218