Chapter VII:

The Schrodinger Wave Equation
and Quantum Mechanics:
The Particle and Wave Characteristics of Matter

The Schrodinger Wave Equation, developed by Austrian theoretical physicist Erwin Schrodinger in 1925, defines the quantum mechanical characteristics of the electrons orbiting the nucleus and can be used to define the positions of the protons and neutrons within the “gravitational potential well of attraction” of the nucleus.  The Schrodinger Wave Equation assumes the particle and wave duality characteristic of the electrons, protons, and neutrons.  The “DeBroglie wave function” is the Schrodinger Wave Equation assumed wave function for the electrons, protons, and neutrons.  The Schrodinger Wave Equation defines the total energy of the system analyzed.  In words, the Schrodinger Wave Equation states that the Kinetic Energy (energy of motion of the particle) plus the Potential Energy (stored energy within the particle) equals the Total Energy of the particle.  The Schrodinger Wave Equation, therefore, provides a Quantum Mechanical approach to evaluating the total energy of the proton or neutron in the nucleus or the electron orbiting the nucleus.  The Schrodinger Wave Equation is a spatial dependent and time dependent differential equation.  Classical Mechanics excluding Quantum Mechanics states:

Total Energy  =  Kinetic Energy  +  Potential Energy

TE  =  KE  +  PE

This equation assumes the velocity of electrons, protons, or neutrons are “non-relativistic” which means their velocities are much, much, less than the speed of light.  Hence, it is acceptable to use mv²/2 for Kinetic Energy in this equation.  Albert Einstein’s Special Theory of Relativity need only be considered if velocities are greater than 10% of the speed of light.  The Classical Physics form of the Schrodinger Wave Equation is acceptable when applied to the electrons orbiting the nucleus because the velocities of the electrons are “non-relativistic.”  The Classical Physics form of the Schrodinger Wave Equation may or may not be acceptable for evaluating the protons or neutrons bound to the nucleus of the atom.  The velocities of the protons and neutrons are non-relativistic which is acceptable.  However, if Gravity is the “Strong Nuclear Force,” holding the protons and neutrons together in the nucleus, General Relativistic effects may have to be considered.  To confirm whether or not General Relativity applies to the Nuclear Gravitation Field, a Classical Quantum Mechanical analysis must be performed, first, using the Classical Schrodinger Wave Equation.  When an object or particle velocity is much, much less than the speed of light, the following relationship between the Classical Kinetic Energy of the object or particle and the Momentum of the object or particle can be developed.  Momentum is equal to mass times velocity or “mv” and is usually designated by “P.”  The variable “m” represents the mass of the object or particle and the variable “v” represents the velocity of the object or particle.  If momentum is squared, such that momentum squared = P² = m²v², then Classical Kinetic Energy can be represented as follows:

For any atomic or subatomic particles having rest mass, if the particles have velocities much, much less than that of the speed of light, the Schrodinger Wave Equation can be used in tandem with Classical Physics to determine various properties of the particles such as position, velocity, momentum, etc., as a function of time.  The Schrodinger Wave Equation assumes that particles such as the electrons, protons, or neutrons, are a wave function in addition to being particles with rest mass.  The three dimensional time dependent Schrodinger Wave Equation, displayed in spherical coordinates, is provided below:

Total Energy  =  Kinetic Energy  +  Potential Energy

Reference:  “Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles,” Robert Eisberg and Robert Resnick, Page 256, Equation 7-11

The Total Energy of the particle is represented by the first partial differential of the wave as a function of position and distance from the origin of interest and with respect to time.  The Kinetic Energy of the particle is the second partial differential of the spatial dependence of the wave as a function of position and distance from the origin and time.  The Potential Energy of the particle is a wave function of the particle position and distance from the origin of interest and time.  The terms and variables of the Schrodinger Wave Equation include “i” which represents the square root of negative 1, [-1]^1/2; “h-bar” equals the constant h/2π where “h” represents Planck’s constant equal to 6.626×10^-34 Joule-sec; m represents the mass of the particle of interest; V(r,θ,φ) represents the Potential Energy function, a function of both position and distance from the origin of interest; ψ(r,θ,φ,t) represents the assumed wave function of the particle, a wave function of both the position and distance from the origin of interest and of time; “r” represents the distance of the particle from the origin of interest (from 0 to infinity); the angle “θ” represents the azimuthal angle (from 0 to 2π Radians) in the horizontal plane in space of the particle relative to the origin of interest; and the angle “φ” represents the vertical or altitude angle (from -π/2 to +π/2 Radians) of the particle from the origin of interest.

NOTE:	The Kinetic Energy term, with its [h-bar]2/2m, in the Schrodinger Wave Equation has a format that looks very similar to the term P2/2m, the Classical Kinetic Energy as follows:

The Schrodinger Wave Equation is used to determine the discrete, or quantized, energy levels available for the electrons to “orbit” the nucleus.  These “orbits” or, more accurately, “electron clouds of potential physical position” require the electrons to have discrete energies because of the wavelike characteristics of electrons.  The discrete energy levels force the electrons to have one or more complete wavelengths in their “orbits” about the nucleus -- the “DeBroglie wavelength.”  The electrons cannot exist as bound electrons to the atom unless they each contain one of the discrete energies defined by the Schrodinger Wave Equation solutions.

NOTE:

French physicist Louis DeBroglie postulated his theory of the wave nature of matter in 1925, which, became one of the foundations of Quantum Mechanics.  DeBroglie expanded upon German theoretical physicist Max Planck's theory of the equivalence of the energy of a quantum of light to the frequency or wavelength of that light (see Chapter X) and Albert Einstein's theory of the equivalence of Energy and matter (see Chapter V) in order to develop his equivalence of the velocity of matter to its wavelength as follows: E  =  hν  =  hc/λ E  =  mc2 mc2  =  hc/λ λ  =  h/mc If light with an equivalent mass, m, has a wavelength inversely proportional to the velocity of light, c, then the wavelength of a particle, such as an electron, proton, or neutron, should also have a wavelength inversely proportional to its velocity as follows. λ  =  h/mv Where “E” represents energy, “h” represents Planck's constant, “ν” represents the frequency of light, “λ” represents either the wavelength of light or the wavelength of a particle of matter, “m” represents either the equivalent mass of a quantum of light or the mass of a particle of matter, “c” represents the velocity of light, and “v” represents the velocity of the particle of matter.

The discrete energy levels for the electrons are based upon the wave characteristic of the electrons.  The electrons must have the exact applicable multiple of a wavelength to fill a specific energy level about the nucleus.  The Potential Energy function in the Schrodinger Wave Equation, which ultimately defines those energy levels, includes the potential associated with the Electrostatic Force of Attraction of the negatively charged electrons to the nucleus containing the positively charged protons.  These discrete electron energy levels establish the various chemical characteristics of each of the elements and define the pattern of the “Periodic Table of the Elements” as displayed in Table 7-1.  Elements with the same electron configurations in their outermost energy levels fall into “element families” defined by the vertical columns of the Periodic Table of the Elements and determine the chemical properties of those elements in their respective families.  The Inert, or Noble Gas, family of elements in the far right column of the “Periodic Table of the Elements” (Table 7-1) do not readily react with other elements and are found in a “monatomic” state because their outermost electron energy levels (orbitals) are filled to capacity.  These elements “have no desire” to either give up an electron or take on another electron because their outer electron energy level is filled to capacity.  The electron configuration for the family of Noble Gases, which have a completely filled outer electron energy level, or shell, are as follows:

Helium	Element #2	1s2
Neon	Element #10	1s2 2s2 2p6
Argon	Element #18	1s2 2s2 2p6 3s2 3p6
Krypton	Element #36	1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6
Xenon	Element #54	1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6
Radon	Element #86	1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6

The “element numbers” for these Noble Gases represent “magic numbers” for the electrons.  All of the Noble Gases, except Helium, have both a full outer s-Orbital and a full outer p-Orbital.  Helium is the exception because the first energy level only includes an s-Orbital.

The Schrodinger Wave Equation determines the Total Energy of a particle.  In this case, the Schrodinger Wave Equation is being used to determine the total energy of the electron orbiting the nucleus, by adding the Potential Energy and the Kinetic Energy of a particle.  The Schrodinger Wave Equation, as applied to the electrons orbiting the nucleus, assumes the electron has a wave characteristic.

Total Energy  =  Kinetic Energy  +  Potential Energy

Where the potential function V(r,θ,φ) for the electron interaction with the nucleus is the following function:

Where “Z” represents the number of protons in the nucleus which defines the Element Number (see Table 7-1, “Periodic Table of the Elements”); “e” represents the magnitude of the electrostatic charge of one proton (or electron) equal to 1.6022×10^-19 Coulomb; “1/4πε₀” represents the constant of proportionality to relate the electrostatic charge of the nucleus to the Nuclear Electric Field Potential propagating omni-directionally outward from the nucleus; and “r” represents the distance between the center of charge of the nucleus and the electron of interest.  The Nuclear Electric Field Potential is equal to Force divided by the representative unit charge of an electron.

Table 7-1:  Periodic Table of the Elements

Group	1	2		3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Period	s-Orbitals			d-Orbitals										p-Orbitals
1	1 H	2 He																1 H	2 He
2	3 Li	4 Be												5 B	6 C	7 N	8 O	9 F	10 Ne
3	11 Na	12 Mg												13 Al	14 Si	15 P	16 S	17 Cl	18 Ar
4	19 K	20 Ca		21 Sc	22 Ti	23 V	24 Cr	25 Mn	26 Fe	27 Co	28 Ni	29 Cu	30 Zn	31 Ga	32 Ge	33 As	34 Se	35 Br	36 Kr
5	37 Ru	38 Sr		39 Y	40 Zr	41 Nb	42 Mo	43 Tc	44 Ru	45 Rh	46 Pd	47 Ag	48 Cd	49 In	50 Sn	51 Sb	52 Te	53 I	54 Xe
6	55 Cs	56 Ba	*	71 Lu	72 Hf	73 Ta	74 W	75 Re	76 Os	77 Ir	78 Pt	79 Au	80 Hg	81 Tl	82 Pb	83 Bi	84 Po	85 At	86 Rn
7	87 Fr	88 Ra	**	103 Lr	104 Rf	105 Db	106 Sg	107 Bh	108 Hs	109 Mt	110 UUN	111 UUU	112 UUB	113 UUT	114 UUQ	115 UUP	116 UUH	117 UUS	118 UUO
			f-Orbitals														NOTE: The existence of either Element 117 or Element 118 has not yet been officially observed.
*  Lanthanoid Series			57 La	58 Ce	59 Pr	60 Nd	61 Pm	62 Sm	63 Eu	64 Gd	65 Tb	66 Dy	67 Ho	68 Er	69 Tm	70 Yb
**  Actinoid     Series			89 Ac	90 Th	91 Pa	92 U	93 Np	94 Pu	95 Am	96 Cm	97 Bk	98 Cf	99 Es	100 Fm	101 Md	102 No

References:  http://www.webelements.com/index.html
http://chemlab.pc.maricopa.edu/periodic/113.html

Table 7-2:  Electron Configuration and Energy Levels for the Periodic Table of the Elements

Group	1	2		3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Period	s-Orbitals			d-Orbitals										p-Orbitals
1	1s1	1s2																1s1	1s2
2	2s1	2s2												2p1	2p2	2p3	2p4	2p5	2p6
3	3s1	3s2												3p1	3p2	3p3	3p4	3p5	3p6
4	4s1	4s2		3d1	3d2	3d3	3d4	3d5	3d6	3d7	3d8	3d9	3d10	4p1	4p2	4p3	4p4	4p5	4p6
5	5s1	5s2		4d1	4d2	4d3	4d4	4d5	4d6	4d7	4d8	4d9	4d10	5p1	5p2	5p3	5p4	5p5	5p6
6	6s1	6s2	*	5d1	5d2	5d3	5d4	5d5	5d6	5d7	5d8	5d9	5d10	6p1	6p2	6p3	6p4	6p5	6p6
7	7s1	7s2	**	6d1	6d2	6d3	6d4	6d5	6d6	6d7	6d8	6d9	6d10	7p1	7p2	7p3	7p4	7p5	7p6
			f-Orbitals
*  Lanthanoids			4f1	4f2	4f3	4f4	4f5	4f6	4f7	4f8	4f9	4f10	4f11	4f12	4f13	4f14
**   Actinoids			5f1	5f2	5f3	5f4	5f5	5f6	5f7	5f8	5f9	5f10	5f11	5f12	5f13	5f14

References:  http://www.webelements.com/index.html
http://chemlab.pc.maricopa.edu/periodic/113.html

NOTE:

Electrons fill the electron energy levels starting from left to right along each row and by rows from top to bottom.  Hydrogen (H), with a 1s1 electron configuration, and Helium (He), with a 1s2 electron configuration, are placed at the top of the Periodic chart on both sides for Table 7-1 and Table 7-2 because the first electron energy level consists only of an “s-Orbital”.  Hydrogen can take on either the characteristic of an Alkali Metal or a Halogen.  Helium is a Noble Gas.

Figure 7-1:  The Order of Electron Fill of the Electron Energy Levels Around the Nucleus

Reference: http://www.chemtutor.com/struct.htm#con

Figure 7-1, “The Order of Electron Fill of the Electron Energy Levels Around the Nucleus,” provides another visual representation of the order the electrons fill the orbitals as is provided in Table 7-2, “Electron Configuration and Energy Levels for Periodic Table of the Elements.”

Note in Table 7-1, “Periodic Table of the Elements,” that several elements, such as Helium, Oxygen, and Calcium, are highlighted with a “red border” around them.  In like manner as the “element numbers” for the Noble Gases represent “electron magic numbers” indicating the outermost ground state energy level for the given element is filled to electron capacity, the nucleus also contains quantized energy levels with “proton magic numbers” and “neutron magic numbers” indicating outermost ground state energy levels for the protons and/or neutrons are filled to capacity.  The elements highlighted with the “red border” represent the elements with their outermost proton energy levels filled to capacity.  In other words, these highlighted element numbers, which define the elements by number of protons, also represent the proton magic numbers.  The number of protons determine the element.  The nucleus of the atom contains discrete, or quantized, energy levels for the protons and the neutrons in the same manner the atom has discrete, or quantized, energy levels for the electrons to orbit the nucleus.  If Gravity and the “Strong Nuclear Force” are the same, then the potential function for the Schrodinger Wave Equation for the nucleus should take on a similar form to that for evaluating the electron energy level configurations.  The Schrodinger Wave Equation can be used to evaluate the “Strong Nuclear Force” assuming it is the same as Gravity, for a given nucleon, either a proton or a neutron, relative to the rest of the nucleons in the nucleus is as follows:

“G” represents the Universal Gravitation Constant, “Z” represents the number of protons in the nucleus, “M_p” represents the mass of a proton, “N” represents the number of neutrons in the nucleus, “M_n” represents the mass of a neutron, and “ψ(r,θ,φ,t)” is the assumed wave function for the proton or neutron of interest.  The Nuclear Gravitation Field Potential function, V(r,θ,φ), for the nucleon interaction with the nucleus is given below:

It might seem improper to initially assume the “1/r²” Nuclear Gravitation Field Potential function in the Schrodinger Wave Equation in order to demonstrate Gravity is the same force as the “Strong Nuclear Force” because the analysis assumes the “Strong Nuclear Force” and Gravity are the same force.  However, one must assume the “1/r²” function to properly evaluate the results of the Schrodinger Wave Equation to determine if it is reasonable for Gravity to be the same as the “Strong Nuclear Force.”  Using any other potential function in the Schrodinger Wave Equation would serve no purpose.

Figure 7-2, “Table of the Nuclides,” provides the “proton magic numbers” and the “neutron magic numbers” in the nucleus.  The variable for the Nuclear Gravitation Field Potential function is the same as the variable for the Nuclear Electric Field Potential function, or 1/r².  Without any further analysis, one would expect the form of the solution of the Schrodinger Wave Equation to be the same, therefore, the “magic numbers” should be the same for the protons or the neutrons in the nucleus as they are for the electrons.  However, Figure 7-2, “Table of the Nuclides,” indicates the “magic numbers” for protons and neutrons are different than for the electrons as noted by Table 7-1, “Periodic Table of the Elements.”  If the “magic numbers” were the same, one would expect the Noble Gases to contain the “proton magic numbers” for the nucleus, also.  Helium (He) is the only element that contains both “magic numbers” for the protons and for the neutrons in its nucleus along with the “magic number” for its electrons.  The “proton magic numbers” are 2 for Helium (He), 8 for Oxygen (O), 20 for Calcium (Ca), 28 for Nickel (Ni), 50 for Tin (Sn), 82 for Lead (Pb), and 114 for Element 114 (UnUnQuadium or UUQ).  The “neutron magic numbers” are 2, 8, 20, 28, 50, 82, and 126.  As Table 7-1, “Periodic Table of the Elements,” indicates, the magic numbers for the electrons are 2, 10, 18, 36, 54, 86, and 118.  These are the atomic numbers of all the Noble Gases from Helium to Radon.  Element 118 is expected to be a Noble Gas based upon its outer electron orbital configuration.  On the surface, the different values of the “magic numbers” for the electrons in the atom and the “magic numbers” for the protons and neutrons in the nucleus seems to indicate that the “Strong Nuclear Force” and Gravity cannot be the same.  However, there are other effects that must be considered if the “Strong Nuclear Force” and Gravity are the same force.  The effects of the Nuclear Gravitation Field that can alter the “magic numbers” will be evaluated in Chapter X.

Z = Number of Protons = Vertical Axis     N = Number of Neutrons = Horizontal Axis

Figure 7-2:  Table of the Nuclides

Reference: http://atom.kaeri.re.kr

Index and Direct Links to Other Chapters of Nuclear Gravitation Field Theory
and Nuclear Gravitation Field Theory Home Page/Table of Contents:

Nuclear Gravitation Field Theory

1. Purpose for Evaluation of the Strong Nuclear Force and the Force of Gravity



2. Executive Summary



3. The Classical Physics Evaluation of Electrostatics and Gravity
1. The Electrostatic Repulsion Force
2. Newton’s Law of Gravity - The Attractive Force of Masses
3. Comparison of Electrostatic Repulsion and Gravitational Attraction



4. Nuclear Gravitation Field Theory:  Major Stumbling Blocks to Overcome



5. New Theory Results Must Equal Old Theory Results When and Where Applicable
1. Newton’s Law of Gravity as It Applies to Large Masses and Nuclear Gravitation Field Theory
2. Kepler’s Laws, Gravity, and Nuclear Gravitation Field Theory



6. Structure of the Nucleus of the Atom



7. The Schrodinger Wave Equation and Quantum Mechanics - The Particle and Wave Characteristics of Matter



8. Nuclear Gravitation Field Theory Versus Accepted Strong Nuclear Force Overcoming Electrostatic Repulsion



9. Comparison of the Nuclear Gravitation Field to the Gravitational Field of the Sun and the Gravitational Field of a Neutron Star



10. Quantum Mechanics, General Relativity, and the Nuclear Gravitation Field Theory



11. Properties of the Strong Nuclear Force, Nuclear Properties of Bismuth, and the Nuclear Gravitation Field Theory



12. Conclusion



13. Appendix A:  References



14. Appendix B:  Background of the Author

Index and Direct Hyperlinks to the Other Web Pages on this Website:

1. Gravity Warp Drive Home Page



2. Nuclear Gravitation Field Theory  (Specific Chapter Links are Provided on this Web Page)



3. Purchase e-Books



4. History of My Research and Development of the Nuclear Gravitation Field Theory



5. “The Zeta Reticuli Incident” by Terence Dickinson



6. Supporting Information for the Nuclear Gravitation Field Theory



7. Government Scientist Goes Public



8. “Sport Model” Flying Disc Operational Specifications



9. Design and Operation of the “Sport Model” Flying Disc Anti-Matter Reactor



10. Element 115



11. Bob Lazar’s Gravity Generator



12. United States Patent Number 3,626,605:  “Method and Apparatus for Generating a Secondary Gravitational Force Field”



13. United States Patent Number 3,626,606:  “Method and Apparatus for Generating a Dynamic Force Field”



14. V. V. Roschin and S. M. Godin:  “Verification of the Searl Effect”



15. The Physics of Star Trek and Subspace Communication:  Science Fiction or Science Fact?



16. Constellation:  Reticulum



17. Reticulan Extraterrestrial Biological Entity



18. Zeta 2 Reticuli:  Home System of the Greys?



19. UFO Encounter and Time Backs Up



20. UFO Testimonies by Astronauts and Cosmonauts and UFO Comments by Presidents and Top U.S. Government Officials



21. Pushing the Limits of the Periodic Table



22. General Relativity



23. Rethinking Relativity



24. The Speed of Gravity - What the Experiments Say



25. Negative Gravity



26. The Bermuda Triangle:  Space-Time Warps



27. The Wright Brothers



28. Website Endorsements and Favorite Quotes



29. Sponsors of This Website



30. Romans Road to Eternal Life In Jesus Christ

© Copyright Kenneth F. Wright, April 10, 2000.
All rights reserved.  No portion of this document may be
reproduced in any form without written permission of the author.