Quantum Mechanics in Natural Planck Units
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The meaning of quantum mechanics becomes clearer when we restate Planck's constant and the gravitational constant in natural Planck units. These units reveal hidden structure that improves our understanding of physics and gives new meaning to fundamental ideas.
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Quantum Mechanics in Natural Planck Units
- 1. Quantum Mechanics In natural Planck units David Humpherys, https://orcid.org/0000-0001-7375-7897, https://www.preprints.org/manuscript/202006.0017/v2
- 2. Modern physics is formulated largely upon two constants: gravitational constant Planck’s constant 6.674 x 10-11 m3/kgs2 1.055 x 10-34 kgm2/s
- 3. But replacing these constants with natural units of length, mass, and time reveals hidden structure. ℏ = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 𝐺𝐺 = 𝑙𝑙𝑃𝑃 𝑚𝑚𝑃𝑃 𝑐𝑐2 gravitational constant Planck’s constant 6.674 x 10-11 m3/kgs2 1.055 x 10-34 kgm2/s …where 𝑐𝑐 = 𝑙𝑙𝑃𝑃 𝑡𝑡𝑃𝑃
- 4. The structure of natural formulas The Planck units offer more granular information than Planck’s constant and the gravitational constant. For example, the formula for gravitational acceleration can be written in Planck units as: 𝒈𝒈 = − 𝑮𝑮𝑮𝑮 𝒓𝒓𝟐𝟐 = 𝒍𝒍𝑷𝑷 𝟑𝟑 𝒎𝒎𝑷𝑷𝒕𝒕𝑷𝑷 𝟐𝟐 𝑴𝑴 𝒓𝒓𝟐𝟐 = 𝒍𝒍𝑷𝑷 𝒓𝒓 𝑴𝑴 𝒎𝒎𝑷𝑷 𝒍𝒍𝑷𝑷 𝒓𝒓 𝒍𝒍𝑷𝑷 𝒕𝒕𝑷𝑷 𝟐𝟐 Replacing the gravitational constant with natural Planck units reveals the following structure:
- 5. A Planck unit potential in the unit dimensions you are solving for; in this case, acceleration. One or more dimensionless ratios quantifying the physical properties of a particle or system in proportion to the Planck scale. Natural formulas stated in Planck units have two parts. PLANCK UNIT POTENTIAL DIMENSIONLESS RATIOS OF PROPORTIONALITY 1 2
- 6. GRAVITATIONAL CONSTANT FORMULA INPUTS DIMENSIONLESS RATIOS PLANCK UNIT POTENTIAL The gravitational constant gives correct answers only because its composite value embodies Planck units in the right proportions.
- 7. ƛ𝑪𝑪 = ℏ 𝑚𝑚𝑚𝑚 = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑐𝑐 𝒍𝒍𝑷𝑷 Compton wavelength ƛ = ℏ 𝑚𝑚𝑚𝑚 = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝒍𝒍𝑷𝑷 de Broglie wavelength Momentum Photon energy 𝒑𝒑 = ℏ ƛ = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 ƛ = 𝑙𝑙𝑃𝑃 ƛ 𝒑𝒑𝑷𝑷 𝑬𝑬 = ℏc ƛ = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐2 ƛ = 𝑙𝑙𝑃𝑃 ƛ 𝑬𝑬𝑷𝑷 Mechanical properties of elementary particles and systems are quantified by dimensionless ratios and Planck unit potentials. THE STRUCTURE OF NATURAL FORMULAS Formulas and descriptions are non-relativistic
- 8. Gravitational acceleration Gravitational force Gravitational energy Escape velocity 𝒈𝒈 = − 𝐺𝐺𝐺𝐺 𝑟𝑟2 = 𝑙𝑙𝑃𝑃 3 𝑚𝑚𝑃𝑃𝑡𝑡𝑃𝑃 2 𝑀𝑀 𝑟𝑟2 = 𝑙𝑙𝑃𝑃 𝑟𝑟 𝑀𝑀 𝑚𝑚𝑃𝑃 𝑙𝑙𝑃𝑃 𝑟𝑟 𝒍𝒍𝑷𝑷 𝒕𝒕𝑷𝑷 𝟐𝟐 𝑭𝑭 = 𝐺𝐺𝐺𝐺𝐺𝐺 𝑟𝑟2 = 𝑙𝑙𝑃𝑃 3 𝑚𝑚𝑃𝑃𝑡𝑡𝑃𝑃 2 𝑀𝑀𝑀𝑀 𝑟𝑟2 = 𝑙𝑙𝑃𝑃 𝑟𝑟 𝑀𝑀 𝑚𝑚𝑃𝑃 𝑙𝑙𝑃𝑃 𝑟𝑟 𝑚𝑚 𝑚𝑚𝑃𝑃 𝒍𝒍𝑷𝑷𝒎𝒎𝑷𝑷 𝒕𝒕𝑷𝑷 𝟐𝟐 𝑼𝑼 = − 𝐺𝐺𝐺𝐺𝐺𝐺 𝑟𝑟 = 𝑙𝑙𝑃𝑃 3 𝑚𝑚𝑃𝑃𝑡𝑡𝑃𝑃 2 𝑀𝑀𝑀𝑀 𝑟𝑟 = 𝑙𝑙𝑃𝑃 𝑟𝑟 𝑀𝑀 𝑚𝑚𝑃𝑃 𝑚𝑚 𝑚𝑚𝑃𝑃 𝒍𝒍𝑷𝑷 𝟐𝟐 𝒎𝒎𝑷𝑷 𝒕𝒕𝑷𝑷 𝟐𝟐 𝒗𝒗𝒆𝒆 = 2𝐺𝐺𝐺𝐺 𝑟𝑟 = 2 𝑙𝑙𝑃𝑃 3 𝑚𝑚𝑃𝑃𝑡𝑡𝑃𝑃 2 𝑀𝑀 𝑟𝑟 = 2 𝑙𝑙𝑃𝑃 𝑟𝑟 𝑀𝑀 𝑚𝑚𝑃𝑃 𝒍𝒍𝑷𝑷 𝒕𝒕𝑷𝑷 Schwarzschild radius 𝒓𝒓𝒔𝒔 = 2 𝐺𝐺𝐺𝐺 𝑐𝑐2 = 2 𝑙𝑙𝑃𝑃 𝑚𝑚𝑃𝑃 𝑐𝑐2 𝑀𝑀 𝑐𝑐2 = 2 𝑀𝑀 𝑚𝑚𝑃𝑃 𝒍𝒍𝑷𝑷 * The natural formula reveals a hidden quantity of Planck mass in the numerator and denominator * * Gravitational properties of elementary particles and systems are quantified by dimensionless ratios and Planck unit potentials. THE STRUCTURE OF NATURAL FORMULAS
- 9. Wavelength
- 10. de Broglie Wavelength ƛ 𝑓𝑓 𝑥𝑥 ƛ = ℏ 𝑝𝑝 = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 𝑚𝑚0𝑣𝑣 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 A particle’s wavelength can be determined from its momentum. For matter particles, ratios of rest mass and velocity give the correct proportions.
- 11. 𝑓𝑓 𝑥𝑥 ƛ𝐶𝐶 = ℏ 𝑝𝑝 = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 𝑚𝑚0𝑣𝑣 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑐𝑐 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 The speed of light is a special case of the wavelength formula. The Compton and de Broglie wavelength formulas converge at this limit. Compton Wavelength Note that: The Compton wavelength is the minimum limit of a matter particle’s wavelength.
- 12. ƛ limit as 𝑣𝑣 → 𝑐𝑐 ƛ𝐶𝐶 For matter particles, Compton and de Broglie wavelengths converge at the velocity limit.
- 13. The ratio of Compton wavelength to de Broglie wavelength is equal to the ratio of velocity to the speed of light. WAVELENGTH- VELOCITY RELATION
- 14. Consider the ratio of Compton wavelength to de Broglie wavelength; for a given quantity of rest mass, the ratio between the two wavelengths is equal to the ratio of velocity to the speed of light. Why? wavelength-velocity relation The Compton and de Broglie wavelength formulas describe a proportional relationship between wavelength and velocity. ƛ = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝒄𝒄 𝒗𝒗 𝑙𝑙𝑃𝑃 ƛ𝑪𝑪 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝒄𝒄 𝒄𝒄 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 ƛ = 𝑣𝑣 𝑐𝑐
- 15. The wavelength-velocity relation is evident in the properties of charged leptons. The relationship holds for different quantities of rest mass and velocity. lepton ƛ𝐶𝐶 m 𝑣𝑣 m/s ƛ m ƛ𝐶𝐶 ƛ 𝑣𝑣 𝑐𝑐 3.862 x 10-13 2,997,924 3.862 x 10-11 0.0100 0.0100 1.868 x 10-15 1,000 5.599 x 10-10 3.336 x 10-6 3.336 x 10-6 1.111 x 10-16 100,000,000 3.329 x 10-16 0.3336 0.3336 arbitrary ℏ 𝑝𝑝 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 CODATA ƛ𝑪𝑪 ƛ = 𝒗𝒗 𝒄𝒄
- 16. ƛ ƛ𝐶𝐶 𝑙𝑙𝑃𝑃 The wavelength-velocity relation is demonstrated by a ground state electron. In this case, the ratio of proportionality is the fine-structure constant. 1 ~137 2,187,691 m/s 1 ~137 5.292 x 10-11 m electron velocity electron wavelength 1 ~137
- 17. The ratio of Planck length to Compton wavelength is equal to the ratio of rest mass to Planck mass. LENGTH-MASS SYMMETRY
- 18. ƛ𝐶𝐶 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑙𝑙𝑃𝑃 Given 𝑐𝑐 𝑐𝑐 = 1, arrange the Compton wavelength formula as an equivalence between ratios of length and mass to the Planck scale. Why? 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 = 𝑚𝑚0 𝑚𝑚𝑃𝑃 ƛ𝐶𝐶 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑐𝑐 𝑙𝑙𝑃𝑃 Length-mass symmetry The wavelength formulas show that ratios of Compton wavelength and rest mass to the Planck scale are equivalent.
- 19. Length-mass symmetry is evident in the Compton wavelengths and rest masses of the charged leptons. 𝒍𝒍𝑷𝑷 ƛ𝑪𝑪 = 𝒎𝒎𝟎𝟎 𝒎𝒎𝑷𝑷 lepton ƛ𝐶𝐶 m 𝑚𝑚0 kg 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 𝑚𝑚0 𝑚𝑚𝑃𝑃 3.862 x 10-13 9.109 x 10-31 4.185 x 10-23 4.185 x 10-23 1.868 x 10-15 1.884 x 10-28 8.654 x 10-21 8.654 x 10-21 1.111 x 10-16 3.168 x 10-27 1.455 x 10-19 1.455 x 10-19 CODATA CODATA
- 20. Due to length-mass symmetry, the product of Compton wavelength and rest mass is conserved. lepton ƛ𝐶𝐶 m 𝑚𝑚0 kg ƛ𝐶𝐶𝑚𝑚0 kgm 3.862 x 10-13 9.109 x 10-31 3.518 x 10-43 1.868 x 10-15 1.884 x 10-28 3.518 x 10-43 1.111 x 10-16 3.168 x 10-27 3.518 x 10-43 ƛ𝑪𝑪𝒎𝒎𝟎𝟎 = 3.52 x 10-43 kgm CODATA CODATA
- 21. Wavelength-mass is conserved to the Planck scale. So, the product of a wavelength-mass pair is equal to the product of Planck length and Planck mass, or 3.52 x 10-43 kgm. ƛ𝐶𝐶𝑚𝑚0 3.52 x 10-43 kgm Rest mass is inversely proportional to Compton wavelength Compton wavelength is inversely proportional to rest mass
- 22. 𝛽𝛽𝑚𝑚 Dimensionless ratios create a natural unit scale in which the Planck units are equal to 1. We can characterize each ratio on a scale of 0 to 1. 𝛽𝛽 = dimensionless ratio on a scale of 0 to 1 type of ratio REST MASS RATIO 1 0 natural unit scale
- 23. 𝛽𝛽𝑚𝑚 Rest mass ratio The rest mass ratio is the ratio of Planck length to Compton wavelength, and the ratio of rest mass to Planck mass. ƛ𝐶𝐶 𝑚𝑚0 𝛽𝛽𝑚𝑚
- 24. 𝛽𝛽𝑣𝑣 Velocity ratio The velocity ratio is the ratio of velocity to the speed of light. 𝛽𝛽𝑣𝑣 ƛ T ƛ T ƛ𝐶𝐶
- 25. Momentum
- 26. 𝑝𝑝 𝑝𝑝 = ℏ ƛ = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 ƛ = 𝑙𝑙𝑃𝑃 ƛ 𝑝𝑝𝑃𝑃 𝑓𝑓 𝑥𝑥 The quantum mechanical momentum formula can be stated in Planck units as: Quantum mechanical momentum
- 27. A particle with Planck length wavelength has Planck momentum. As wavelength increases, momentum decreases inversely. ƛ = 15𝑙𝑙𝑃𝑃 01 02 03 04 05 06 07 08 09 10 14 11 12 13 15 ƛ = 𝑙𝑙𝑃𝑃 00 𝑝𝑝 = 𝑙𝑙𝑃𝑃 𝑙𝑙𝑃𝑃 𝑝𝑝𝑃𝑃 = 𝑝𝑝𝑃𝑃 𝑝𝑝 = 𝑙𝑙𝑃𝑃 15𝑙𝑙𝑃𝑃 𝑝𝑝𝑃𝑃 = 1 15 𝑝𝑝𝑃𝑃
- 28. 𝑝𝑝 The classical momentum formula is the product of rest mass and velocity. Classical momentum
- 29. electrons have rest mass, photons do not Do classical and quantum mechanical momentum formulas describe the same phenomenon? photons move at light speed, electrons do not
- 30. Q: Is there a consistent description of momentum for classical and quantum systems? A: (yes, there is)
- 31. Classical formula Quantum mechanical formula 𝑝𝑝 = 𝑚𝑚𝑚𝑚 𝑝𝑝 = ℏ ƛ rest mass: 9.109 x 10-31 kg velocity: 2,187,691 m/s reduced Planck constant: 1.055 x 10-34 kgm2/s reduced wavelength: 5.292 x 10-11 m 𝑝𝑝 = (1.055 x 10-34 kgm2/s) / (5.292 x 10-11 m) = 1.99 x 10-24 kgm/s 𝑝𝑝 = (9.109 x 10-31 kg) (2,187,691 m/s) = 1.99 x 10-24 kgm/s First, we can show that the quantum and classical formulas applied to a ground state electron produce the same result.
- 32. Classical formula Quantum mechanical formula Next, we can show that ratios of rest mass and velocity are equal to the ratio of wavelength. = 𝑚𝑚0 𝑚𝑚𝑃𝑃 = 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 = = 𝑣𝑣 𝑐𝑐 = ƛ𝐶𝐶 ƛ = 𝜷𝜷𝒎𝒎 𝜷𝜷𝒗𝒗 9.109 x 10-31 kg 2.176 x 10-8 kg = 4.185 x 10-23 = 𝑙𝑙𝑃𝑃 ƛ = 2,187,691 m/s 299,792,458 m/s = .0073 = 3.054 x 10-25 𝜷𝜷𝝀𝝀 = 3.054 x 10-25 1.616 x 10-35 m 5.292 x 10-11 m 𝜷𝜷𝒎𝒎 𝜷𝜷𝒗𝒗 𝜷𝜷𝝀𝝀 𝑝𝑝 = 𝑚𝑚𝑚𝑚 𝑝𝑝 = ℏ ƛ = 𝑙𝑙𝑃𝑃 ƛ
- 33. lepton 𝑣𝑣 𝛽𝛽𝑚𝑚 𝛽𝛽𝑝𝑝 𝛽𝛽𝑚𝑚 𝛽𝛽𝑝𝑝 𝛽𝛽λ 2,997,924 4.185 x 10-23 0.0100 4.185 x 10-25 4.185 x 10-25 1,000 8.654 x 10-21 3.336 x 10-6 2.887 x 10-26 2.887 x 10-26 100,000,000 1.455 x 10-19 0.3336 4.855 x 10-20 4.855 x 10-20 𝑚𝑚0 𝑚𝑚𝑃𝑃 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 𝑙𝑙𝑃𝑃 ƛ , ƛ𝐶𝐶 ƛ arbitrary 𝜷𝜷𝒎𝒎 𝜷𝜷𝒑𝒑 = 𝜷𝜷𝝀𝝀 The charged leptons demonstrate that the ratio of rest mass and velocity is equal to the ratio of wavelength.
- 34. 𝑚𝑚0 Rest mass quantifies the Compton wavelength Velocity quantifies the ratio of Compton wavelength to de Broglie wavelength Length-mass symmetry Wavelength-velocity relation 𝑣𝑣 Rest mass and velocity quantify particle wavelength.
- 35. 𝑚𝑚0 = 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 𝑚𝑚𝑃𝑃 𝑣𝑣 = ƛ𝐶𝐶 ƛ 𝑐𝑐 𝑝𝑝 Replacing rest mass and velocity with equivalent ratios of wavelength shows how the two momentum formulas are related.
- 36. Momentum is therefore the same phenomenon for both matter and radiation. The quantum mechanical formula correctly describes both; however, for large particle systems, it is pragmatic to calculate momentum using rest mass and velocity. The classical momentum formula calculates the average wavelength of constituent matter particles using the bulk properties of rest mass and velocity. 𝑓𝑓 𝑥𝑥 𝑝𝑝 = 𝑚𝑚0𝑣𝑣 = 𝑙𝑙𝑃𝑃 ƛ𝐶𝐶 𝑚𝑚𝑃𝑃 ƛ𝐶𝐶 ƛ 𝑐𝑐 = 𝑙𝑙𝑃𝑃 ƛ 𝑝𝑝𝑃𝑃 = ℏ ƛ
- 37. particle 𝑚𝑚0 kg 𝑣𝑣 m/s ƛ m 𝑝𝑝 kgm/s 9.109 x 10-31 2,997,925 3.862 x 10-11 2.731 x 10-24 1.884 x 10-28 14,499 3.862 x 10-11 2.731 x 10-24 3.168 x 10-27 862 3.862 x 10-11 2.731 x 10-24 - 299,792,458 3.862 x 10-11 * 2.731 x 10-24 Rest mass and velocity give the right proportions for calculating the momentum of matter particles, but only wavelength gives consistent meaning to both matter and radiation. CODATA electron velocity chosen arbitrarily; muon & tau velocities chosen to match electron wavelength ℏ 𝑝𝑝 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 ℏ ƛ = 𝑙𝑙𝑃𝑃 ƛ 𝑝𝑝𝑃𝑃 * photon wavelength chosen to match lepton wavelengths
- 38. The textbook description of momentum as mass in motion is misleading. The momentum of a cement truck barreling down the highway represents a static property—the strength of the truck’s particle wavelengths which have greater kinetic energy potential the shorter they are (just like photons). Velocity as a measure of displacement describes the truck’s kinetic energy. A proportional change in particle wavelengths and velocity produces a squared amount of change in kinetic energy.
- 39. Energy
- 40. 𝐸𝐸 𝑓𝑓 𝑥𝑥 𝐸𝐸 = ℏc ƛ = 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐2 ƛ = 𝑙𝑙𝑃𝑃 ƛ 𝐸𝐸𝑃𝑃 The quantum mechanical formula for photon energy can be restated in natural Planck units. The formula shows that a photon’s energy is proportional to its wavelength. Quantum mechanical energy
- 41. 𝐸𝐸𝐾𝐾 The simplified natural kinetic energy formula is therefore: We can define a natural kinetic energy formula building on the momentum equation. Inserting a one-half coefficient and the particle’s velocity produces the formula. 𝐸𝐸𝐾𝐾 momentum energy
- 42. It is easy to show that the natural kinetic energy formula gives the same result as the classical formula. lepton 𝑚𝑚0 kg 𝑣𝑣 m/s ƛ m 𝐸𝐸 kgm2/s2 𝐸𝐸 kgm2/s2 9.109 x 10-31 2,997,924 3.862 x 10-11 4.094 x 10-18 4.094 x 10-18 1.884 x 10-28 1,000 5.599 x 10-10 9.418 x 10-23 9.418 x 10-23 3.168 x 10-27 100,000,000 3.329 x 10-16 1.584 x 10-11 1.584 x 10-11 classical formula natural formula CODATA 1 2 𝑙𝑙𝑃𝑃 ƛ 𝑣𝑣 𝑐𝑐 𝐸𝐸𝑃𝑃 ℏ 𝑝𝑝 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 1 2 𝑚𝑚0𝑣𝑣2 arbitrary
- 43. We can compare the structure of matter and radiation by writing the photon energy formula with a coefficient of 1 and the particle’s velocity, the speed of light. The natural energy formulas for matter and radiation reveal a remarkably similar structure. 𝒉𝒉𝒉𝒉 𝝀𝝀 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒗𝒗𝟐𝟐
- 44. For matter & radiation, kinetic energy is a function of a particle’s wavelength potential (momentum) and its velocity. 2-PART ENERGY MECHANISM
- 45. spin* PART 1: A particle’s kinetic energy potential inversely proportional to its wavelength. PART 2: A particle’s velocity. 𝐸𝐸𝐾𝐾 𝐸𝐸 For matter and radiation, energy consists of two parts. 2 1 𝒉𝒉𝒉𝒉 𝝀𝝀 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒗𝒗𝟐𝟐 * Quantum mechanical spin can account for the kinetic energy coefficients, distributing the energy of half-spin particles over an extended cycle.
- 46. An equal distribution of energy across a particle’s wavelength and velocity generates a squared amount of kinetic energy. ƛ = ½ = 1 = 2 = 1 𝑣𝑣 𝑣𝑣 ƛ part 1: ƛ = 1 part 2: v = 1 KE = 1 part 1: ƛ = ½ part 2: v = 2 KE = 4 Doubling velocity reduces the wavelength in half
- 47. The charged leptons demonstrate that when velocity doubles, particle wavelength decreases by half, and kinetic energy increases 4x. lepton 𝑣𝑣1 m/s 𝑣𝑣2 m/s ƛ 1 m ƛ 2 m 𝐸𝐸1 kgm2/s2 𝐸𝐸2 kgm2/s2 2,997,924 5,995,848 3.862 x 10-11 1.931 x 10-11 4.094 x 10-18 1.637 x 10-17 8,293,910 16,587,820 6.751 x 10-14 3.375 x 10-14 6.478 x 10-15 2.591 x 10-14 23,092,348 46,184,696 1.442 x 10-15 7.209 x 10-16 8.446 x 10-13 3.378 x 10-12 ℏ 𝑝𝑝 = 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 1 2 𝑚𝑚0𝑣𝑣2 = 1 2 𝑙𝑙𝑃𝑃 ƛ 𝑣𝑣 𝑐𝑐 𝐸𝐸𝑃𝑃 arbitrary 2𝑣𝑣1 𝒗𝒗𝟐𝟐 = 𝟐𝟐𝒗𝒗𝟏𝟏 ƛ𝟐𝟐 = ½ƛ𝟏𝟏 E𝟐𝟐 = 𝟒𝟒𝑬𝑬𝟏𝟏
- 48. 𝐸𝐸 𝐾𝐾𝐾𝐾* Every elementary particle has the same energy budget—the Planck energy. This budget can be spent in two ways: 1. diluting the particle’s energy over space 2. diluting the particle’s energy over time Another way to look at it… 1. The longer a particle’s wavelength, the more diluted its energy potential becomes. This potential is equal to the ratio of one Planck length to the particle’s wavelength. 2. There is no temporal dilution of a photon’s energy as it moves at light speed. Due to rest mass, the kinetic energy of matter is temporally diluted by the ratio of its velocity to the speed of light. 𝒄𝒄 𝒄𝒄 *KE is further reduced by half. 𝑙𝑙𝑃𝑃 ƛ 𝑙𝑙𝑃𝑃 ƛ 𝒗𝒗 𝒄𝒄
- 49. The relationship between momentum and energy The 2-part energy mechanism clarifies the physical meaning of momentum and energy. Momentum is the energy potential of a particle’s wavelength and kinetic energy is the realization of that potential in motion. Momentum is analogous to a payload quantified by the concentration of a particle’s wavelength, while velocity delivers the payload as energy.
- 50. Because velocity is not a physical representation of momentum, we need to re-evaluate whether unit dimensions of momentum are correct. ? 𝑝𝑝
- 51. The physical meaning of momentum The concept of momentum pre-dates quantum theory when wave-like attributes of matter first became known. For Newton and others who described momentum prior to the 20th century, the quantity 𝑣𝑣2 simply matched the observational data. Multiplying rest mass by reveal that velocity is not a physical representation of momentum, and that unit dimension M is a more consistent way of representing the strength of a particle’s wavelength for all types of particles. Isaac Newton Louis de Broglie velocity-squared correctly modeled the proportionality between rest mass and energy, but the physical meaning of redundant unit dimensions LT-1 remained unexplained. After Louis de Broglie explained the wave-like nature of matter, it was possible to identify the distinct roles of wavelength and velocity in determining kinetic energy. Natural Planck unit formulas
- 52. Mass
- 53. 𝑚𝑚 To represent the momentum of matter and radiation in unit dimension M, we’ll define a quantity of inertial mass 𝑚𝑚. Inertial mass is inversely proportional to a particle’s wavelength.
- 54. Inertial mass is easy to calculate from rest mass or momentum. We can treat inertial mass as a simple unit conversion from unit dimensions of momentum to mass. 𝑚𝑚 = 𝑚𝑚0 𝑣𝑣 𝑐𝑐 = 𝑝𝑝 𝑐𝑐
- 55. The ratio of Planck length to wavelength is equal to the ratio of inertial mass to Planck mass. LENGTH-MASS SYMMETRY
- 56. Length-mass symmetry (general form) With inertial mass, we can make length-mass symmetry a general principle in which Compton wavelength and rest mass is a special case. ƛ 𝑚𝑚 ƛ𝐶𝐶 𝑚𝑚0
- 57. 3.52 x 10-43 kgm ƛ𝑚𝑚 Mass is inversely proportional to wavelength Wavelength is inversely proportional to mass The general form of length-mass symmetry applies to every combination of wavelength and inertial mass, and it works for both matter and radiation.
- 58. particle 𝑚𝑚0 kg 𝑣𝑣 m/s ƛ m 𝑚𝑚 kg ƛ𝑚𝑚 kg 9.109 x 10-31 2,997,924 3.862 x 10-11 9.109 x 10-33 3.518 x 10-43 1.884 x 10-28 14,499 3.862 x 10-11 9.109 x 10-33 3.518 x 10-43 3.168 x 10-27 862 3.862 x 10-11 9.109 x 10-33 3.518 x 10-43 - 299,792,458 3.862 x 10-11 9.109 x 10-33 3.518 x 10-43 Mass is the right unit dimension for quantifying the density of a particle’s wavelength. It gives consistent meaning to matter and radiation. CODATA 𝑚𝑚𝑃𝑃 𝑚𝑚0 𝑐𝑐 𝑣𝑣 𝑙𝑙𝑃𝑃 = ℏ 𝑝𝑝 𝑙𝑙𝑃𝑃 ƛ 𝑚𝑚𝑃𝑃 electron velocity chosen arbitrarily; muon & tau velocities chosen to match electron wavelength chosen to match the leptons
- 59. 𝐸𝐸 = 𝑚𝑚2 + 𝑚𝑚0 2 𝑐𝑐2 Replacing momentum with inertial mass gives the following simplified form of the momentum–energy relation. 𝑓𝑓 𝑥𝑥 𝐸𝐸 = 𝑝𝑝𝑝𝑝 2 + 𝑚𝑚0𝑐𝑐2 2 = 𝑚𝑚2𝑐𝑐4 + 𝑚𝑚0 2 𝑐𝑐4 = 𝑚𝑚2 + 𝑚𝑚0 2 𝑐𝑐2
- 60. radiation matter inertial mass inertial mass + rest mass ƛ 𝑚𝑚 ƛ𝐶𝐶 𝑚𝑚0 T T ƛ T ƛ 𝑚𝑚 ƛ Part 1 Part 2 kinetic energy potential of a particle’s wavelength velocity dx and dt converge at the Compton wavelength limit of the particle’s shortest wavelength dx = dt at every wavelength ƛ𝐶𝐶 ƛ = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 Rest mass reduces the range of a particle’s wavelength and velocity.
- 61. Putting it all together
- 62. 𝛽𝛽𝜆𝜆 = 𝛽𝛽𝑚𝑚 𝛽𝛽𝑝𝑝 The wavelength ratio is equal to the product of rest mass and velocity ratios. 𝛽𝛽𝑝𝑝 = 𝛽𝛽𝑣𝑣 The momentum ratio is equal to the velocity ratio. The relationship between the four mechanical ratios is shown in the illustration below. These ratios describe the mechanical properties of elementary particles and systems. 2 1 Note the following relationships:
- 63. 𝛽𝛽𝜆𝜆 Wavelength ratio The wavelength ratio is the ratio of Planck length to particle wavelength, and the ratio of inertial mass to Planck mass. 𝛽𝛽𝜆𝜆 ƛ 𝑚𝑚
- 64. 𝛽𝛽𝑝𝑝 Momentum ratio The momentum ratio is the ratio of Compton wavelength to de Broglie wavelength, and the ratio of inertial mass to rest mass. 𝛽𝛽𝑝𝑝 ƛ 𝑚𝑚 ƛ𝐶𝐶 𝑚𝑚0
- 65. The simple, natural formulas for quantifying mechanical properties of elementary particles are summarized below. ƛ = 𝑙𝑙𝑃𝑃 𝛽𝛽𝜆𝜆 𝑚𝑚 = 𝛽𝛽𝜆𝜆 𝑚𝑚𝑃𝑃 𝑝𝑝 = 𝛽𝛽𝜆𝜆 𝑝𝑝𝑃𝑃 T = 𝑡𝑡𝑃𝑃 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸 = 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 𝐸𝐸 = ½ 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 𝑚𝑚0 = 𝛽𝛽𝑚𝑚 𝑚𝑚𝑃𝑃 wavelength mass momentum period energy Dimensionless ratios 𝛽𝛽𝜆𝜆 , 𝛽𝛽𝑣𝑣 𝛽𝛽𝑚𝑚 𝛽𝛽𝑝𝑝 , , wavelength, inertial mass Compton wavelength, rest mass wavelength ratio mass ratio velocity
- 66. particle 𝑣𝑣 m/s ƛ m 𝑚𝑚 kg 𝑝𝑝 kgm/s 𝑇𝑇 s 𝐸𝐸 kgm2/s2 2,997,924 3.862 x 10-11 9.109 x 10-33 2.731 x 10-24 1.288 x 10-17 4.094 x 10-18 1,000 5.599 x 10-10 6.283 x 10-34 1.884 x 10-25 5.599 x 10-13 9.418 x 10-23 100,000,000 3.329 x 10-16 1.057 x 10-27 3.168 x 10-19 3.329 x 10-24 1.584 x 10-11 299,792,458 1.935 x 10-8 1.818 x 10-35 5.451 x 10-27 6.454 x 10-17 1.634 x 10-18 𝑙𝑙𝑃𝑃 𝛽𝛽𝜆𝜆 𝛽𝛽𝜆𝜆 𝑚𝑚𝑃𝑃 𝛽𝛽𝜆𝜆 𝑝𝑝𝑃𝑃 𝑡𝑡𝑃𝑃 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 ½ 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 Natural formulas give the same results as classical and quantum formulas while yielding new insights into the physical meaning of quantum physics. arbitrary
- 67. The Rydberg energy formula showcases the four mechanical operators. The formula calculates the ground state energy of an electron. RYDBERG CONSTANT hc RYDBERG ENERGY ½ 𝛽𝛽𝑚𝑚 𝛽𝛽𝑝𝑝 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 ½ 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃
- 68. Length-mass symmetry yields the conservation of action in traditional unit dimensions of momentum. CONSERVATION OF ACTION
- 69. Since ƛ𝑚𝑚 is conserved, multiplying ƛ𝑚𝑚 by the constant 𝑐𝑐 also conserves Planck’s constant, 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐. 3.518 x 10-43 kgm ƛ𝑚𝑚 ƛ𝑚𝑚𝑐𝑐 1.055 x 10-34 kgm (Planck’s constant)
- 70. Conservation of action is an alternative expression of length-mass symmetry. It can be stated in terms of momentum and energy. ƛ𝑝𝑝 𝐸𝐸𝑇𝑇 1.055 x 10-34 kgm 1.055 x 10-34 kgm * * Converting length-mass symmetry into momentum unit dimensions conserves Planck’s constant.
- 71. The known properties of elementary particles show that wavelength-mass is conserved, which in turn conserves wavelength-momentum and energy-time. particle 𝑣𝑣 m/s ƛ m 𝑚𝑚 kg ƛ𝑚𝑚 kgm 𝑝𝑝 kgm/s 𝑇𝑇 s 𝐸𝐸 kgm2/s2 ƛ𝑝𝑝, 𝐸𝐸𝐸𝐸 * kgm2/s2 2,997,924 3.862 x 10-11 9.109 x 10-33 3.518 x 10-43 2.731 x 10-24 1.288 x 10-17 4.094 x 10-18 1.055 x 10-34 1,000 5.599 x 10-10 6.283 x 10-34 3.518 x 10-43 1.884 x 10-25 5.599 x 10-13 9.418 x 10-23 1.055 x 10-34 100,000,000 3.329 x 10-16 1.057 x 10-27 3.518 x 10-43 3.168 x 10-19 3.329 x 10-24 1.584 x 10-11 1.055 x 10-34 299,792,458 1.935 x 10-8 1.818 x 10-35 3.518 x 10-43 5.451 x 10-27 6.454 x 10-17 1.634 x 10-18 1.055 x 10-34 arbitrary 𝑙𝑙𝑃𝑃 𝛽𝛽𝜆𝜆 𝛽𝛽𝜆𝜆 𝑚𝑚𝑃𝑃 𝛽𝛽𝜆𝜆 𝑝𝑝𝑃𝑃 𝑡𝑡𝑃𝑃 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 ½ 𝛽𝛽𝜆𝜆 𝛽𝛽𝑣𝑣 𝐸𝐸𝑃𝑃 * 2𝐸𝐸𝐸𝐸 for matter
- 72. The Heisenberg uncertainty principle is a statement about the incompatibility of simultaneous wave-like and point-like descriptions. Δ𝑥𝑥Δ𝑝𝑝 Δ𝑥𝑥 𝑚𝑚0 Δ𝑣𝑣 A common form of the uncertainty principle says that the uncertainty in a particle’s position and momentum is greater than or equal to one-half ℏ We can re-state the uncertainty principle in Planck units while breaking it down into components of length, mass, and velocity. Rest mass and velocity also give us the wavelength. ≥ ℏ 2 ≥ 𝑙𝑙𝑃𝑃𝑚𝑚𝑃𝑃𝑐𝑐 2
- 73. The following table breaks down each component of the uncertainty principle and applies it to an electron. The left-hand side of the inequality must be larger than the right-hand side after all three components are considered. 2 1 𝑚𝑚0 𝑚𝑚𝑃𝑃 4.23 x 10-23 Δ𝑣𝑣 𝑐𝑐 Δ𝑥𝑥 < < 𝑣𝑣 𝑐𝑐 The electron’s rest mass is 23 orders of magnitude less than the Planck mass. So far, the left hand-side of the inequality is much smaller than the right-hand side. The electron’s velocity is less than the speed of light which puts the inequality further into the red. Let’s say we know the electron’s velocity, which means we also know its wavelength. Given 1 and 2, the uncertainty in the particle’s position is at least half of the particle’s wavelength. The uncertainty principle uses the de Broglie formula to quantify wavelength. 𝑙𝑙𝑃𝑃 𝑐𝑐 𝑣𝑣 The uncertainty principle incorporates the de Broglie wavelength formula ≥ 4.23 x 10-23 1 3 𝜟𝜟𝒙𝒙 𝒎𝒎𝟎𝟎 𝜟𝜟𝒗𝒗 𝒍𝒍𝑷𝑷𝒎𝒎𝑷𝑷𝒄𝒄 𝟐𝟐 ≥ contribution to inequality ( ) 1 2
- 74. David Humpherys, https://orcid.org/0000-0001-7375-7897, https://www.preprints.org/manuscript/202006.0017/v2 https://www.youtube.com/channel/UCpLIfTjYwdfT6vGPUGc9nLA https://illuminating.science/ More than a century after Max Planck proposed his constant of proportionality, the physical meaning of ℏ and 𝐺𝐺 is widely unknown. Please share this presentation with others seeking a better physical understanding of quantum mechanics.