In classical physical space, particles have definite positions and momenta, not probabilities of positions and momenta. If physical space has no external forces, positions and momenta are independent. If physical space has force fields, position change changes momentum in only one way, according to energy conservation. Because particles have definite positions and momenta, and classical configuration space has only real numbers, classical configuration space has no real-number/imaginary-number interactions and so no waves. The Hamiltonian function represents energy as a function of momentum (kinetic energy) and space (potential energy) coordinates.
In quantum-mechanical physical space, particles have probabilities of positions and momenta. Quantum-mechanical physical space has energy conservation, but positions and momenta are not independent, so energy-conservation equation (Schrödinger equation) and S-matrix theory, which relate kinetic-energy change and momentum to potential-energy change and position, have complex numbers. Exponentials with complex-number exponents represent cosine and sine waves. (Maxwell's equations relate kinetic-energy change and momentum to potential-energy change and position, and solutions are electromagnetic waves.) Frequency is time derivative. Wave number is spatial derivative. The time derivative introduces an imaginary number to multiply the time derivative to give a real number.
Quantum-mechanical configuration space (phase space) has complex-number particle position and momentum coordinates. Along each configuration-space dimension, real and imaginary numbers interact to make helical scalar waves {matter wave}| {de Broglie wave} {probability wave}.
scalar
Electromagnetic waves are vector waves, because electric and magnetic forces and fields have direction, electromagnetic waves propagate in a direction, and energy travels in that direction. Matter waves are scalar, because they are not about forces or fields, have no energy, and do not propagate and so do not travel and are standing waves. Scalar waves have amplitude but no direction.
phase space
Matter waves are not in physical space.
wavelength
Wavelength determines possible particle positions and momenta, at maximum-displacement positions. Frequency and phase affect amplitude.
amplitude and probability
Amplitude determines probability that particle is at that position or momentum.
frequency and kinetic energy
Particle kinetic energy E determines matter-wave frequency f: E = h * f, where h is Planck constant. For higher energies, matter waves have higher frequencies and lower wavelengths. Particle momentum p determines matter-wave wavelength w: h = p * w. Theoretical matter-wave velocity v increases with particle kinetic energy: v = f * w = (E/h) * (h/p) = E/p.
transverse wave
Real and imaginary number interactions make transverse waves around each phase-space dimension.
length
Matter waves are in configuration space, which has infinitely-long dimensions, so matter waves are infinitely long. By uncertainty principle, matter waves extend through all space, but with low amplitude outside physical system.
no propagation and no energy
Because they are infinitely long, matter waves do not propagate, are standing waves, and have no travel, no velocity, no energy, and no leading or trailing edge. Matter waves resonate in phase space.
positions, points, and intervals
Waves require one wavelength to be a wave, so there is no definite position. For waves, positions cannot be points but are one-wavelength or half-wavelength intervals.
solidity
Matter waves have width of at least one wavelength, so they cause matter to spread over space, not be at points. Matter waves make matter have area, and matter appears solid.
momentum and position
In quantum mechanics, unlike classical mechanics, momentum and position are not independent, because amplitude relates to position, frequency relates to momentum, wave amplitude-change rate relates to wave frequency-change rate, wavelength relates to position uncertainty, and amplitude-change rate relates to momentum uncertainty.
particle sizes
Large objects have high matter-wave frequencies. At high frequencies, matter-wave properties are undetectable, because wavelengths are too small, so classical mechanics applies. Small objects have low matter-wave frequencies, so atomic particles have detectable quantum properties.
waves and quanta
Resonating waves have fundamental frequency and harmonic overtones. Particles have matter waves with harmonic frequencies. Harmonic frequencies correspond to a series of positions or energy/momentum levels, separated by equal amounts (quantum).
Waves change frequency without passing through intermediate frequencies. No intermediate frequencies means no intermediate positions or energies/momenta. Matter waves explain why particles have discrete energy levels, separated by quanta, and why, during energy-level transitions, particles never have in-between energy levels. Particles also have discrete locations, separated by quantum distances.
physical systems
In free space, particle matter waves have a small range of frequencies and superpose to make a wave packet. Particle systems superpose particle matter waves to make system matter waves. Non-interacting particles have dependent matter waves that add non-linearly (entangle). In atoms and molecules, electrons, neutrons, and protons have phase-space matter waves that represent transitions among atomic orbits.
Electrons cannot be near nucleus, because then electron matter-wave interacts with proton matter-wave, and atom collapses.
philosophy
Perhaps, matter waves are particles, only associate with particles, are mathematical descriptions, or are all that observers can know.
Physical Sciences>Physics>Quantum Mechanics>Waves
5-Physics-Quantum Mechanics-Waves
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Date Modified: 2022.0224