Dynamical mechanics methods {quantum mechanics}| determine particle momentum through space {particle trajectory} or particle energy through time. In quantum mechanics, physical systems have states of objects, their properties, and events. Quantum-mechanics states are in phase space (coordinate space), rather than physical space. Phase space includes all particle positions and momenta and so includes physical space. Relativistic phase space includes physical space-time. Phase-space states represent discrete energy, momentum, angular momentum, length, time, and mass quanta. Adding energy, momentum, angular momentum, length, time, or mass increases energy, momentum, angular momentum, length, time, or mass by quantum leaps, not continuously.
particles and fields
Without quantum mechanics, in continuous space and time, particles have properties that cause forces, which make continuous force fields. Particles have continuous energies, times, positions, and momenta. Fields have potentials at all space locations and at all times. Particles have zero energy. By statistical mechanics, with or without quantum mechanics, energy tends to spread to all positions equally (energy equipartition). Because continuous fields have many more locations than particles, energy tends to go from particles into fields over time.
Without quantum mechanics, in continuous space and time, electrons orbiting atomic nuclei interact with other nuclei and electrons and their orbits decay, As they spiral into nucleus, they emit electromagnetic waves to maintain conservation of energy. As electrons become closer to nucleus, they emit higher-frequency electromagnetic waves because force is stronger. By energy equipartition, electrons eventually fall into nucleus, ending all atoms. However, the universe has stable atoms.
As they absorb electromagnetic waves from space and emit them to space, hot objects in cooler space tend to transfer heat to space, because absorption concentrates on one location but emission spreads to all other locations and so never returns to the location. By energy equipartition, all frequencies have equal probability. Without quantum mechanics, higher frequencies carry away most energy, and hot objects cool quickly. However, experiments show a Boltzmann distribution of frequencies, with higher frequencies having lower probabilities. Hot objects cool slower.
In quantum mechanics, particles have discrete states and discontinuous fields. In quantum mechanics, length, time, and mass have quanta, with non-zero minimum (Planck) length, time, and mass. Minimum length makes maximum momentum. Minimum time makes maximum energy. Minimum mass makes maximum frequency. For each property, particles have one quantum plus a number of quanta, up to a maximum number. Atom electrons must have minimum energy and so do decay into atom nuclei. Higher states have lower probabilities, because higher states are harder to reach. High-frequency electromagnetic waves are fewer. Quantum mechanics results in Boltzmann frequency distribution and observed slower cooling. Quantum mechanics matches atomic- and subatomic-particle behavior.
energy quanta
Particle-collision, light-absorption, and light-emission experiments show that particles absorb or emit energy in non-zero minimum amounts (quantum). Charge accelerations differ by a number of quanta. Minimum electromagnetic-wave energy varies directly with frequency. Higher energy means more electromagnetic waves. Particle and particle-system energy is one quantum plus a number of energy quanta. Energy levels differ by quanta and are discrete, not continuous.
particle wave
Simultaneously sending particles through two slits makes almost the same target patterns as sending electromagnetic waves, suggesting that particles behave like waves. Large particles at ordinary energies have very high-frequency waves with imperceptible wavelengths. Atomic- and subatomic-particles have low-frequency waves with observable wavelengths. Particles have waves in phase space, not in space-time. Particle waves have fundamental frequency, at lowest energy level, and higher harmonic frequencies at higher energy levels. Particle waves have discrete, not continuous, frequencies at overtones of the fundamental frequency. Wave energy varies directly with frequency, so particle energy levels are discrete.
Quantum-mechanics wave equations describe potential and kinetic energies in particle and force-field systems. Equation solutions are periodic wavefunctions that describe particle and field positions and momenta.
ground state
Particles cannot have zero energy, because they cannot have zero motion, because they have phase-space waves and waves propagate. Particles have a lowest energy state (ground state), which corresponds to the lowest-frequency (fundamental) particle wave. All other energy states are quantum amounts higher than ground state.
determinism
Previous particle positions and momenta determine future phase-space states. Quantum wave and particle mechanics is deterministic and follows normal causality.
observation
Observations interact with particle to put particle in one observed state. Observation immediately and discontinuously selects observed state from among possible states. From that moment, phase-space again follows determinism and causality. Old system wavefunction "collapses" to nothing, and new system wavefunction, about both system and measuring apparatus, begins.
Scientific experiments try to isolate observer from experimental system to prevent interactions. However, in experiments involving small things, observation has to cause disturbance and perturb observed system. Observer and observed become one new system. Observation causes measurement uncertainty about observed system.
Perception theory is about physical events that cause state perception. Quantum mechanics has no perception theory. Quantum theory does not describe observer, only observed system. Quantum mechanics leaves open the possibility of perceiving state superposition. However, people and instruments detect only states and never observe state superpositions. Quantum-mechanical waves are in phase space, not physical space.
particle systems
Without quantum mechanics, particle systems follow classical statistical Markov processes, such as diffusion and Brownian motion, in space-time. Diffusion and Brownian motion apply energy and momentum conservation to many particles. Many particles follow many paths (path distribution), completely determined by previous positions and momenta. Particles tend to have the highest-probability energy distribution.
Quantum-mechanics equations (Schrödinger equation) also derive from energy and momentum conservation and are similar to diffusion and Brownian-motion equations. Quantum-mechanics equations are about coordinate space or phase space, rather than physical space. Quantum-mechanics functions (wavefunction) are complex-number functions, which relate trigonometric and exponential functions. Because particles can be anywhere along infinite dimensions, wavefunctions are over infinite space and number of possible system phase-space states is infinite. In phase space, particles have possible trajectories (path distribution), each with different probability. Previous positions and momenta determine only probabilities of later paths. The sum of all probabilities equals 1 = 100% that the particle is somewhere. All possible states exist simultaneously and evolve independently. In quantum mechanics, particle measurement causes only one phase-space state/path to be observed. That state tends to have the highest probability.
particle systems: superposition
In quantum mechanics, particles have wavefunctions. Particle systems have wavefunctions that are particle-wavefunction linear combinations (superposition), just as electromagnetic waves superpose. Particle wavefunctions evolve independently, just as electromagnetic waves are independent. Waves do not have multiplicative or dependent effects on each other.
Because wavefunctions are complex-number functions, wavefunctions can add in two ways, constructive interference, A + B, and destructive interference, A - B = B - A. It is like positive and negative momentum, as in a reverberating system. Both superpositions are possible phase-space states, with probabilities.
particle systems: normalization
For linear equations, dividing all terms by any number results in equivalent equations, with same equation solutions. Dividing by any number only changes term coefficients/weights. Therefore, only coefficient/weight ratios determine equation meaning. Making sum of squared coefficients/weights equal one makes total probability 1 = 100% (normalization). Normalizing weights reflects the physical meaning of quantum-mechanics linear equations, that all state probabilities add to 100%.
particle systems: action at a distance
Physical processes can create two particles simultaneously, making two-particle systems. Both particles share one system wavefunction and have related energy levels. Measurement on one particle immediately affects system wavefunction and, by conservation laws, determines states of both particles, even if other particle is far away. State determination happens faster than light speed (action at a distance), appearing to send information faster than light and so violate relativity. However, observer knowledge of newly determined state happens only after information travels at light speed back to observer, so relativity is intact.
particle systems: energy partitioning
As in classical mechanics, for a specific total energy, quantum-mechanics phase-space wave interactions transfer total energy among wave frequencies so that energy distribution (Boltzmann distribution) is the wave-frequency distribution with maximum number of system states. Only waves with frequencies that make their energy less than half total energy can be in the distribution.
wave-particle duality
Quantum mechanics combines ideas about particles and waves (wave-particle duality). Particles have energies. Waves have wavelengths and positions. To calculate energies, quantum mechanics uses particle properties. To calculate positions, quantum mechanics uses wave properties.
waves
Quantum-mechanical particle phase-space waves extend infinitely in space and time. Wave equations have no initial conditions or boundaries.
waves: wave packets
Superposing many similar-frequency waves cancels amplitudes in most places but increases amplitude in a small space-time interval (wave packet). Wave packets are particles. Particles have many similar-frequency waves. Wave-packet frequency varies directly with particle energy. Wave-packet amplitude varies directly with particle-wave phase range, because narrower phase range makes higher amplitude.
waves: particle energies
Waves have frequency and wavelength. Quantum-mechanics wave-equation solutions are particle wavefunctions. Periodic solutions are true at lowest frequency and at all integer multiples (harmonics/overtones) of that frequency, because those waves have the same phase. Other frequencies have different phases and are not solutions. Wavefunction frequency varies directly with particle energy. Waves with higher frequency have higher energy, because field change is more when wavelength is less, so momentum change is more. Therefore, wavefunctions represent a series of possible particle energies. Energy levels differ by energy quanta. Particle-wave frequencies and energies are not continuous but discrete.
waves: position
Waves have frequency and wavelength and occupy space and time intervals. Waves cannot be at points. Particles and fields are waves and so have no definite position.
waves: shapes
Reverberations cause resonance and standing waves. Standing waves have different shapes depending on space boundaries and spin and orbit rotations. String waves have nodes at ends. For overtones of fundamental frequency, string waves have nodes at regular intervals. Molecule electrons have spherical s orbits and p, d, and f orbits with nodes. Wave shapes reflect average field density and probabilities that particle is in those space regions. In dynamic systems, wave shapes can vary over time.
waves: uncertainty
If wave has higher wavelength, position interval is wider and is less certain, but momentum and energy change, measured as less steep wave slope, is slower and so more certain. If wave has lower wavelength, position interval is narrower and is more certain, but momentum and energy change, measured as steeper wave slope, is faster and so less certain. Therefore, both position and momentum cannot be specific, and one or the other, or both, are uncertain.
waves: complex-number exponential functions
Wavefunctions are sine or cosine (trigonometric) functions. For frequency f, amplitude A, and position x, field = A * sin(2 * pi * f / x). For period t, field = A * sin(2 * pi * x / t). For wavelength l, field = A * sin(2 * pi * x / l).
Because e^(i*a) = cos(a) + i * sin(a), where a is angle in radians and is real, wavefunctions are complex-number exponential functions. sin(a) = (e^i * a - e^-i * a) / (2*i). cos(a) = (e^i * a + e^-i * a) / 2. Phase-space particle wavefunctions can superpose constructively and destructively, because they are complex-number functions.
quantum mechanics and projective geometry
Quantum mechanics has elements of projective geometry, which account for its non-local properties because projective geometry has no distance or between-ness. (General relativity is about metric geometry. Quantum mechanics is relativistic, so it also has metric properties.)
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Date Modified: 2022.0224