Abstract Euclidean or non-Euclidean space {configuration space} {phase space, quantum mechanics} can have any number of dimensions and discrete or continuous points, with vectors from origin to points.
physical space and classical configuration space
Particles have center-of-gravity positions and momenta. In three-dimensional physical space, particle positions have three coordinates. Positions are real numbers, over an infinite range. In three-dimensional physical space, particle momenta have three coordinates. Momenta are real numbers, over an infinite range. Classical configuration space has six dimensions for each particle. In three-dimensional physical space, one particle has six-dimension configuration space: three dimensions for space coordinates and three dimensions for momentum coordinates. Two particles have twelve-dimension configuration space. For an N-particle system, classical configuration space has 6*N dimensions. Systems must have a finite number of particles, because universe is not infinitely big. Classical configuration space has Euclidean topology.
Phase space represents particle positions and momenta. For one particle, particle physical-space position coordinates can be the same as particle configuration-space position coordinates. For more than one particle, particle physical-space position coordinates are put on different configuration-space dimensions. For one particle, particle physical-space momentum coordinates are the same as measured in physical space at that position. For more than one particle, particle physical-space momentum coordinates are put on different configuration-space dimensions. In general, configuration space includes physical space for only one particle.
Particle positions and momenta are independent dimensions, because particles are independent. In classical physical space, a particle has a real-number density function, and particles have independent real-number density functions that add to make system density function.
To simplify, assume one particle and that the y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along the x-axis. For no external forces and so constant momentum, configuration space has a straight-line trajectory parallel to the x-axis. For constant external force in the positive direction along the x-axis and so increasing momentum, configuration space has a straight-line trajectory with positive slope to the x-axis. For two particles under the same conditions, configuration space has four independent dimensions and two independent straight-line trajectories.
To account for rotations and angular momenta, configuration space can have three more dimensions for each particle.
quantum mechanics
In quantum mechanics, particle positions and momenta have three complex-number coordinates. Configuration space has six dimensions for each particle, but each dimension has two dependent components: real and imaginary. If particles interact, particle dimensions are not independent. For example, when processes create two photons, photon spins entangle.
In quantum-mechanics configuration space, the system density function is not the sum of particle complex-number wave functions. Quantum-mechanical configuration space has non-Euclidean topology.
states
Configuration-space points represent all possible physical-system states. Assume one particle and that y-axis and z-axis positions and momenta are zero, so configuration space has x-axis perpendicular to x-momentum-axis. Assume that one particle moves in the positive direction along x-axis. For no external forces and so constant momentum, quantum-mechanical configuration space has evenly-spaced points along a straight-line trajectory parallel to x-axis. For constant external force in the positive direction along x-axis and so increasing momentum, quantum-mechanical configuration space has unevenly-spaced points along a straight-line trajectory with positive slope to x-axis. Assume that particle is inside a box, and particle has elastic collisions with box walls, then particle has higher probability of being in the box than outside.
Number of possible states is infinite, because matter waves are infinitely long, because configuration-space dimensions are infinite. Particle positions are anywhere along dimension, because matter waves are infinitely long. Particle momenta are anywhere along dimension, because mass can increase indefinitely.
states: lattice
In continuous physical space, number of positions is infinite. Using a lattice of points, separated by a fixed distance, makes number of positions over an interval finite, for computer calculation.
time
Over time, system coordinates stay orthogonal, and states that are orthogonal stay orthogonal. Scalar products stay constant {unitary evolution, spaces}. Relations between vectors do not change.
time: steps
Over continuous time, number of times is infinite. Using time steps, separated by a fixed interval, can make number of times over an interval finite.
momentum or energy levels
Over continuous momentum or energy, number of levels is infinite. Using quanta, separated by a fixed interval, can make number of levels over an interval finite.
spin angular momentum levels
Spin angular momenta can be 0, +1/2, -1/2, 1, -1, +3/2, -3/2, and so on. For particle systems, total spin angular-momentum levels can be 0 (0, +1/2, -1/2, 1, -1, +3/2, or -3/2, and so on), 2 (+1/2 or -1/2), 3 (+1, 0, or -1), 4 (+3/2, +1/2, -1/2, or -3/2), and so on.
waves
Classical configuration space has no matter waves, because it has only real numbers and so no real-number/imaginary-number interactions. Quantum-mechanical configuration space has complex numbers and resonating matter waves. Complex-number wavefunctions represent all possible particle positions and momenta, or energies and times, and their probabilities. Matter waves cause space, time, energy, and momentum quanta and the uncertainty principle. Possible configuration-space points are possible particle states (state vector), because they are wavefunction solutions. Matter waves only relate to electromagnetic waves for a system with one photon. Matter waves are not in physical space, do not travel, and have no energy.
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Date Modified: 2022.0224