Particle energy E and particle phase-space wave frequency f (in cycles per second) are directly proportional by constant h {Planck constant}| {Planck's constant}: E = h * f, so h = E / f. Planck constant unit is energy times time, and action in physics is energy times time, so Planck constant is quantum of action {quantum of action} {action quantum}. h = 6.626 * 10^-34 Joule-seconds or 4.136 * 10^-15 eV-s.
Particle momentum p and particle phase-space wave wavelength w are inversely proportional by Planck constant: h = p * w. For light, E = h * f = h * c / w, so h = p * w. Momentum times distance is action in physics.
For angular frequency, radians per second, Planck constant divides by 2 * pi {reduced Planck constant} {Dirac constant} {h-bar}: h-bar = h / (2 * pi). h-bar is the quantum of angular momentum.
In quantum mechanics, phase space includes particle positions and momenta and so includes physical space. Particle systems have phase-space waves that determine probabilities of particle positions and momenta at times. In bounded space regions, such as atoms, molecules, and boxes, particles have resonating phase-space waves, with stationary points at boundaries, whose frequencies are harmonics. For example, a particle in a box has phase-space waves, with stationary points at box walls, which have fundamental frequency, twice fundamental frequency, thrice fundamental frequency, and so on. Phase-space wave frequencies determine energies, so system energies are discrete and in series: E0, E1, E2, and so on. Energy-level differences are quanta that are functions of fundamental frequency.
Because energy has quanta, momentum and angular momentum (including spin) have quanta. Electron experiments have determined the angular-momentum quantum unit to be h-bar / (2)^0.5. Momentum has quantum: h / (phase-space wave wavelength). Energy has quantum: h * (phase-space wave frequency). Electron experiments have determined that action has quanta, so energy times time, and momentum times distance, have quanta.
Because a continuous quantity times a discontinuous quantity would make a continuous quantity, for action to have quanta, time and length must have quanta. The quantum-mechanical uncertainty principle depends on particle-wave properties, relates indeterminacies in particle energy and time (or momentum and position), and so relates energy uncertainty to time uncertainty: dE * dt >= h. In space-time, maximum particle energy is where particle gravity has quantum effects and makes space-time discontinuous: 1.22 * 10^19 GeV. By the uncertainty principle, minimum time is then 10^-43 seconds (and minimum length is 10^-35 meters).
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Date Modified: 2022.0224