The quantum mechanics wave equation, which relates kinetic and potential energy to total energy, has complex-number, single-valued, continuous, and finite solutions {wavefunction}|. The wave equation, and its wavefunction solutions, are about abstract phase space, which includes space-time and describes system momenta and position or energy and time states. Wavefunctions represent possible physical-system energy levels and positions, and their probabilities. Wavefunctions correlate particle energies and times or particle momenta and positions. Wavefunctions typically depend on position, because energy includes potential energy. Wavefunctions typically depend on time, because energy includes kinetic energy. Wavefunctions are not physical waves and have no energy or momenta, but mathematically represent system properties.
Wavefunction is about infinite-dimensional abstract Hilbert space, in which wavefunction rotates as a unitary function and is deterministic.
energy and frequency
Because particle matter waves resonate in physical systems, wavefunctions have fundamental frequency and harmonics of fundamental frequency. System energy levels depend on wavefunction frequencies. System energy levels are discrete, and quanta separate energy levels. High-frequency waves have high energies. System boundary conditions set used or injected energy and wave fundamental frequency and harmonics of fundamental frequency.
amplitude, intensity, and probability
Wavefunction amplitudes are complex numbers that reflect physical-system position, time, energy, or momentum relations. Probabilities that particles are at locations depend on wavefunction amplitude for that location. Probabilities are linear and add, so probability of a set of states is sum of state probabilities. Wavefunction amplitudes can normalize, so sum of all state probabilities is one.
Intensity is absolute value of wavefunction-amplitude squared: wavefunction complex conjugate times position vector times wavefunction. Squared amplitude eliminates imaginary numbers and so is only real numbers. Absolute value makes only positive numbers. Intensities and energies are only discrete real positive numbers (eigenvalue). Amplitude squared absolute value relates to particle cross-section, collision frequency, and scattering-angle probabilities, and so to state probabilities.
wavelength
Waves have wavelength and so cannot be at a point but must spread over one wavelength. Particles have wave properties and can be at any point in region one-wavelength wide. Regions have wave amplitudes and so probabilities that particle is there.
resonance
In systems, reflected matter waves add constructively, and superpositions make standing-wave harmonic frequencies. Other frequencies cancel. Resonating fundamental wave has wavelength equal to system length or diameter and lowest-frequency. Fundamental-wave harmonic frequencies determine discrete possible particle energy levels.
deterministic
Wavefunctions are deterministic.
one particle
A one-particle system has a fundamental matter wave and its harmonics that determine possible particle positions and momenta. Harmonic wavefunctions are orthogonal/independent and linearly superpose. For particles with small momentum range and small position range moving along a straight line, wavefunctions are helices around the line with almost no amplitude at line ends and rising amplitude then falling amplitude near particle location.
one particle: definite momentum
For definite particle momentum along a straight line, position wavefunctions are helices around the line. If particle is at a well-defined position, helical waves have short wavelengths. If particle is at widespread positions, helical waves have long wavelengths.
one particle: no momentum
If particle has no momentum, momentum wavefunction is a straight line, and position wavefunction is constant.
one particle: definite position
For definite particle position along a straight line, momentum wavefunctions are helices around the line. If particle has high momentum, helical waves have short wavelengths. If particle has low momentum, helical waves have long wavelengths.
one particle: no position
If particle can be anywhere along a straight line, position wavefunction is a straight line, and momentum wavefunction is constant.
Physical Sciences>Physics>Quantum Mechanics>Wavefunction
5-Physics-Quantum Mechanics-Wavefunction
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224