In classical mechanics, particles have definite physical-space positions and momenta (velocities) through time. Particles have trajectories through physical space-time. For one-particle systems in physical three-dimensional space, classical configuration spaces have six continuous, infinite, and orthogonal dimensions: three for position and three for momentum. Classical configuration spaces have trajectories of successive states.
In quantum mechanics, particles do not have definite physical-space positions and momenta through time. Particle positions and momenta are functions of system energy, momenta, position, and time. Particles do not have trajectories through physical space-time but can be at any position and any momentum in physical space-time. For one-particle systems in physical three-dimensional space, quantum-mechanics configuration spaces have six continuous, infinite, and not necessarily orthogonal dimensions. Quantum-mechanics configuration-space points have scalar displacements that can vary over time. Frequency varies directly with particle energy. Adjacent-point scalar displacements vary over a wavelength. Wavelength varies inversely with particle momentum. Matter waves do not propagate or travel and so have no energy and are scalar waves. Maximum displacements (amplitudes) differ at different points, varying with system energy, momenta, position, and time. Matter waves have complex-number amplitudes because space-time has time coordinate of opposite sign from space coordinates, because of energy and momentum conservation, and because complex-number amplitudes result in constant-amplitude waves. Real-number waves travel outward and lose amplitude with distance. In the complex plane, multiplying by i rotates pi/2 radians (90 degrees). Complex-numbers represent rotation, frequency, phase, and magnitude.
Constants can be matrices.
Quantum-mechanics complex-number wave equations {Schrödinger equation}| relate energies and times. Schrödinger equations are similar to diffusion equations, but with a complex-number term, which makes them wave equations. Schrödinger equations require an imaginary term because they are about space-time and time has opposite sign to space components. Hermitian operators act on possible system-state Hilbert space to define observable quantities. Operator eigenvalues are possible physical-quantity measurements. Hamiltonian is total system-energy operator.
Isolated systems have constant total energy. By energy conservation, Schrödinger equations set constant total energy equal to potential energy plus kinetic energy. Potential energy varies with position. Kinetic energy varies with momentum. For waves, kinetic energy E varies directly with frequency f, and momentum p varies inversely with wavelength l: E = hf and p = h/l. Potential energy Wavefunction solutions represent system energy-level probabilities.
phase space
Physical systems have particles within boundaries. Particles have positions and momenta. Abstract phase space represents all particle positions and momenta. Particles deterministically follow trajectories through phase space. Particles have a succession of states (state vector) in phase space.
matter waves
Particles have matter waves. Matter waves resonate in phase space with harmonic wavelengths. Matter waves describe particle trajectories through phase space.
matter waves and particle energies
Matter-waves have frequencies, which determine particle energies. Waves must have frequency to be waves, so wave energy cannot be zero. Lowest-frequency resonating fundamental wave has lowest ground-state energy. Resonating waves also have fundamental-frequency overtones. Wave frequencies are not continuous but discrete. Particle energy levels are not continuous but discrete and separated by energy quanta. Energy-level differences decrease with higher frequency. Higher frequency waves have higher energy and have lower probability. Wave frequencies can increase indefinitely.
transitions
Schrödinger equations describe conservation of energy in particle systems and phase spaces and relate particle energies and times. Schrödinger equations have wavefunction solutions that define possible different-probability particle energy levels over time.
Schrödinger equations are about particle energy-level transitions. Particle can go from one energy level to another along infinitely many paths. For example, particle can go directly from one energy level to another {direct channel} or go to higher energy level and then drop down to lower energy level {cross channel}. Particles have matter waves, and each transition changes matter waves to a different frequency. For cross channels, net transition is superposition of matter-wave transitions.
transitions: probability
Going from one energy level to another has a probability that depends on energy difference and starting energy. Schrödinger-equation wavefunction solutions have transition complex-number amplitudes. For cross channels, total amplitude is complex-number sum of all transition amplitudes. Transition probabilities are absolute values of squared amplitudes. Squaring complex numbers makes real numbers. Absolute values make positive numbers. Therefore, transition probabilities are positive real numbers.
transitions: renormalization
Because number of paths is infinite, transition-probability sum seems infinite. However, higher frequencies have lower probabilities, so amplitude renormalization can make probability sum equal 1 = 100%.
energy
Potential energy PE is force F from field E times distance ds: PE = F * ds = E * H, where H is wavefunction. Kinetic energy KE depends on mass m and velocity v: KE = 0.5 * m * v^2 = 0.5 * (1/m) * p^2, where momentum p = m * v. Momentum squared is (h / (2 * pi))^2 times second derivative of wavefunction, because momentum squared depends on velocity squared: KE * H = 0.5 * (1/m) * (h /(2 * pi))^2 * (d^2)H / (dx)^2, where H is wavefunction, (d^2) is second derivative, h is Planck constant, p is momentum, and m is mass. Schrödinger equation sets sum of wavefunction potential-energy and kinetic-energy operators equal to wavefunction total energy operator {Hamiltonian operator}.
operators
Momentum over position, or energy over time, is physical action. Momentum and position operators, or energy and time operators, are commutative.
time
Wavefunctions can change over time (time-dependent Schrödinger equation).
frequency
Frequency is partial derivative of wavefunction with time.
spin
Schrödinger equation does not include particle spin, because waves cannot account for spin.
relativity
Schrödinger equation does not include relativistic effects, because waves cannot account for relativity.
Physical Sciences>Physics>Quantum Mechanics>Theory>Equations
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Date Modified: 2022.0224