Instruments can measure momentum, position, energy, and time by absorbing energy and using clocks and rulers. However, instruments cannot simultaneously or precisely measure both particle momentum and position {uncertainty principle}| {Heisenberg uncertainty principle} {indeterminacy principle}, because measuring one alters information about the other. Instruments cannot simultaneously or precisely measure both particle energy and time, because they relate to momentum and position.
situation
The uncertainty principle is about measurement precision on one particle at one time and place. The uncertainty principle does not apply to different measurements on same particle over time. The uncertainty principle does not apply to simultaneous momentum and position, or energy and time, measurements on different particles.
wave packet
Particles have wavefunctions, so measurements are about wave packets. As particle moves through time and space, total uncertainty increases, because wave packet spreads out.
wave properties
Uncertainty follows from wave properties, because wave position and momentum, or time and energy, inversely relate. Energy and momentum depend on wave frequency. Position and time depend on wave amplitude. Measuring wave frequency or wavelength precisely prevents measuring wave amplitude precisely. Measuring wave amplitude precisely prevents measuring wave frequency or wavelength precisely. If momentum or position is specific, position or momentum must be uncertain. If energy or time is specific, time or energy must be uncertain.
At space points, wavefunctions that have high amplitude have precise position and timing. However, wavefunction slope is steep, so amplitude change between nearby points is large, so velocity change, momentum change, and energy change are large and so uncertain at that position. See Figure 1.
Wavefunctions with wide wave packets have large uncertainty. Wavefunction slopes are not steep, and amplitude change at nearby points is small, so velocity change, momentum change, and energy change are small in that region. Momentum is precise, while position is imprecise. Alternatively, energy is precise, while timing is imprecise. See Figure 2.
Waves that have just one frequency and wavelength have one momentum and energy. Only one wave can have no superposition and no cancellation anywhere in space or time, making wave equally present throughout all space and time, and so completely uncertain in position and time. See Figure 3.
Wavefunctions that have almost all frequencies and wavelengths have precise position and time, because waves cancel everywhere, except one space or time point. Wavefunctiond that have almost all frequencies and wavelengths have almost all momentum and energy levels, making wave momentum and energy very uncertain. See Figure 4.
Waves that have some frequencies and wavelengths have moderate uncertainty in momentum and energy and moderate uncertainty in position and time, because waves cancel, except at moderate-size wave packet.
Waves with two or three frequencies and wavelengths have beat frequencies where waves superpose. Beat frequency makes precise momentum and energy, but time and position are uncertain. See Figure 5.
Waves with harmonic frequencies and wavelengths have beat frequencies where waves superpose. Beat frequencies make precise momentum and energy, but time and position are uncertain.
measurement processes
Besides wavefunction effects, physical processes limit precision. To find precise frequency for energy and momentum takes time and space, so position and time information are uncertain. To find precise position and time takes high amplitude, so position and time information are uncertain. Uncertainty's physical cause is discontinuity, whereas uncertainty's quantum-mechanical cause is wave-particle duality, because particles are about momentum and energy and waves are about position and time, as shown above.
mathematics
Quantum of action is h, and energy over time is action. Therefore, energy uncertainty dE times time uncertainty dt equals at least Planck constant divided by 4 * pi: dE * dt >= h / (4 * pi).
dE = F * dx = (dp / (4 * pi * dt)) * dx, so dE * dt * (4 * pi) = dp * dx. Position uncertainty dx times momentum uncertainty dp equals at least Planck constant: dx * dp >= h.
dx = 4 * pi * dF, and dp = dN / 2. Phase uncertainty dF times phonon number uncertainty dN equals Planck constant divided by 2 * pi: dF * dN = h / (2 * pi).
energy levels
Electrons in lower atomic orbitals have higher frequency, kinetic energy, and angular momentum and lower time period and orbital diameter. Electrons in higher atomic orbitals have lower frequency, kinetic energy, and angular momentum and higher time period and orbital diameter. Therefore, higher orbitals have higher position uncertainty and lower momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to adjacent higher energy level, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to adjacent lower energy level, decreasing position uncertainty and increasing momentum uncertainty.
For low-orbital and high-orbital electrons, photon absorption can cause electronic transition to non-adjacent higher energy levels, increasing position uncertainty and decreasing momentum uncertainty. For low-orbital and high-orbital electrons, photon emission can cause electronic transition to non-adjacent lower energy levels, decreasing position uncertainty and increasing momentum uncertainty.
Besides fundamental Heisenberg uncertainty, electron, proton, and neutron configuration changes affect measured amounts. Electronic transitions conserve energy, momentum, and angular momentum, so absorption and emission do not necessarily have the same photon frequency. Electrons cannot transition to same orbital.
two particles
Though instruments cannot measure either's time or energy, instruments can measure two particles' energy difference and time difference precisely and simultaneously. Such measurement can define one-ness and two-ness.
confinement
By uncertainty principle, particles confined to smaller regions or times have greater momentum and energy. In confined regions, even in vacuum, energy is high, allowing particle creation and annihilation.
matrices
In quantum mechanics, particle position and momentum are quantized and so are matrices (not scalars or vectors), with complex-number elements. Because particles have probabilities of being anywhere in space, matrix rows and columns have infinite numbers of elements, and matrices are square matrices. In quantum mechanics, position and momentum are not necessarily independent, but depend on the whole particle system.
Matrices represent electronic transitions between energy levels. Matrix rows are one energy level, and matrix columns are the other energy level. Matrix elements represent the probability of that electronic transition. Matrix elements are periodic to represent the possible quanta. The diagonal represents transitions between the same energy level and so has value zero. Near the diagonal represents transitions between adjacent energy levels and so has higher values. Far from the diagonal represents transitions between non-adjacent energy levels and so has lower values. Energy levels have ground state and no upper limit, so the matrices have infinite numbers of elements. There is no zero energy level.
For non-infinite-dimension square matrices with real elements, PQ = QP (commutative). For infinite-dimension and/or non-square and/or complex-number-element matrices, PQ <> QP (non-commutative). Matrix multiplication is typically non-commutative.
In quantum mechanics, particle action is the product of the momentum P and position Q matrices: action = PQ. For infinite-dimension square matrices with complex-number elements, PQ - QP = -i*h*I, where I is identity matrix and h is Planck constant, because action has Planck-constant units and complex number multiplication rotates the axes by pi/2 radians.
Though electrons and protons have strong electrical attraction, and outside electrical attractions and repulsions can disturb atom orbitals, electrons do not spiral into protons and collapse atoms. Because particles have matter waves, by the uncertainty principle, orbiting electrons cannot spiral into atomic nucleus {atom, stability}. See Figure 1.
waves
Particles have matter waves, whose harmonic frequencies relate to particle energy levels.
uncertainty
Waves by definition must be at least one wavelength long. Therefore, particle waves have location uncertainty of at least one wavelength. Particle waves have time uncertainty of at least one period, which is one wavelength divided by light speed. Particle waves have momentum uncertainty of at least Planck constant divided by wavelength. Particle waves have energy uncertainty of at least Planck constant divided by period. Particle waves make the uncertainty principle.
energy
By uncertainty principle, particles must move, and so they cannot have zero energy. Particles cannot have zero energy because they cannot have zero motion, because that violates conservation of both energy and momentum. Lowest particle energy is first-quantum-level ground-state energy.
orbits
Electron orbits have quantum distances from nucleus and take quantum durations to orbit nucleus. In lowest orbital, electron position uncertainty has same diameter as orbital. Electron can be anywhere in that region around nucleus. In lowest orbital, electron time uncertainty is same period as orbital rotation. Electron can be anywhere in that interval. In lowest orbital, electron is already at closest possible distance and smallest possible time.
transitions
From lowest orbital, electrons cannot go to lower orbits, because there are no lower energy levels. They cannot lose more energy, because if energy decreases then time increases, by uncertainty principle, making orbital go higher. They cannot lose more distance because if distance decreases then energy must increase, by uncertainty principle, making orbital go higher. Therefore, lowest orbital has lowest energy, smallest distance, and shortest time. Lowest orbital already includes nucleus region, so it cannot be smaller.
kinetic and potential energy
In quantum mechanics and classical mechanics, electric-field positions relate to potential energies. In quantum mechanics, unlike classical mechanics, kinetic energy cannot completely convert to potential energy, and vice versa. Kinetic energy and potential energy have minimum energy level and cannot be zero.
energy quantum
First energy quantum is difference between ground-state energy and next-highest-orbital energy. Second energy quantum is difference between next-highest-orbital energy and third-orbital energy. Energy quanta are not equal. Energy quanta decrease at higher orbitals. Energy quanta relate to wave harmonic frequencies. Higher adjacent wave frequencies have smaller energy differences.
atom nucleus
Atomic nucleus occupies only 10^-5 volume inside lowest-electron-orbital volume. Nucleus protons and neutrons have energy, momentum, position, and time uncertainty and so have ground-state energies. Nucleus protons and neutrons have quantum energy levels.
Lowest-orbital electrons and highest-orbital neutrons and protons never collide, because electrons have lower orbiting energies, and higher orbital radii, than neutrons and protons.
electron-proton collision
At high-enough energy and beam collimation, electrons can collide with atomic nuclei, because increased energy can narrow position, by uncertainty principle. Such electrons are not orbiting, so this situation is not about atom stability.
Particle in enclosed space {particle in box} must have velocity, because particle has fixed position, so uncertainty is in momentum. If enclosed space is smaller, velocity must be more.
Electric field and magnetic field cannot be at rest {quantum fluctuation}, because then they have precise position and precise zero momentum and so violate uncertainty principle. All fields have random motion, even in vacuum where net energy is zero.
At quantum level, empty-space field fluctuation {vacuum polarization}| is infinite.
Two parallel uncharged metal plates attract each other by reducing vacuum-energy fluctuations and number of wavelengths between them {Casimir effect} {Casimir force}: energy density = c / d^4, where c is constant and d is plate distance. Energy at plate is zero. Interior energy density decreases, so exterior energy density increases and pushes plates together. Fewer particle histories with closed time-like loops are between plates.
Particles cannot be at rest {zero point motion}|, because then they have precise position and precise zero momentum and so violate uncertainty principle. All particles have random motion, even in vacuum where net energy is zero.
5-Physics-Quantum Mechanics-Waves
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Date Modified: 2022.0225