For objective measurement, events {observable} must be independent of where or when they happen. Objective measurements cannot be functions of space or time coordinates.
Measurements need reference points, such as x=0, and measurement units, such as meter. By relativity, objective measurements cannot be functions of reference points or units.
Measured state is orthogonal to all other possible states, because if one state happens, others do not. Measured state can be along coordinate {primitive measurement}.
Measurements in systems with no waves, or with waves with no phase differences, can have any order {commuting measurement}. Primitive measurements commute, because they are not about phase, only about yes or no. Measurements in systems with waves and phase differences depend on sequence {non-commuting measurement}. Most measurements do not commute, because they find value or probability.
subjective measurement
In quantum mechanics, time and space are not continuous but have quanta. In phase space, momenta relate to positions, and energies relate to times, so events are functions of space and time coordinates. Because positions and lengths relate to momenta, events are functions of reference points and units. Objective measurement is not possible. Quantum mechanics has only subjective measurement.
interaction
To measure particle size, light must have wavelength less than particle diameter and so high frequency and energy. High energy can change particle momentum. Higher energy increases momentum uncertainty.
To measure particle momentum, light must have low energy, to avoid deflecting particle, and so long wavelength. Longer wavelength increases location uncertainty.
Measuring position requires different-frequency light wave than measuring momentum, so experiments cannot find both position and momentum simultaneously (uncertainty principle).
wavefunction collapse
Measuring disturbs particle and creates a new system of observer, instrument, and particle, with a new wavefunction. At actual measurement, the new system wavefunction collapses to zero. Measuring allows observing only one particle property.
operator
Momentum, energy, angular momentum, space, or time functions {operator, wavefunction} operate on wavefunction to find discrete positive real values (eigenvalue) of momentum, energy, angular momentum, space, or time, which are all possible outcomes, each with probability. Direct measurements project onto space or time coordinate or energy or momentum vector.
Projection operators operate on wavefunction and project onto space or time coordinate, or energy or momentum vector, to give discrete positive real values (eigenvalue) {direct measurement}, which are all possible outcomes, each with probability. Experimenters can know possible measured values and predict probabilities. However, values may be less than quantum sizes and so not measurable. Operators have same dimensions as particle.
Alternatively, experimenters can prepare a quantum system in a known initial state, have particle interact with prepared quantum system, separate particle and prepared quantum system, and then measure quantum-system state {indirect measurement}. Indirect measurements require entangling particle and prepared quantum system, to couple their states. Wavefunction collapse puts quantum system into a state that indirectly determines particle state. Quantum system can have same or more dimensions as particle.
Operators on wavefunctions produce discrete positive real values (eigenvalue) {positive-operator-valued measure} {positive operator-valued measure} (POVM).
Operators {projection operator, measurement} on wavefunctions can project values onto measurement axis.
5-Physics-Quantum Mechanics-Wavefunction-Collapse-Measurement
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Date Modified: 2022.0225