Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
These are excerpts and elaborations from my book "The Nature of Consciousness"
In classical Physics, a quantity (such as the position or the mass) is both an attribute of the state of the system and an observable (a quantity that can be measured by an observer). Quantum Theory makes a sharp distinction between states and observables. If the system is in a given state, an observable can assume a range of values (so called “eigenvalues”), each one with a given probability. The evolution over time of a system can be viewed as due (according to Heisenberg) to time evolution of the observables or (according to Schroedinger) to time evolution of the states.
An observer can measure at the same time only observables that are compatible. If the observables are not compatible, they stand in a relation of mutual indeterminacy: the more accurate a measurement of the one, the less accurate the measurement of the other. Position and momentum are, for example, incompatible. This is a direct consequence of the wave-particle dualism: only one of the two natures is "visible" at each time. One can choose which one to observe (whether the particle, that has a position, or the wave, that has a momentum), but cannot observe both aspects at the same time.
Precisely, Heisenberg’s famous "uncertainty principle" states that there is a limit to the precision with which we can measure, at the same time, certain pairs of quantities, notably the momentum and the position of a particle. If one measures the momentum, then it cannot measure the position, and viceversa. Technically speaking: the product of uncertainties in position and in momentum must be greater than Planck’s constant. This is actually a direct consequence of Einstein's equation that related the wavelength and the momentum (or the frequency and the energy) of a light wave: if coordinates (wavelength) and momentum are related, they are no longer independent quantities. Einstein never believed in this principle, but he was indirectly the one who discovered it.
A similar principle applies to other incompatible observables, for example between time and energy: one cannot measure energy precisely at a precise instant in time. Either the time or the amount of energy has to be imprecise. Hence, in theory, violations of energy conservation can occur… but we cannot observe them. The more the energy missing (unaccounted for), the faster it will be returned (the shorter the period of time before it is accounted for). If you try to measure energy at a precise time, then no information is known on how much energy is there.
The wave function contains the answers to all the questions that can be asked about a system, but not all those questions can be asked simultaneously. If they are asked simultaneously, the replies will not be precise.
The degree of uncertainty is proportional to the Planck constant. This implies that there is a limit to how small a physical system can be, because, below a quantity proportional to the Planck constant and called "Planck length", the physical laws of Quantum Theory stop working altogether. The Planck scale (10–33 cm, i.e. the shortest possible length, and 10–43 sec, i.e. the time it takes for a light beam to cross the Planck length, i.e. the shortest possible time tick) is the scale at which space-time is no longer a continuum but becomes a grid of events separated by the Planck distance. What happens within a single cell of the grid, is beyond the comprehension of Physics. As the US physicist John Wheeler suggested in the 1950s, even the very notions of space and time stop making sense in this "quantum foam".
Note that Heisenberg does not forbid precise measurements of "compatible observables", for example of position, charge and spin. It only applies to "incompatible observables", which are couples: position/momentum, energy/time, electric field/magnetic field, angle/angular momentum, etc.
The uncertainty predicted by Quantum Theory (and verified by countless experiments in countless laboratories) has been sometimes interpreted as a consequence of the fact that, at the microscopic level, one cannot pretend that a measurement is “objective” at all: a measurement is an interaction between two systems, which, like all interactions, affects both. But that is not quite where Heisenberg’s calculations came from. They originate, as everything else, from Planck’s constant.
For the record, there had been other “principles of uncertainty” in Physics, and an important one in Mathematics, the one discovered by Joseph Fourier in the 19th century that a signal cannot be simultaneously localized both in time and in frequency: for example, there is a limit to the precision of the simultaneous measurement of the duration and frequency of a sound.
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