Uncertainty relations     

Uncertainty relations are fundamental relations of quantum mechanics that establish the accuracy limit for the simultaneous determination of the so-called additional physical quantities that characterize a system (for example, coordinate and momentum). In a simplified formulation, these relations state that additional physical quantities cannot be precisely determined at the same time.

The uncertainties of the relationship are a consequence of the dual, particle-wave nature of the particles of matter, a reflection of the probabilistic (statistical) essence of quantum mechanics.     The uncertainties of the relationship have the form of inequalities, for example, ΔxΔp> ћ = h / 2π, where Δx is the uncertainty of the coordinate (particle or system), Δp is the uncertainty of its momentum, and h = 6.6 · 10-34 Js. S = 4.1 · 10-15 e.s. - Planck's constant. This shows that the product of the uncertainties of the coordinate and momentum cannot be less than ћ, and this boundary cannot be overcome by any improvement in the methods of observation. An increase in the accuracy of determining the coordinate inevitably leads to a loss of accuracy in determining the momentum. The maximum accuracy of the simultaneous determination of the coordinate and momentum is given by the relation Δx · Δp ≈ ћ.    

 Another important pair of additional physical quantities is energy E and time t. The uncertainty relation for them has the form ΔЕ · Δt> ћ. This relation for a relativistic system or particles (moving at a speed close to the speed of light c) can be obtained from the uncertainty relation for the coordinate and momentum by a simple transformation: Δx / s · Δpс = ΔtΔЕ> ћ. The obtained ratio for time and energy can be interpreted as follows. In order to determine the energy of a particle (system) with an accuracy of ΔЕ, it is necessary to carry out measurements over a period of time Δt> ћ / ΔЕ. A consequence of this relationship is the possibility of virtual (unobservable) processes that underlie the mechanism of particle interaction in quantum field theory. The two particles interact, exchanging with a violation of the energy balance by ΔЕ a virtual (unobservable) interaction carrier that exists for a time Δt <ћ / ΔЕ.     

Another interpretation of the relation ΔЕΔt ≈ ћ is related to the concept of the lifetime of an unstable (decaying state of a system or particle). So, if a quantum system in a discrete energy state lives on average time τ ≈ Δt, then the energy width of the level Г is given by the relation Г ≈ ΔЕ ≈ ћ / Δt ≈ ћ / τ.     

Due to the extreme smallness of the Planck constant ћ, the uncertainty relations play practically no role for macroscopic bodies.