In quantum physics, the Heisenberg uncertainty principle or just Uncertainty principle (sometimes also the Heisenberg indeterminacy principle - a name given to it by Niels Bohr) states that one cannot measure values (with arbitrary precision) of certain conjugate quantities, which are pairs of observables of a single elementary particle. These pairs include the position, and momentum. Mathematics provides a positive lower bound for the product of the uncertainties of measurements of the conjugate quantities. The uncertainty principle is one of the cornerstones of quantum mechanics and was discovered by Werner Heisenberg in 1927. The principle follows from the mathematical definition of operators in quantum mechanics; it is represented by a set of theorems of functional analysis. It is often confused with the observer effect. Two-dimensional graphical representation of the Heisenberg Uncertainty Principle for a Gaussian sinusoidal wave-function of a quantum particle in the third energy level of a one-dimensional infinite potential well. The Uncertainty Principle was developed as an answer to the question: How does one measure the location of an electron around a nucleus? The Heisenberg uncertainty principle provides a quantitative relationship between the uncertainties of the hypothetical infinitely precise measurements of p and x as measured by the sizes of their distributions in the following way: If the particle state is such that the first measurement yields a dispersion of values Δx, then the second measurement will have a distribution of values whose dispersion Δp is at least inversely proportional to Δx. For the limiting case, the constant of proportionality is derivable using commutator arithmetic. It is equal to Planck's constant divided by 4π. Every measured particle in quantum mechanics exhibits wavelike behavior, so there is an exact, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals Formulation and characteristics Measurements of position and momentum taken in several identical copies of a system in a given state will vary according to known probability distributions. This is the fundamental postulate of quantum mechanics. If we compute the uncertainty Δx of the position measurements and the standard deviation Δp of the momentum measurements, then Where is the reduced Planck's constant (Planck's constant divided by 2π). Heisenberg did not just use any arbitrary number to describe the minimum standard deviation between position and momentum of a particle. Heisenberg knew that particles behaved like waves and he knew that the energy of any wave is the frequency multiplied by Planck's constant. The term Copenhagen interpretation of quantum mechanics was often used interchangeably with and as a synonym for Heisenberg's Uncertainty Principle by detractors who believed in fate and determinism and saw the common features of the Bohr-Heisenberg theories as a threat. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics (i.e. it was not accepted by Einstein or other physicists such as Alfred Landé), the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form—but rather as a collection of probabilities, or potentials. The Uncertainty Principle is frequently, but incorrectly, confused with the "observer effect", wherein the observation of an event changes the event. The observer effect is an important effect in many fields, from electronics to psychology and social science. In software programming, a Heisenbug is a software error that disappears or alters its characteristics when it is researched.
