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Uncertainty principle mathematically follows from definition of Operators in Quantum Mechanics . It is often confused with the Observer Effect . OVERVIEW Sinusoidal Wavefunction of a quantum particle in the third Energy Level of a one-dimensional Infinite Potential Well .]] The concept of Probability Distribution s pervades the science of Measurement . Until the beginning of the discovery of quantum physics, it was thought that the only uncertainty in measurement was caused by the limitations of a measuring tool's precision. But it is now understood that no treatment of any scientific subject, Experiment , or measurement is said to be Accurate without disclosing the nature of the probability distribution (sometimes called the Error ) of the measurement. Uncertainty is the characterization of the relative narrowness or broadness of the distribution function applied to a physical Observation . Illustrative of this is an experiment in which a Particle is prepared in a definite State and two successive measurements are performed on the particle. The first one measures the particle's Position and the second immediately after measures its Momentum . Each time the experiment is performed, some value ''x'' is obtained for position and some value ''p'' is obtained for momentum. Depending upon the Precision of the instrument taking the measurements, each successive measurement of the positions and momenta respectively should be nearly identical, but in practice they will exhibit some deviation due to constraints of measurement using a real world instrument that is not infinitely precise. However, Heisenberg showed that, even in theory with a hypothetical infinitely precise instrument, no measurement could be made to arbitrary Accuracy of both the position and the momentum of a physical object. The Heisenberg uncertainty principle (developed in an essay published in 1927) 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 Δ''x'', then the second measurement will have a distribution of values whose dispersion Δ''p'' is at least Inversely Proportional to Δ''x''. For the limiting case, the constant of proportionality is derivable using Commutator arithmetic. It is equal to Planck's Constant divided by 4. This stipulates that the product of the uncertainties in position and velocity is equal to or greater than about Joule - Second s. Therefore, the product of the uncertainties only becomes significant for regimes where the uncertainty in position or momentum measurements is small. Thus, the uncertainty principle governs the observable nature of atoms and subatomic particles while its effect on measurements in the macroscopic world is negligible and can be usually ignored. The Heisenberg uncertainty relations are a theoretical bound over all measurements. They hold for so-called ideal measurements, sometimes called Von Neumann measurements. They hold even more so for non-ideal or Landau measurements. WAVE-PARTICLE DUALITY AND THE RELATIONSHIP TO THE UNCERTAINTY PRINCIPLE A fundamental consequence of the Heisenberg Uncertainty Principle is that no physical phenomena can be to arbitrary accuracy described as a "classic point particle" or as a Wave but rather the microphysical situation is best described in terms of Wave-particle Duality . The uncertainty principle, as initially considered by Heisenberg, is concerned with cases in which ''neither'' the wave nor the point particle descriptions are fully and exclusively appropriate, such as a Particle In A Box with a particular energy value. Such systems are characterized ''neither'' by one unique "position" (one particular value of distance from a potential wall) ''nor'' by one unique value of momentum (including its direction). Any Observation that determines either a position or a momentum of such a waveparticle to arbitrary accuracy - known as Wavefunction Collapse - is subject to the condition that the width of the wavefunction collapse in position, multiplied by the width of the wavefunction collapse in momentum, is constrained by the principle to be greater than or equal to Planck's constant divided by 4. Every measured particle in quantum mechanics exhibits wavelike behaviour so there is an exact, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals. For example, in a time-varying Signal such as a Sound Wave , it is meaningless to ask about the Frequency spectrum at a single moment in Time because the measure of frequency is the measure of a repetition recurring over a period of time. In order to determine the frequencies accurately, the signal needs to be sampled for a finite (non zero) time. This necessarily implies that time precision is lost in favor of a more accurate measurement of the frequency spectrum of a signal. This is analogous to the relationship between momentum and position, and there is an equivalent formulation of the uncertainty principle which states that the uncertainty of Energy of a wave (directly proportional to the frequency) is inversely proportional to the uncertainty in time with a constant of proportionality identical to that for position and momentum. Common incorrect explanation of the uncertainty principle The uncertainty principle in Quantum Mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Heisenberg himself may have initially offered explanations which suggested this view. That this disturbance does not describe the essence of the uncertainty principle in current theory has been demonstrated above. The fundamentally non-classical characteristics of the uncertainty Measurements in quantum mechanics were clarified due to the EPR Paradox which arose from Einstein attempting to show flaws in quantum measurements that used the uncertainty principle. Instead of Einstein succeeding in showing uncertainty was flawed, Einstein guided researchers to examine more closely what uncertainty measurements meant and led to a more refined understanding of uncertainty. Prior to the publication of the EPR paper in 1935, a measurement was often visualized as a physical disturbance inflicted directly on the measured system, being sometimes illustrated as a thought experiment called Heisenberg's Microscope . For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant Entangled Particle . 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 Distribution s. This is ''the'' fundamental postulate of quantum mechanics. If we compute the uncertainty Δ''x'' of the position measurements and the standard deviation Δ''p'' of the momentum measurements, then : where 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. In a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. This actually is equivalent to a circle of 360 degrees, or 2 radians. Therefore, dividing h by 2 describes a constant that when multiplied by the frequency of a wave gives the energy of one radian. Heisenberg took 1/2 of '''' as his standard deviation. This can be written as '''' over 2 as above or it can be written using simple algebra and multiplying 1/2 x h/(2) as h/(4). Normally one will see '''' over 2 as this is simpler. Two years earlier in 1925 when Heisenberg had developed his Matrix Mechanics the difference in position and momentum were already showing up in the formula. In developing matrix mechanics Heisenberg was measuring amplitudes of position and momentum of particles such as the electron that have a period of 2, like a cycle in a wave, which are called Fourier Series variables. When amplitudes of position and momentum are measured and multiplied together, they give intensity. However, Heisenberg found that when the position and momentum were multiplied together in that respective order, and then, there was a difference or deviation in intensity between them of h/(2). In other words, they did not commute. In 1927, to develop the standard deviation for the uncertainty principle, Heisenberg took the Gaussian Distribution or bell curve for the imprecision in the measurement of the position q of a moving electron to the corresponding bell curve of the measured momentum p. That gave the minimum standard deviation to be 1/2 of h/(2), or, . In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of Normally Distributed Variables , leads to a larger lower bound of h/(2) for the product of the uncertainties. Note that this inequality allows for several possibilities: the state could be such that ''x'' can be measured with high precision, but then ''p'' will only approximately be known, or conversely ''p'' could be sharply defined while ''x'' cannot be precisely determined. In yet other states, both ''x'' and ''p'' can be measured with "reasonable" (but not arbitrarily high) precision. Expression of finite available amount of Fisher information The uncertainty principle alternatively derives as an expression of the Cramér-Rao Inequality of classical measurement theory. This is in the case where a particle position is measured. See Stam (1959). The mean-squared particle momentum enters as the Fisher Information in the inequality. See also Extreme Physical Information . Common observables which obey the uncertainty principle An uncertainty relation arises between ''any'' two observable quantities that can be defined by non- Commuting operators. This means that the uncertainty principle arises in measuring the position and the velocity of an object, or in measuring the position and momentum of an object.
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