Information AboutHeisenberg Uncertainty Principle |
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In Quantum Physics , the outcome of even an ideal Measurement of a system is not Deterministic , but instead is characterized by a Probability Distribution , and the larger the associated Standard Deviation is, the more "uncertain" we might say that that characteristic is for the system. The Heisenberg uncertainty principle gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously. More precisely, the product of the standard deviations , where is the Reduced Planck Constant . The principle generalizes to many other pairs of quantities besides position and momentum (for example, angular momentum about two different axes), and can be derived directly from the Axioms of quantum mechanics. Note that the uncertainties in question are characteristic of the mathematical quantities themselves. In any real-world measurement, there will be ''additional'' uncertainties created by the non-ideal and imperfect measurement process. The uncertainty principle holds true regardless of whether the measurements are ideal (sometimes called Von Neumann measurements) or non-ideal ( Landau measurements). Note also that the product of the uncertainties, of order 10−35 Joule - Second s, is so small that the uncertainty principle has negligible effect on objects of Macroscopic scale, despite its importance for atoms and subatomic particles. The uncertainty principle was an important step in the development of Quantum Mechanics when it was discovered by Werner Heisenberg in 1927 . It is often confused with the Observer Effect . WAVE-PARTICLE DUALITY See Also: Wave–particle duality A fundamental postulate of quantum mechanics, which manifests itself in the Heisenberg Uncertainty Principle, is that no physical phenomenon 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 Heisenberg uncertainty principle is a consequence of this picture. The amplitude of the wave associated with a particle corresponds to its position, and the wavelength (more precisely, its Fourier Transform ) is inversely proportional to Momentum . In order to localize the wave so as to have a sharp peak (i.e., a small position uncertainty), it is necessary to incorporate waves with very short wavelengths, corresponding to high momenta in all directions, and therefore a large momentum uncertainty. Indeed, the Heisenberg Uncertainty Principle is equivalent to a theorem in Functional Analysis that the standard deviation of the squared absolute value of a function, times the standard deviation of the squared absolute value of its Fourier transform, is at least 1/(16π&2) (Folland and Sitaram, Theorem 1.1). A helpful analogy can be drawn between the wave associated with a quantum-mechanical particle and a more familiar wave, the time-varying Signal associated with, say, 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. Indeed, in order for a signal to have a relatively well-defined frequency, it must persist for a long period of time, and conversely, a signal that occurs at a relatively well-defined moment in time (i.e., of short duration) will necessarily encompass a broad Frequency Band . This is, indeed, a close mathematical analogue of the Heisenberg uncertainty principle. See also Complementarity (physics) . UNCERTAINTY PRINCIPLE VERSUS OBSERVER EFFECT shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the wavelength λ of the incoming light.]] The uncertainty principle in Quantum Mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum, and vice versa—i.e., that the uncertainty principle is a manifestation of the Observer Effect . Indeed, Heisenberg himself may have initially offered explanations which suggested this view. Prior to the more modern understanding, 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. Equating the uncertainty principle and the observer effect mischaracterizes the way Measurement In Quantum Mechanics is understood. Consider a hypothetical experiment in which a physicist prepares an Ensemble of 2N particles in the same way. Suppose further that the physicist is using perfect measuring equipment and that N is sufficiently large so that the net result is Statistically Significant . For the first N particles of this ensemble, the position would be measured and recorded, giving a probability distribution for position. For the remaining N particles, momentum would be measured, giving a probability distribution for momentum. Finally, the product of the standard deviations would be computed, giving a value of at least . If the position and momentum had been measured subsequently for the same particle, then the results of the second measurement would not reflect the original state, due to a correct application of the observer effect. But in this experiment, no such claim is made. The physicist never attempts to measure the position and momentum of a single particle but measures them for a ''different'' set of N particles from the same initial state. One measurement cannot affect the other. Moreover, although each measurement Collapses the quantum state of the particle, the probability distribution resulting from these measurements will correctly reflect the quantum state as it existed before the measurement. Consequently, the uncertainty principle should be considered an intrinsic smearing of statistical information instead of a limitation on measuring equipment.1 The EPR Paradox is one indication that it is wrong to view the uncertanty principle as simply a measurement directly disturbing a particle. This "paradox" shows that a measurement can be performed on a particle without disturbing it directly, by performing a measurement on a distant Entangled Particle . In any case, it is now understood that the uncertainties in the system exist prior to and independent of the measurement, and the uncertainty principle is therefore independent of the observer effect. GENERALIZATION, PRECISE FORMULATION, AND ROBERTSON-SCHRöDINGER RELATION Measurements of position and momentum taken in several identical copies of a system in a given state will each vary according to a Probability Distribution characteristic of the system’s state. This is ''the'' fundamental postulate of Quantum Mechanics . If we compute the standard deviations Δ''x'' and Δ''p'' of the position and momentum measurements, then : where More generally, given any Hermitian Operators ''A'' and ''B'', and a system in the state ψ, there are probability distributions associated with the Measurement of each of ''A'' and ''B'', giving rise to standard deviations Δψ''A'' and Δψ''B''. Then | ||
| In 1926, Dirac Offered A Precise Definition And Derivation Of This Uncertainty Relation, As Coming From A Relativistic Quantum Theory Of "events" But The Better-known, More Widely-used, Correct Formulation | "http://daarbnarodru/mandtamm-enghtml" class="copylinks" target="_blank">was given only in 1945 by L I Mandelshtam and I E Tamm , as follows For a quantum system in a non-stationary state <math>\psi
angle</math> and an observable <math>B</math> represented by a self-adjoint operator <math>\hat B</math>, the following formula holds: |
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