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Probability amplitude
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<p>In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, a probability amplitude is a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> used for describing the behaviour of systems. The square of the <a href="/facts/Absolute_value/dwJBpEs5">modulus</a> of this quantity at a point in space represents a <a href="/facts/Probability_density_function/zvfybna4">probability density</a> at that point. </p><p>Probability amplitudes provide a relationship between the <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> vector of a system and the results of observations of that system, a link that was first proposed by <a href="/facts/Max_Born/6VnUDWhA">Max Born</a>, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the <a href="/facts/Copenhagen_interpretation/DvMFrYJa">Copenhagen interpretation</a> of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as <a href="/facts/Atomic_emission_spectroscopy/5s0wg1sX">emissions from atoms</a> being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 <a href="/facts/Nobel_Prize_in_Physics/o6w1nCbr">Nobel Prize in Physics</a> for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and <a href="/facts/Quantum_measurement/VY7jPfP0">quantum measurements</a>, were vigorously contested at the time by the original physicists working on the theory, such as <a href="/facts/Erwin_Schr%C3%B6dinger/wCprepF2">Schrödinger</a> and <a href="/facts/Albert_Einstein/CNTet8qu">Einstein</a>. It is the source of the mysterious consequences and philosophical difficulties in the <a href="/facts/Interpretations_of_quantum_mechanics/WEjlzVPl">interpretations of quantum mechanics</a>—topics that continue to be debated even today. </p>