path of an electron through a cloud chamber." In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. Heisenberg wrote:It can be expressed in its simplest form as follows: One can never know with perfect accuracy both of those two important factors which determine the movement of one of the smallest particles—its position and its velocity. It is impossible to determine accurately both the position and the direction and speed of a particle at the same instant. In his celebrated 1927 paper, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture he refined his principle: Kennard in 1927 first proved the modern inequality: where , and , are the standard deviations of position and momentum. Heisenberg only proved relation () for the special case of Gaussian states. Terminology and translation Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit" ("indeterminacy"), to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit" ("uncertainty"). When the English-language version of Heisenberg's textbook, The Physical Principles of the Quantum Theory, was published in 1930, however, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter. Heisenberg's microscope The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violating the uncertainty principle is by utilizing the observer effect of an imaginary microscope as a measuring device. He imagines an experimenter trying to measure the position and momentum of