derive the uncertainty principle
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Heisenberg published his derivation of the uncertainty principle in 1927. At that time, physicists were still struggling to understand the new quantum formalism in terms of physical concepts. Under what circumstances, for example, could one properly apply the concepts of wave and particle to quantum mechanical systems? While Bohr was developing his ideas about complementarity, Heisenberg took a more mathematical approach by analyzing the theory itself to see what limits it placed on the precision with which one could simultaneously determine the values of two conjugate quantities, such as position and momentum.
Heisenberg prefaced his derivation by a physical argument analyzing the trajectory of an electron, illustrating why a measurement of position at some point in the path implies an uncertainty in momentum, and vice versa. Specifically, considering the measurement of the electron using a microscope, he argues that, due to the Compton effect, the product of the uncertainties of position and momentum is approximately h. He attributes this to the relation pq−qp=−iℏ , then proceeds with the formal mathematical derivation of the uncertainty principle from this relation.
The mathematical details of Heisenberg's original derivation can be found on p.69 of Quantum Theory and Measurement, ed. Wheeler and Zurek. In essence, he writes down the wave function for a particle in position representation with position uncertainty q1 , then shows that the corresponding wave function in momentum representation has a momentum uncertainty of p1 , where p1q1=h .