Another look through Heisenberg’s microscope

Heisenberg introduced his famous uncertainty relations in a seminal 1927\npaper entitled "The Physical Content of Quantum Kinematics and Mechanics". He\nmotivated his arguments with a gedanken experiment, a gamma ray microscope to\nmeasure the position of a particle. A primary result was that, due to the\nquantum nature of light, there is an inherent uncertainty in the determinations\nof the particle's position and momentum dictated by an indeterminacy relation,\n$\\delta q \\delta p \\sim h$. Heisenberg offered this demonstration as "a direct\nphysical interpretation of the [quantum mechanical] equation $\\textbf{pq} -\n\\textbf{qp} = -i\\hbar$" but considered the indeterminacy relation to be much\nmore than this. He also argued that it implies limitations on the very meanings\nof position and momentum and emphasized that these limitations are the source\nof the statistical character of quantum mechanics. In addition, Heisenberg\nhoped but was unable to demonstrate that the laws of quantum mechanics could be\nderived directly from the uncertainty relation. In this paper, we revisit\nHeisenberg's microscope and argue that the Schr\\"odinger equation for a free\nparticle does indeed follow from the indeterminacy relation together with\nreasonable statistical assumptions.\n