Another look through Heisenberg's microscope

TL;DR

This paper revisits Heisenberg's microscope thought experiment and demonstrates that the Schrödinger equation for a free particle can be derived from the uncertainty principle combined with statistical assumptions.

Contribution

It provides a new derivation of the Schrödinger equation from the uncertainty relation and statistical assumptions, expanding on Heisenberg's original ideas.

Findings

Schrödinger equation follows from uncertainty relations

Supports the physical interpretation of quantum mechanics

Links indeterminacy to fundamental quantum laws

Abstract

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