Another look through Heisenberg’s microscope

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, δqδp ∼ h. Heisenberg offered this demonstration as "a direct physical interpretation of the [quantum mechanical] equation pq − qp = −ih" 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 Schrödinger equation for a free particle does indeed follow from the indeterminacy relation together with reasonable statistical assumptions.

Diferentes proposições do princípio da incerteza para posição e momentum: integrando formalismo matemático, fenomenologia e interpretações no ensino da teoria quântica

“…Uma simplificação corrigida do experimento de Heisenberg é proposta por [18] e está representada na Figura 1.…”

“…Embora exista uma ampla literatura voltada para discussões acerca do Princípio da Incerteza, incluindo diferentes derivações e generalizações do princípio [7][8][9][10][11][12][13][14][15][16][17], estudos voltados para discussões fenomenológicas, centrados no experimento mental do microscópio de raios-gama proposto por Heisenberg em 1927 [18][19][20][21] e, ainda, trabalhos que buscam discutir conceitualmente o Princípio da Incerteza, suas implicações teóricas e possíveis interpretações, [22][23][24][25], a maioria dos trabalhos costuma ser voltada a pesquisadores e estudiosos da área, tornando pouco provável sua utilização como instrumento auxiliar para o ensino da Teoria Quântica. Na literatura nacional já existem estudos voltados para a discussão de aspectos matemáticos [26] e filosóficointerpretativos do Princípio da Incerteza [27].…”

A Quantum Story

“…The Hamilton-Jacobi formalism provides an approach in which such uncertainties can be included in the fundamental equations of classical mechanics [24,25]; although, it is usually far more convenient to deal with them in the analysis of a measurement rather than as fundamental facet of the theory. 15 An interesting aside is that by combining the statistical Hamilton-Jacobi formalism of classical mechanics with the Heisenberg uncertainty relations, one can generate a plausible route to Schrödinger's equation and the concomitant wave nature of particles [26,27]. One can even construe statistical relations in classical physics in terms of classical indeterminacy relations δx > 0 and δp > 0 [28].…”

Popper’s conjecture with angular slits and twisted light

“…Uncertainty relations are core to many branches of physics, spanning space-time bandwidth considerations in ultrafast laser pulses to spatial frequency limits in the analysis of finite apertured optical systems. In keeping with these classical optical examples, it should be noted that Heisenberg’s position-momentum uncertainty principle was itself derived from considerations of an optical microscope 1 and extrapolated to a more general meaning in the context of quantum mechanics. More recently the emergence of structured light 2 has seen the renewed testing of these ideas, with orbital angular momentum (OAM) 3 – 5 as a key driver.…”