A full quantum analysis of the Stern–Gerlach experiment using the evolution operator method: analyzing current issues in teaching quantum mechanics

Abstract

To the quantum mechanics specialists community it is a well-known fact that the famous original Stern–Gerlach experiment (SGE) produces entanglement between the external degrees of freedom (position) and the internal degree of freedom (spin) of silver atoms. Despite this fact, almost all textbooks on quantum mechanics explain this experiment using a semiclassical approach, where the external degrees of freedom are considered classical variables, the internal degree is treated as a quantum variable, and Newton's second law is used to describe the dynamics. In the literature there are some works that analyze this experiment in its full quantum mechanical form. However, astonishingly, to the best of our knowledge the original experiment, where the initial states of the spin degree of freedom are randomly oriented coming from the oven, has not been analyzed yet in the available textbooks using the Schrödinger equation (to the best of our knowledge there is only one paper that treats this case: Hsu et al (2011 Phys. Rev. A 83 012109)). Therefore, in this contribution we use the time-evolution operator to give a full quantum mechanics analysis of the SGE when the initial state of the internal degree of freedom is completely random, i.e. when it is a statistical mixture. Additionally, as the SGE and the development of quantum mechanics are heavily intermingled, we analyze some features and drawbacks in the current teaching of quantum mechanics. We focus on textbooks that use the SGE as a starting point, based on the fact that most physicist do not use results from physics education research, and comment on traditional pedagogical attitudes in the physics community.

Nonlocal single particle steering generated through single particle entanglement

“…Here, we analyze how the Stern-Gerlach experiment works in a quantum mechanical way when we send individual atoms one by one. As it was demonstrated in references 32,33 the evolution of the wave function in the Stern-Gerlach experiment is given by:…”

“…SGE featuring the Einstein's boxes. Where the red box is the oven, the blue box represents the magnet, the red dot represents the fact that there is not classical trajectories, see references 32,33 . Alice could communicate with Bob by using the classical channel in magenta.…”

“…However, it was until 1927 that the scientific community began to realize that what the Stern-Gerlach experiment (SGE) really proved was the existence of the internal spin 31 . From then on, the SGE has been a fundamental tool for the development of quantum mechanics; which, as it was explained in references 32,33 , it is an entanglement device. In fact, the quantum attributes of the SGE and a new explanation for how it works and has been given in many papers, see for example reference 34 ; in particular, in references 32,33,35 the Schrödinger equation for this experiment has been solved and the violation of local realism was proved in reference 36 .…”

“…From then on, the SGE has been a fundamental tool for the development of quantum mechanics; which, as it was explained in references 32,33 , it is an entanglement device. In fact, the quantum attributes of the SGE and a new explanation for how it works and has been given in many papers, see for example reference 34 ; in particular, in references 32,33,35 the Schrödinger equation for this experiment has been solved and the violation of local realism was proved in reference 36 . It is well worth mentioning that recent experimental evidence confirms the existence of the superposition of the wavepackets 37,38 .…”

Single-photon steering

“…The state given by Eq. ( 1) is an entangled state between the internal (polarization) and external (path) DoFs of the photon, it is equivalent to the following entangled state given in references 31,32 :…”