Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

A mechanical model for teaching quantum mechanics

Carl Frederick

Published Mar 1, 1978• 6 citations
Publisher

Abstract

The wave function ψ in quantum mechanics is a rather abstract quantity with which the physics student often has trouble. It is instructionally advantageous therefore, to have a mechanical model which can be described both by the quantum-mechanical formalism and also by derivations from ’’first principles.’’ It is shown that a particle traveling in a helical path gives rise to such a model.

DOI
10.1119/1.11132

Related papers

Quantum mechanics

Stephen W. McKnight, Christos Zahopoulos

The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.

Cambridge University Press eBooks•Aug 10, 2015•0 citations

A First Course in Quantum Mechanics

Ahmad Amer

Based on years of teaching experience, this book provides a clearly structured and accessible introduction to the fundamental principles of quantum mechanics, tailored for undergraduate students. The text builds a strong conceptual and mathematical foundation while promoting intuitive understanding and critical thinking. Students progress from basic concepts such as wave mechanics to more advanced topics within matrix mechanics. Complex concepts are presented in a diligent and understandable way, achieving a balance between conceptual insight and mathematical rigor without oversimplifying the subject. Thoroughly class-tested, this book is designed to support both classroom instruction and independent study. It will serve as a great tool for undergraduates studying introductory and intermediate quantum mechanics, as well as modern physics and quantum systems for engineers.

CRC Press eBooks•Nov 5, 2025•0 citations

Quantum Mechanics

P. J. E. Peebles

This book explains the often counterintuitive physics of quantum mechanics, unlocking this key area of physics for students by enabling them to work through detailed applications of general concepts and ideas. The book states general principles first in terms of wave mechanics and then in the standard abstract linear space formalism. It offers a detailed discussion of measurement theory — an essential feature of quantum mechanics — and emphasizes the art of numerical estimates. Along the way, the book provides a wealth of physical examples together with numerous problems, some easy, some challenging, but all of them selected because they are physically interesting. The book is an essential resource for advanced undergraduates and beginning graduate students in physics.

Princeton University Press eBooks•May 14, 2019•0 citations

Deriving the time-independent Schrödinger equation

Jonathan Gorard

Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.

Physics Education•Sep 14, 2016•3 citations

Introduction

Konstantin K Likharev

This introductory chapter briefly reviews the major experimental motivations for quantum mechanics, and then discusses its simplest formalism—the Schrödinger’s wave mechanics. Much of this material (perhaps besides the last section) may be found in undergraduate textbooks, so that the discussion is rather brief, and focused on the most important conceptual issues.

Institute of Physics eBooks•May 1, 2019•0 citations

Test on the effectiveness of the sum over paths approach in favoring the construction of an integrated knowledge of quantum physics in high school

Massimiliano Malgieri, Pasquale Onorato, Anna De Ambrosis

In this paper we present the results of a research-based teaching-learning sequence on introductory quantum physics based on Feynman's sum over paths approach in the Italian high school.Our study focuses on students' understanding of two founding ideas of quantum physics, wave particle duality and the uncertainty principle.In view of recent research reporting the fragmentation of students' mental models of quantum concepts after initial instruction, we collected and analyzed data using the assessment tools provided by knowledge integration theory.Our results on the group of n ¼ 14 students who performed the final test indicate that the functional explanation of wave particle duality provided by the sum over paths approach may be effective in leading students to build consistent mental models of quantum objects, and in providing them with a unified perspective on both the photon and the electron.Results on the uncertainty principle are less clear cut, as the improvements over traditional instruction appear less significant.Given the low number of students in the sample, this work should be interpreted as a case study, and we do not attempt to draw definitive conclusions.However, our study suggests that (i) the sum over paths approach may deserve more attention from researchers and educators as a possible route to introduce basic concepts of quantum physics in high school, and (ii) more research should be focused not only on the correctness of students' mental models on individual concepts, but also on the ability of students to connect different ideas and experiments related to quantum theory in an organized whole.

Physical Review Physics Education Research•Jan 3, 2017•29 citations
The wave function ψ in quantum mechanics is a rather abstract quantity with which the physics student often has trouble. It is instructionally advantageous therefore, to have a mechanical model which can be described both by the quantum-mechanical formalism and also by derivations from ’’first principles.’’ It is shown that a particle traveling in a helical path gives rise to such a model.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
Based on years of teaching experience, this book provides a clearly structured and accessible introduction to the fundamental principles of quantum mechanics, tailored for undergraduate students. The text builds a strong conceptual and mathematical foundation while promoting intuitive understanding and critical thinking. Students progress from basic concepts such as wave mechanics to more advanced topics within matrix mechanics. Complex concepts are presented in a diligent and understandable way, achieving a balance between conceptual insight and mathematical rigor without oversimplifying the subject. Thoroughly class-tested, this book is designed to support both classroom instruction and independent study. It will serve as a great tool for undergraduates studying introductory and intermediate quantum mechanics, as well as modern physics and quantum systems for engineers.
This book explains the often counterintuitive physics of quantum mechanics, unlocking this key area of physics for students by enabling them to work through detailed applications of general concepts and ideas. The book states general principles first in terms of wave mechanics and then in the standard abstract linear space formalism. It offers a detailed discussion of measurement theory — an essential feature of quantum mechanics — and emphasizes the art of numerical estimates. Along the way, the book provides a wealth of physical examples together with numerous problems, some easy, some challenging, but all of them selected because they are physically interesting. The book is an essential resource for advanced undergraduates and beginning graduate students in physics.
Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.
In this paper we present the results of a research-based teaching-learning sequence on introductory quantum physics based on Feynman's sum over paths approach in the Italian high school.Our study focuses on students' understanding of two founding ideas of quantum physics, wave particle duality and the uncertainty principle.In view of recent research reporting the fragmentation of students' mental models of quantum concepts after initial instruction, we collected and analyzed data using the assessment tools provided by knowledge integration theory.Our results on the group of n ¼ 14 students who performed the final test indicate that the functional explanation of wave particle duality provided by the sum over paths approach may be effective in leading students to build consistent mental models of quantum objects, and in providing them with a unified perspective on both the photon and the electron.Results on the uncertainty principle are less clear cut, as the improvements over traditional instruction appear less significant.Given the low number of students in the sample, this work should be interpreted as a case study, and we do not attempt to draw definitive conclusions.However, our study suggests that (i) the sum over paths approach may deserve more attention from researchers and educators as a possible route to introduce basic concepts of quantum physics in high school, and (ii) more research should be focused not only on the correctness of students' mental models on individual concepts, but also on the ability of students to connect different ideas and experiments related to quantum theory in an organized whole.