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

The Physics Teacher

Quantum Mechanics for Beginning Physics Students

Mark B. Schneider

Published Sep 17, 2010• 6 citations
Publisher

Abstract

The past two decades of attention to introductory physics education has emphasized enhanced development of conceptual understanding to accompany calculational ability. Given this, it is surprising that current texts continue to rely on the Bohr model to develop a flawed intuition, and introduce correct atomic physics on an ad hoc basis. For example, Halliday, Resnick, and Walker1 describe the origin of atomic quantum numbers as such: “The restrictions on the values of the quantum number for the hydrogen atom, as listed in Table 39-2, are not arbitrary but come out of the solution to Schrödinger's equation.” They give no further justification, but do point out the values are in conflict with the predictions of the Bohr model.

DOI
10.1119/1.3488198

Related papers

Quantum Mechanics

Ana Marı́a Cetto, Luis de la Peña

Offering a detailed account of the key concepts and mathematical apparatus of quantum mechanics, this textbook is an ideal companion to both undergraduate and graduate courses. The formal and practical aspects of the subject are explained clearly alongside examples of modern applications, providing students with the tools required to thoroughly understand the theory and apply it. The authors provide an intuitive conceptual framework that is grounded in a coherent physical explanation of quantum phenomena, established over decades of teaching and research in quantum mechanics and its foundations. The book's educational value is enhanced by the inclusion of examples and exercises, with solutions available online, and an extensive bibliography is provided. Notes throughout the text provide fascinating context on the tumultuous history of quantum mechanics, the people that developed it, and the questions that still remain at its center. This title is also available as Open Access on Cambridge Core.

Cambridge University Press eBooks•Dec 18, 2025•1 citation

Hydrogen Atom

Arjun Berera, Luigi Del Debbio

Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.

Cambridge University Press eBooks•Oct 21, 2021•0 citations

Puzzling over quantum mechanics

Alastair I. M. Rae

What is it about quantum mechanics that continues to make its interpretation so difficult and controversial? It is now more than 70 years after its inception and at least half a century since the verification of its predictions of physical phenomena that range from the motion of electrons in solids to the basic structure and properties of the atomic nucleus. After reading the first half of this book one could be forgiven for believing that it has been just our unwillingness or inability to understand properly what Niels Bohr said from the beginning, and to update these ideas in the light of later experience. The aim of the present book is to do just this: to restate the Copenhagen interpretation and to show us how it has been developed to take modern results and ideas into account.

Physics World•Feb 1, 1995•0 citations

Methodological Aspects of Teaching Quantum Mechanics Using the Example of The Hydrogen Atom

M. K.

This article explores methodological aspects of teaching quantum mechanics in higher education within the framework of physics and mathematics programs. Particular attention is given to the challenge’s students face in mastering fundamental concepts due to the high level of abstraction inherent in the subject. The hydrogen atom model is considered as a focal example – being both conceptually significant and visually illustrative in demonstrating the principles of quantum theory. The core emphasis is placed on the interpretation of the wave function and its physical meaning through the concept of probability density. Methods for visualizing electron wave states are discussed, alongside graphical representations of probability densities corresponding to various quantum numbers. The relationship between the analytical solutions of the Schrodinger equation and the physical interpretation of these results is substantiated. Special attention is also devoted to developing students’ understanding of the probabilistic nature of quantum systems, as well as their ability to apply mathematical tools for modeling and interpreting quantum states. Didactic approaches are proposed to enhance students’ abstract thinking and intuitive grasp of quantum principles.

International Journal of Pedagogics•Jul 1, 2025•0 citations

Introduction and overview

Stephen L. Adler

Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.

Cambridge University Press eBooks•Aug 26, 2004•2 citations

Schrödinger and the development of wave mechanics

Robert Golub, S. K. Lamoreaux

Abstract While, as Feynman said, nobody can really understand the origin of the Schrödinger equation, we investigate several lines of thought leading to such an equation. We show some simple applications of the equation. An important application was Schrödinger’s solution for the hydrogen atom. This provided a theoretical basis for Bohr’s ad hoc model. We conclude with Schrödinger’s demonstration of the equivalence of his wave mechanics with Heisenberg’s matrix mechanics.

•Feb 28, 2023•0 citations
The past two decades of attention to introductory physics education has emphasized enhanced development of conceptual understanding to accompany calculational ability. Given this, it is surprising that current texts continue to rely on the Bohr model to develop a flawed intuition, and introduce correct atomic physics on an ad hoc basis. For example, Halliday, Resnick, and Walker1 describe the origin of atomic quantum numbers as such: “The restrictions on the values of the quantum number for the hydrogen atom, as listed in Table 39-2, are not arbitrary but come out of the solution to Schrödinger's equation.” They give no further justification, but do point out the values are in conflict with the predictions of the Bohr model.
Offering a detailed account of the key concepts and mathematical apparatus of quantum mechanics, this textbook is an ideal companion to both undergraduate and graduate courses. The formal and practical aspects of the subject are explained clearly alongside examples of modern applications, providing students with the tools required to thoroughly understand the theory and apply it. The authors provide an intuitive conceptual framework that is grounded in a coherent physical explanation of quantum phenomena, established over decades of teaching and research in quantum mechanics and its foundations. The book's educational value is enhanced by the inclusion of examples and exercises, with solutions available online, and an extensive bibliography is provided. Notes throughout the text provide fascinating context on the tumultuous history of quantum mechanics, the people that developed it, and the questions that still remain at its center. This title is also available as Open Access on Cambridge Core.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
What is it about quantum mechanics that continues to make its interpretation so difficult and controversial? It is now more than 70 years after its inception and at least half a century since the verification of its predictions of physical phenomena that range from the motion of electrons in solids to the basic structure and properties of the atomic nucleus. After reading the first half of this book one could be forgiven for believing that it has been just our unwillingness or inability to understand properly what Niels Bohr said from the beginning, and to update these ideas in the light of later experience. The aim of the present book is to do just this: to restate the Copenhagen interpretation and to show us how it has been developed to take modern results and ideas into account.
This article explores methodological aspects of teaching quantum mechanics in higher education within the framework of physics and mathematics programs. Particular attention is given to the challenge’s students face in mastering fundamental concepts due to the high level of abstraction inherent in the subject. The hydrogen atom model is considered as a focal example – being both conceptually significant and visually illustrative in demonstrating the principles of quantum theory. The core emphasis is placed on the interpretation of the wave function and its physical meaning through the concept of probability density. Methods for visualizing electron wave states are discussed, alongside graphical representations of probability densities corresponding to various quantum numbers. The relationship between the analytical solutions of the Schrodinger equation and the physical interpretation of these results is substantiated. Special attention is also devoted to developing students’ understanding of the probabilistic nature of quantum systems, as well as their ability to apply mathematical tools for modeling and interpreting quantum states. Didactic approaches are proposed to enhance students’ abstract thinking and intuitive grasp of quantum principles.
Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.