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arXiv:2007.13478 · European Journal of Physics

Building confidence in the Dirac δ -function

Asim Gangopadhyaya, Constantin Rasinariu

Published Aug 30, 2018• 2 citations
arXivHTMLPDF

Abstract

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac $δ$-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function $ψ(x)$. Since the first derivative of $ψ(x)$ is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac $δ$-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac $δ$-function and highlights its efficiency and elegance.

DOI
10.1088/1361-6404/aaddcd

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In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac δδδ-function. Namely, we compute the expectation value of the Hamiltonian of a free-particle in a state described by a triangular wave function ψ(x)ψ(x)ψ(x). Since the first derivative of ψ(x)ψ(x)ψ(x) is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero --an unphysical answer. This problem provides a pedagogical application of the Dirac δδδ-function. By arriving at the same result via alternate pathways, this exercise reinforces students' confidence in the Dirac δδδ-function and highlights its efficiency and elegance.
Quantum mechanics is challenging even for advanced undergraduate and graduate students.In the Schrödinger representation, the wave function evolves in time according to the time dependent Schrödinger equation.The time dependence of the wave function gives rise to time dependence of the expectation value of observables.We have been exploring the difficulties that advanced undergraduate and graduate students have with time dependence of expectation values in quantum mechanics.We have developed and administered conceptual free response and multiplechoice questions to students to investigate these difficulties.We also interviewed 23 students individually using a thinkaloud protocol to obtain a better understanding of the rationale behind students' written responses.We find that many students struggle with time dependence of expectation values of observables.We discuss some findings.
Dirac notation is often used in upper level quantum mechanics courses, but students struggle with this representation.To investigate the difficulties that advanced students (i.e., upper-level undergraduate and graduate students) have while translating state vectors in Dirac notation to wave functions in position and momentum representations, we administered free-response and multiple-choice questions and conducted individual interviews with students.We find that students display common difficulties with these topics.
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This popular undergraduate quantum mechanics textbook is now available in a more affordable printing from Cambridge University Press. Unlike many other books on quantum mechanics, this text begins by examining experimental quantum phenomena such as the Stern-Gerlach experiment and spin measurements, using them as the basis for developing the theoretical principles of quantum mechanics. Dirac notation is developed from the outset, offering an intuitive and powerful mathematical toolset for calculation, and familiarizing students with this important notational system. This non-traditional approach is designed to deepen students' conceptual understanding of the subject, and has been extensively class tested. Suitable for undergraduate physics students, worked examples are included throughout and end of chapter problems act to reinforce and extend important concepts. Additional activities for students are provided online, including interactive simulations of Stern-Gerlach experiments, and a fully worked solutions manual is available for instructors.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.