Abstract

We administered written free-response and multiple-choice questions and conducted individual interviews to investigate the difficulties that upper-level undergraduate and graduate students have with quantum states while translating state vectors in Dirac notation to wave functions in position and momentum representations. We find that students share common difficulties with translating a state vector written in Dirac notation to the wave function in position or momentum representation.

Highlights

  • Learning quantum mechanics is challenging [1,2,3,4,5,6]

  • Dirac notation is commonly used in upper-level quantum mechanics

  • Student difficulties were investigated by administering open-ended and multiple-choice questions to upper-level undergraduate and graduate students, observing difficulties on in-class quizzes and exams, and conducting individual interviews with students enrolled in quantum mechanics courses

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Summary

INTRODUCTION

Dirac notation is commonly used in upper-level quantum mechanics. One goal of upper-level quantum mechanics courses involves helping students become facile at using Dirac notation. We investigate student difficulties in translating a state vector in Dirac notation to a wave function in position or momentum representation. The generic quantum state |Ψ⟩ in Dirac notation contains all information about the system. To represent the generic state |Ψ⟩ as a wave function in the position representation, one must project |Ψ⟩ along position eigenstates |x⟩, i.e., ⟨x|Ψ⟩, where x is a continuous index. A generic state |Ψ⟩ can be represented as a wave function in the momentum representation by projecting |Ψ⟩ along momentum eigenstates |p⟩, i.e., ⟨p|Ψ⟩, where p is a continuous index. A position eigenstate |x′〉 in the momentum representation is a delocalized function of momentum ⟨p|x′⟩ = e−ipx′⁄ħ

METHODOLOGY AND RESULTS
Performance of graduate students
SUMMARY

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