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     1.4 +#+TITLE: Bugs in Quantum Mechanics
     1.5 +#+AUTHOR: Dylan Holmes
     1.6 +#+SETUPFILE: ../../aurellem/org/setup.org
     1.7 +#+INCLUDE:   ../../aurellem/org/level-0.org
     1.8 +
     1.9 +
    1.10 +#Bugs in the Quantum-Mechanical Momentum Operator
    1.11 +
    1.12 +
    1.13 +I studied quantum mechanics the same way I study most subjects\mdash{}
    1.14 +by collecting (and squashing) bugs in my understanding. One of these
    1.15 +bugs persisted throughout two semesters of
    1.16 +quantum mechanics coursework until I finally found
    1.17 +the paper 
    1.18 +[[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
    1.19 +mechanics/]], which helped me stamp out the bug entirely. I decided to
    1.20 +write an article about the problem and its solution for a number of reasons:
    1.21 +
    1.22 +- Although the paper was not unreasonably dense, it was written for
    1.23 +  teachers. I wanted to write an article for students.
    1.24 +- I wanted to popularize the problem and its solution because
    1.25 +  other explanations are currently too hard to find.
    1.26 +- I wanted to check that the bug was indeed entirely
    1.27 +  eradicated. Attempting an explanation is my way of making
    1.28 +  sure.
    1.29 +
    1.30 +* COMMENT
    1.31 + I recommend the
    1.32 +paper not only for students who are learning
    1.33 +quantum mechanics, but especially for teachers interested in debugging
    1.34 +them. 
    1.35 +
    1.36 +* COMMENT
    1.37 +On my first exam in quantum mechanics, my professor asked us to
    1.38 +describe how certain measurements would affect a particle in a
    1.39 +box. Many of these measurement questions required routine application
    1.40 +of skills we had recently learned\mdash{}first, you recall (or
    1.41 +calculate) the eigenstates of the quantity
    1.42 +to be measured; second, you write the given state as a linear
    1.43 +sum of these eigenstates\mdash{} the coefficients on each term give
    1.44 +the probability amplitude.
    1.45 +
    1.46 +* Statement of the Problem
    1.47 +A particle is 
    1.48 +
    1.49 +
    1.50 +
    1.51 +
    1.52 +* COMMENT [TABLE-OF-CONTENTS]