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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 other
    1.25 +  explanations are currently too hard to find. (Even Shankar's
    1.26 +  excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
    1.27 +- I wanted to check that the bug was indeed entirely
    1.28 +  eradicated. Attempting an explanation is my way of making
    1.29 +  sure.
    1.30 +
    1.31 +* COMMENT
    1.32 + I recommend the
    1.33 +paper not only for students who are learning
    1.34 +quantum mechanics, but especially for teachers interested in debugging
    1.35 +them. 
    1.36 +
    1.37 +* COMMENT
    1.38 +On my first exam in quantum mechanics, my professor asked us to
    1.39 +describe how certain measurements would affect a particle in a
    1.40 +box. Many of these measurement questions required routine application
    1.41 +of skills we had recently learned\mdash{}first, you recall (or
    1.42 +calculate) the eigenstates of the quantity
    1.43 +to be measured; second, you write the given state as a linear
    1.44 +sum of these eigenstates\mdash{} the coefficients on each term give
    1.45 +the probability amplitude.
    1.46 +
    1.47 +* The infinite square well potential
    1.48 +
    1.49 +There is a particle in a one-dimensional potential well that is
    1.50 +infinite everywhere except for a well of length \(a\). This means that the
    1.51 +particle exists in a potential[fn:coords][fn:infinity]
    1.52 +
    1.53 +
    1.54 +\(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
    1.55 +}\;x<0\text{ or }x>a.\end{cases}\)
    1.56 +
    1.57 +The Schr\ouml{}dinger equation describes how the particle's state 
    1.58 +\(|\psi\rangle\) will change over time in this system.
    1.59 +
    1.60 +\(\begin{eqnarray}
    1.61 +i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
    1.62 +H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
    1.63 +
    1.64 +This is a differential equation whose solutions are the physically
    1.65 +allowed states for the particle in this system. Physically allowed
    1.66 +states are those that change in physically allowed ways. Like any
    1.67 +differential equation, the Schr\ouml{}dinger equation can be
    1.68 +accompanied by /boundary conditions/\mdash{}conditions that
    1.69 +further restrict which states qualify as physically allowed.
    1.70 +
    1.71 +Whenever possible, physicists impose these boundary conditions:
    1.72 +- The state should be a /continuous function of/ \(x\). This means
    1.73 +  that if a particle is very likely to be /at/ a particular location,
    1.74 +  it is also very likely to be /near/ that location.
    1.75 +- 
    1.76 +
    1.77 +#; physically allowed states are those that change in physically
    1.78 +#allowed ways.
    1.79 +
    1.80 +
    1.81 +** Boundary conditions
    1.82 +Because the potential is infinite everywhere except within the well,
    1.83 +a realistic particle must be confined to exist only within the
    1.84 +well\mdash{}its wavefunction must be zero everywhere beyond the walls
    1.85 +of the well.
    1.86 +
    1.87 +
    1.88 +[fn:coords] I chose my coordinate system so that the well extends from
    1.89 +\(0<x<a\). Others choose a coordinate system so that the well extends from
    1.90 +\(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
    1.91 +situation, they give different-looking answers.
    1.92 +
    1.93 +[fn:infinity] Of course, infinite potentials are not
    1.94 +realistic. Instead, they are useful approximations to finite
    1.95 +potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
    1.96 +of the well\rdquo{} are close enough for your own practical
    1.97 +purposes. Having introduced a physical impossibility into the problem
    1.98 +already, we don't expect to get physically realistic solutions; we
    1.99 +just expect to get mathematically consistent ones. The forthcoming
   1.100 +trouble is that we don't.