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1 #+TITLE: Bugs in Quantum Mechanics
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2 #+AUTHOR: Dylan Holmes
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3 #+SETUPFILE: ../../aurellem/org/setup.org
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4 #+INCLUDE: ../../aurellem/org/level-0.org
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5
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6
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7 #Bugs in the Quantum-Mechanical Momentum Operator
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8
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9
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10 I studied quantum mechanics the same way I study most subjects\mdash{}
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11 by collecting (and squashing) bugs in my understanding. One of these
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12 bugs persisted throughout two semesters of
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13 quantum mechanics coursework until I finally found
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14 the paper
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15 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
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16 mechanics/]], which helped me stamp out the bug entirely. I decided to
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17 write an article about the problem and its solution for a number of reasons:
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18
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19 - Although the paper was not unreasonably dense, it was written for
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20 teachers. I wanted to write an article for students.
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21 - I wanted to popularize the problem and its solution because other
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22 explanations are currently too hard to find. (Even Shankar's
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23 excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
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24 - I wanted to check that the bug was indeed entirely
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25 eradicated. Attempting an explanation is my way of making
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26 sure.
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27
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28 * COMMENT
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29 I recommend the
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30 paper not only for students who are learning
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31 quantum mechanics, but especially for teachers interested in debugging
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32 them.
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33
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34 * COMMENT
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35 On my first exam in quantum mechanics, my professor asked us to
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36 describe how certain measurements would affect a particle in a
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37 box. Many of these measurement questions required routine application
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38 of skills we had recently learned\mdash{}first, you recall (or
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39 calculate) the eigenstates of the quantity
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40 to be measured; second, you write the given state as a linear
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41 sum of these eigenstates\mdash{} the coefficients on each term give
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42 the probability amplitude.
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43
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44 * The infinite square well potential
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45
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46 There is a particle in a one-dimensional potential well that is
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47 infinite everywhere except for a well of length \(a\). This means that the
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48 particle exists in a potential[fn:coords][fn:infinity]
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49
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50
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51 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
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52 }\;x<0\text{ or }x>a.\end{cases}\)
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53
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54 The Schr\ouml{}dinger equation describes how the particle's state
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55 \(|\psi\rangle\) will change over time in this system.
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56
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57 \(\begin{eqnarray}
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58 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
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59 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
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60
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61 This is a differential equation whose solutions are the physically
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62 allowed states for the particle in this system. Physically allowed
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63 states are those that change in physically allowed ways. Like any
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64 differential equation, the Schr\ouml{}dinger equation can be
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65 accompanied by /boundary conditions/\mdash{}conditions that
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66 further restrict which states qualify as physically allowed.
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67
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68 Whenever possible, physicists impose these boundary conditions:
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69 - The state should be a /continuous function of/ \(x\). This means
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70 that if a particle is very likely to be /at/ a particular location,
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71 it is also very likely to be /near/ that location.
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72 -
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73
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74 #; physically allowed states are those that change in physically
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75 #allowed ways.
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76
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77
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78 ** Boundary conditions
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79 Because the potential is infinite everywhere except within the well,
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80 a realistic particle must be confined to exist only within the
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81 well\mdash{}its wavefunction must be zero everywhere beyond the walls
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82 of the well.
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83
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84
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85 [fn:coords] I chose my coordinate system so that the well extends from
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86 \(0<x<a\). Others choose a coordinate system so that the well extends from
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87 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
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88 situation, they give different-looking answers.
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89
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90 [fn:infinity] Of course, infinite potentials are not
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91 realistic. Instead, they are useful approximations to finite
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92 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
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93 of the well\rdquo{} are close enough for your own practical
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94 purposes. Having introduced a physical impossibility into the problem
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95 already, we don't expect to get physically realistic solutions; we
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96 just expect to get mathematically consistent ones. The forthcoming
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97 trouble is that we don't.
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