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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:06:37 -0700
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1 #+TITLE: Bugs in Quantum Mechanics
2 #+AUTHOR: Dylan Holmes
3 #+SETUPFILE: ../../aurellem/org/setup.org
4 #+INCLUDE: ../../aurellem/org/level-0.org
5
6
7 #Bugs in the Quantum-Mechanical Momentum Operator
8
9
10 I studied quantum mechanics the same way I study most subjects\mdash{}
11 by collecting (and squashing) bugs in my understanding. One of these
12 bugs persisted throughout two semesters of
13 quantum mechanics coursework until I finally found
14 the paper
15 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
16 mechanics/]], which helped me stamp out the bug entirely. I decided to
17 write an article about the problem and its solution for a number of reasons:
18
19 - Although the paper was not unreasonably dense, it was written for
20 teachers. I wanted to write an article for students.
21 - I wanted to popularize the problem and its solution because other
22 explanations are currently too hard to find. (Even Shankar's
23 excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
24 - I wanted to check that the bug was indeed entirely
25 eradicated. Attempting an explanation is my way of making
26 sure.
27
28 * COMMENT
29 I recommend the
30 paper not only for students who are learning
31 quantum mechanics, but especially for teachers interested in debugging
32 them.
33
34 * COMMENT
35 On my first exam in quantum mechanics, my professor asked us to
36 describe how certain measurements would affect a particle in a
37 box. Many of these measurement questions required routine application
38 of skills we had recently learned\mdash{}first, you recall (or
39 calculate) the eigenstates of the quantity
40 to be measured; second, you write the given state as a linear
41 sum of these eigenstates\mdash{} the coefficients on each term give
42 the probability amplitude.
43
44 * The infinite square well potential
45
46 There is a particle in a one-dimensional potential well that is
47 infinite everywhere except for a well of length \(a\). This means that the
48 particle exists in a potential[fn:coords][fn:infinity]
49
50
51 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
52 }\;x<0\text{ or }x>a.\end{cases}\)
53
54 The Schr\ouml{}dinger equation describes how the particle's state
55 \(|\psi\rangle\) will change over time in this system.
56
57 \(\begin{eqnarray}
58 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
59 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
60
61 This is a differential equation whose solutions are the physically
62 allowed states for the particle in this system. Physically allowed
63 states are those that change in physically allowed ways. Like any
64 differential equation, the Schr\ouml{}dinger equation can be
65 accompanied by /boundary conditions/\mdash{}conditions that
66 further restrict which states qualify as physically allowed.
67
68 Whenever possible, physicists impose these boundary conditions:
69 - The state should be a /continuous function of/ \(x\). This means
70 that if a particle is very likely to be /at/ a particular location,
71 it is also very likely to be /near/ that location.
72 -
73
74 #; physically allowed states are those that change in physically
75 #allowed ways.
76
77
78 ** Boundary conditions
79 Because the potential is infinite everywhere except within the well,
80 a realistic particle must be confined to exist only within the
81 well\mdash{}its wavefunction must be zero everywhere beyond the walls
82 of the well.
83
84
85 [fn:coords] I chose my coordinate system so that the well extends from
86 \(0<x<a\). Others choose a coordinate system so that the well extends from
87 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
88 situation, they give different-looking answers.
89
90 [fn:infinity] Of course, infinite potentials are not
91 realistic. Instead, they are useful approximations to finite
92 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
93 of the well\rdquo{} are close enough for your own practical
94 purposes. Having introduced a physical impossibility into the problem
95 already, we don't expect to get physically realistic solutions; we
96 just expect to get mathematically consistent ones. The forthcoming
97 trouble is that we don't.