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