Mercurial > dylan
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initial commit of dylan's stuff
author | Robert McIntyre <rlm@mit.edu> |
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date | Mon, 17 Oct 2011 23:17:55 -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. |