annotate bk/bk2.org @ 6:b2f55bcf6853

fixed comments
author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 04:56:15 -0700
parents 44d3dc936f6a
children
rev   line source
rlm@0 1 #+TITLE: Bugs in Quantum Mechanics
rlm@0 2 #+AUTHOR: Dylan Holmes
rlm@0 3 #+SETUPFILE: ../../aurellem/org/setup.org
rlm@0 4 #+INCLUDE: ../../aurellem/org/level-0.org
rlm@0 5
rlm@0 6
rlm@0 7 #Bugs in the Quantum-Mechanical Momentum Operator
rlm@0 8
rlm@0 9
rlm@0 10 I studied quantum mechanics the same way I study most subjects\mdash{}
rlm@0 11 by collecting (and squashing) bugs in my understanding. One of these
rlm@0 12 bugs persisted throughout two semesters of
rlm@0 13 quantum mechanics coursework until I finally found
rlm@0 14 the paper
rlm@0 15 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
rlm@0 16 mechanics/]], which helped me stamp out the bug entirely. I decided to
rlm@0 17 write an article about the problem and its solution for a number of reasons:
rlm@0 18
rlm@0 19 - Although the paper was not unreasonably dense, it was written for
rlm@0 20 teachers. I wanted to write an article for students.
rlm@0 21 - I wanted to popularize the problem and its solution because other
rlm@0 22 explanations are currently too hard to find. (Even Shankar's
rlm@0 23 excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
rlm@0 24 - I wanted to check that the bug was indeed entirely
rlm@0 25 eradicated. Attempting an explanation is my way of making
rlm@0 26 sure.
rlm@0 27
rlm@0 28 * COMMENT
rlm@0 29 I recommend the
rlm@0 30 paper not only for students who are learning
rlm@0 31 quantum mechanics, but especially for teachers interested in debugging
rlm@0 32 them.
rlm@0 33
rlm@0 34 * COMMENT
rlm@0 35 On my first exam in quantum mechanics, my professor asked us to
rlm@0 36 describe how certain measurements would affect a particle in a
rlm@0 37 box. Many of these measurement questions required routine application
rlm@0 38 of skills we had recently learned\mdash{}first, you recall (or
rlm@0 39 calculate) the eigenstates of the quantity
rlm@0 40 to be measured; second, you write the given state as a linear
rlm@0 41 sum of these eigenstates\mdash{} the coefficients on each term give
rlm@0 42 the probability amplitude.
rlm@0 43
rlm@0 44 * The infinite square well potential
rlm@0 45
rlm@0 46 There is a particle in a one-dimensional potential well that is
rlm@0 47 infinite everywhere except for a well of length \(a\). This means that the
rlm@0 48 particle exists in a potential[fn:coords][fn:infinity]
rlm@0 49
rlm@0 50
rlm@0 51 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
rlm@0 52 }\;x<0\text{ or }x>a.\end{cases}\)
rlm@0 53
rlm@0 54 The Schr\ouml{}dinger equation describes how the particle's state
rlm@0 55 \(|\psi\rangle\) will change over time in this system.
rlm@0 56
rlm@0 57 \(\begin{eqnarray}
rlm@0 58 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
rlm@0 59 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
rlm@0 60
rlm@0 61 This is a differential equation whose solutions are the physically
rlm@0 62 allowed states for the particle in this system. Physically allowed
rlm@0 63 states are those that change in physically allowed ways. Like any
rlm@0 64 differential equation, the Schr\ouml{}dinger equation can be
rlm@0 65 accompanied by /boundary conditions/\mdash{}conditions that
rlm@0 66 further restrict which states qualify as physically allowed.
rlm@0 67
rlm@0 68 Whenever possible, physicists impose these boundary conditions:
rlm@0 69 - The state should be a /continuous function of/ \(x\). This means
rlm@0 70 that if a particle is very likely to be /at/ a particular location,
rlm@0 71 it is also very likely to be /near/ that location.
rlm@0 72 -
rlm@0 73
rlm@0 74 #; physically allowed states are those that change in physically
rlm@0 75 #allowed ways.
rlm@0 76
rlm@0 77
rlm@0 78 ** Boundary conditions
rlm@0 79 Because the potential is infinite everywhere except within the well,
rlm@0 80 a realistic particle must be confined to exist only within the
rlm@0 81 well\mdash{}its wavefunction must be zero everywhere beyond the walls
rlm@0 82 of the well.
rlm@0 83
rlm@0 84
rlm@0 85 [fn:coords] I chose my coordinate system so that the well extends from
rlm@0 86 \(0<x<a\). Others choose a coordinate system so that the well extends from
rlm@0 87 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
rlm@0 88 situation, they give different-looking answers.
rlm@0 89
rlm@0 90 [fn:infinity] Of course, infinite potentials are not
rlm@0 91 realistic. Instead, they are useful approximations to finite
rlm@0 92 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
rlm@0 93 of the well\rdquo{} are close enough for your own practical
rlm@0 94 purposes. Having introduced a physical impossibility into the problem
rlm@0 95 already, we don't expect to get physically realistic solutions; we
rlm@0 96 just expect to get mathematically consistent ones. The forthcoming
rlm@0 97 trouble is that we don't.