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author Dylan Holmes <ocsenave@gmail.com>
date Thu, 27 Oct 2011 23:02:59 -0500
parents 44d3dc936f6a
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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
6 #Bugs in Quantum Mechanics
7 #Bugs in the Quantum-Mechanical Momentum Operator
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:
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.
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.
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.
45 * What I thought I knew
47 The following is a list of things I thought were true of quantum
48 mechanics; the catch is that the list contradicts itself.
50 - For any hermitian operator: Eigenstates with different eigenvalues are orthogonal.
51 - For any hermitian operator: Any physically allowed state can be
52 written as a linear sum of eigenstates of the operator.
53 - The momentum operator and energy operator are hermitian, because
54 momentum and energy are measureable quantities.
55 - In vacuum,
56 - the momentum operator has an eigenstate
57 \(p(x)=\exp{(ipx/\hbar)}\) for each value of $p$.
58 - the energy operator has an eigenstate \(|E\rangle =
59 \alpha|p\rangle + \beta|-p\rangle\) for any \(\alpha,\beta\) and
60 the particular choice of momentum $p=\sqrt{2mE}$.
61 - In the infinitely deep potential well,
62 - the momentum operator has eigenstates with the same form $p(x) =
63 \exp{(ipx/\hbar)}$, but because of the boundary conditions on the
64 well, the following modifications are required.
65 - The wavefunction must be zero everywhere outside the well. That
66 is, \(p(x) = \begin{cases}\exp{(ipx/\hbar)},& 0\lt{}x\lt{}a;
67 \\0, & \text{for }x<0\text{ or }x>a \\ \end{cases}\)
68 #0,&\text{for }x\lt{}0\text{ or }x\gt{}a\end{cases}\)
69 - no longer has an eigenstate for each value
70 of $p$. Instead, only values of $p$ that are integer multiples of
71 $\pi a/\hbar$ are physically realistic.
75 * COMMENT:
77 ** Eigenstates with different eigenvalues are orthogonal
79 #+begin_quote
80 *Theorem:* Eigenstates with different eigenvalues are orthogonal.
81 #+end_quote
83 ** COMMENT :
84 I can prove this: if $\Lambda$ is any linear operator, suppose $|a\rangle$
85 and $|b\rangle$ are eigenstates of $\Lambda$. This means that
88 \(
89 \begin{eqnarray}
90 \Lambda |a\rangle&=& a|a\rangle,\\
91 \Lambda|b\rangle&=& b|b\rangle.\\
92 \end{eqnarray}
93 \)
95 If we take the difference of these eigenstates, we find that
97 \(
98 \begin{eqnarray}
99 \Lambda\;\left(|a\rangle-|b\rangle\right) &=& \Lambda |a\rangle - \Lambda |b\rangle
100 \qquad \text{(because $\Lambda$ is linear.)}\\
101 &=& a|a\rangle - b|b\rangle\qquad\text{(because $|a\rangle$ and
102 $|b\rangle$ are eigenstates of $\Lambda$)}
103 \end{eqnarray}\)
106 which means that $a\neq b$.
108 ** Eigenvectors of hermitian operators span the space of solutions
110 #+begin_quote
111 *Theorem:* If $\Omega$ is a hermitian operator, then every physically
112 allowed state can be written as a linear sum of eigenstates of
113 $\Omega$.
114 #+end_quote
118 ** Momentum and energy are hermitian operators
119 This ought to be true because hermitian operators correspond to
120 observable quantities. Since we expect momentum and energy to be
121 measureable quantities, we expect that there are hermitian operators
122 to represent them.
125 ** Momentum and energy eigenstates in vacuum
126 An eigenstate of the momentum operator $P$ would be a state
127 \(|p\rangle\) such that \(P|p\rangle=p|p\rangle\).
129 ** Momentum and energy eigenstates in the infinitely deep well
133 * Can you measure momentum in the infinite square well?
137 ** COMMENT Momentum eigenstates
139 In free space, the Hamiltonian is \(H=\frac{1}{2m}P^2\) and the
140 momentum operator $P$ has eigenstates \(p(x) = \exp{(-ipx/\hbar)}\).
142 In the infinitely deep potential well, the Hamiltonian is the same but
143 there is a new condition in order for states to qualify as physically
144 allowed: the states must not exist anywhere outside of well, as it
145 takes an infinite amount of energy to do so.
147 Notice that the momentum eigenstates defined above do /not/ satisfy
148 this condition.
152 * COMMENT
153 For each physical system, there is a Schr\ouml{}dinger equation that
154 describes how a particle's state $|\psi\rangle$ will change over
155 time.
157 \(\begin{eqnarray}
158 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
159 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
161 This is a differential equation; each solution to the
162 Schr\ouml{}dinger equation is a state that is physically allowed for
163 our particle. Here, physically allowed states are
164 those that change in physically allowed ways. However, like any differential
165 equation, the Schr\ouml{}dinger equation can be accompanied by
166 /boundary conditions/\mdash{}conditions that further restrict which
167 states qualify as physically allowed.
172 ** Eigenstates of momentum
177 #In the infinitely deep well potential $V(x)=0$, the Schr\ouml{}dinger
179 #\(i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle\)
187 * COMMENT
189 #* The infinite square well potential
191 A particle exists in a potential that is
192 infinite everywhere except for a region of length \(a\), where the potential is zero. This means that the
193 particle exists in a potential[fn:coords][fn:infinity]
196 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
197 }\;x<0\text{ or }x>a.\end{cases}\)
199 The Schr\ouml{}dinger equation describes how the particle's state
200 \(|\psi\rangle\) will change over time in this system.
202 \(\begin{eqnarray}
203 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
204 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
206 This is a differential equation; each solution to the
207 Schr\ouml{}dinger equation is a state that is physically allowed for
208 our particle. Here, physically allowed states are
209 those that change in physically allowed ways. However, like any differential
210 equation, the Schr\ouml{}dinger equation can be accompanied by
211 /boundary conditions/\mdash{}conditions that further restrict which
212 states qualify as physically allowed.
215 Whenever possible, physicists impose these boundary conditions:
216 - A physically allowed state ought to be a /smoothly-varying function of position./ This means
217 that if a particle in the state is likely to be /at/ a particular location,
218 it is also likely to be /near/ that location.
220 These boundary conditions imply that for the square well potential in
221 this problem,
223 - Physically allowed states must be totally confined to the well,
224 because it takes an infinite amount of energy to exist anywhere
225 outside of the well (and physically allowed states ought to have
226 only finite energy).
227 - Physically allowed states must be increasingly unlikely to find very
228 close to the walls of the well. This is because of two conditions: the above
229 condition says that the particle is /impossible/ to find
230 outside of the well, and the smoothly-varying condition says
231 that if a particle is impossible to find at a particular location,
232 it must be unlikely to be found nearby that location.
234 #; physically allowed states are those that change in physically
235 #allowed ways.
238 #** Boundary conditions
239 Because the potential is infinite everywhere except within the well,
240 a realistic particle must be confined to exist only within the
241 well\mdash{}its wavefunction must be zero everywhere beyond the walls
242 of the well.
245 [fn:coords] I chose my coordinate system so that the well extends from
246 \(0<x<a\). Others choose a coordinate system so that the well extends from
247 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
248 situation, they give different-looking answers.
250 [fn:infinity] Of course, infinite potentials are not
251 realistic. Instead, they are useful approximations to finite
252 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
253 of the well\rdquo{} are close enough for your own practical
254 purposes. Having introduced a physical impossibility into the problem
255 already, we don't expect to get physically realistic solutions; we
256 just expect to get mathematically consistent ones. The forthcoming
257 trouble is that we don't.