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author Robert McIntyre <rlm@mit.edu>
date Mon, 17 Oct 2011 23:19:18 -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
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 1. For any hermitian operator: Eigenstates with different eigenvalues are orthogonal.
51 2. For any hermitian operator: Any physically allowed state can be
52 written as a linear sum of eigenstates of the operator.
53 3. The momentum operator and energy operator are hermitian, because
54 momentum and energy are measureable quantities.
55 4. In the vacuum potential, the momentum and energy operators have these eigenstates:
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 5. In the infinitely deep potential well, the momentum and energy
62 operators have these eigenstates:
63 - The momentum eigenstates and energy eigenstates have the same form
64 as in the vacuum potential: $p(x) =
65 \exp{(ipx/\hbar)}$ and $|E\rangle = \alpha|p\rangle + \beta|-p\rangle$.
66 - Even so, because of the boundary conditions on the
67 well, we must make the following modifications:
68 + Physically realistic states must be impossible to find outside the well. (Only a state of infinite
69 energy could exist outside the well, and infinite energy is not
70 realistic.) This requirement means, for example, that momentum
71 eigenstates in the infinitely deep well must be
72 \(p(x)
73 = \begin{cases}\exp{(ipx/\hbar)},& \text{for }0\lt{}x\lt{}a;
74 \\0, & \text{for }x<0\text{ or }x>a. \\ \end{cases}\)
75 + Physically realistic states must vary smoothly throughout
76 space. This means that if a particle in some state is very unlikely to be
77 /at/ a particular location, it is also very unlikely be /near/
78 that location. Combining this requirement with the above
79 requirement, we find that the momentum operator no longer has
80 an eigenstate for each value of $p$; instead, only values of
81 $p$ that are integer multiples of $\pi a/\hbar$ are physically
82 realistic. Similarly, the energy operator no longer has an
83 eigenstate for each value of $E$; instead, the only energy
84 eigenstates in the infinitely deep well
85 are $E_n(x)=\sin(n\pi x/ a)$ for positive integers $n$.
87 * COMMENT:
89 ** Eigenstates with different eigenvalues are orthogonal
91 #+begin_quote
92 *Theorem:* Eigenstates with different eigenvalues are orthogonal.
93 #+end_quote
95 ** COMMENT :
96 I can prove this: if $\Lambda$ is any linear operator, suppose $|a\rangle$
97 and $|b\rangle$ are eigenstates of $\Lambda$. This means that
100 \(
101 \begin{eqnarray}
102 \Lambda |a\rangle&=& a|a\rangle,\\
103 \Lambda|b\rangle&=& b|b\rangle.\\
104 \end{eqnarray}
105 \)
107 If we take the difference of these eigenstates, we find that
109 \(
110 \begin{eqnarray}
111 \Lambda\;\left(|a\rangle-|b\rangle\right) &=& \Lambda |a\rangle - \Lambda |b\rangle
112 \qquad \text{(because $\Lambda$ is linear.)}\\
113 &=& a|a\rangle - b|b\rangle\qquad\text{(because $|a\rangle$ and
114 $|b\rangle$ are eigenstates of $\Lambda$)}
115 \end{eqnarray}\)
118 which means that $a\neq b$.
120 ** Eigenvectors of hermitian operators span the space of solutions
122 #+begin_quote
123 *Theorem:* If $\Omega$ is a hermitian operator, then every physically
124 allowed state can be written as a linear sum of eigenstates of
125 $\Omega$.
126 #+end_quote
130 ** Momentum and energy are hermitian operators
131 This ought to be true because hermitian operators correspond to
132 observable quantities. Since we expect momentum and energy to be
133 measureable quantities, we expect that there are hermitian operators
134 to represent them.
137 ** Momentum and energy eigenstates in vacuum
138 An eigenstate of the momentum operator $P$ would be a state
139 \(|p\rangle\) such that \(P|p\rangle=p|p\rangle\).
141 ** Momentum and energy eigenstates in the infinitely deep well
145 * Can you measure momentum in the infinitely deep well?
146 In summary, I thought I knew:
147 1. For any hermitian operator: eigenstates with different eigenvalues
148 are orthogonal.
149 2. For any hermitian operator: any physically realistic state can be
150 written as a linear sum of eigenstates of the operator.
151 3. The momentum operator and energy operator are hermitian, because
152 momentum and energy are observable quantities.
153 4. (The form of the momentum and energy eigenstates in the vacuum potential)
154 5. (The form of the momentum and energy eigenstates in the infinitely deep well potential)
156 Additionally, I understood that because the infinitely deep potential
157 well is not realistic, states of such a system are not necessarily
158 physically realistic. Instead, I understood
159 \ldquo{}realistic states\rdquo{} to be those that satisfy the physically
160 unrealistic Schr\ouml{}dinger equation and its boundary conditions.
162 With that final caveat, here is the problem:
164 According to (5), the momentum eigenstates in the well are
166 \(p(x)= \begin{cases}\exp{(ipx/\hbar)},& \text{for }0\lt{}x\lt{}a;\\0, & \text{for }x<0\text{ or }x>a. \\ \end{cases}\)
168 However, /these/ states are not orthogonal, which contradicts the
169 assumption that (3) the momentum operator is hermitian and (2)
170 eigenstates of a hermitian are orthogonal if they have different eigenvalues.
172 #+begin_quote
173 *Problem 1. The momentum eigenstates of the well are not orthogonal*
175 /Proof./ If $p_1\neq p_2$, then
177 \(\begin{eqnarray}
178 \langle p_1 | p_2\rangle &=& \int_{\infty}^\infty p_1^*(x)p_2(x)dx\\
179 &=& \int_0^a p_1^*(x)p_2(x)dx\qquad\text{ Since }p_1(x)=p_2(x)=0\text{
180 outside the well.}\\
181 &=& \int_0^a \exp{(-ip_1x/\hbar)\exp{(ip_2x/\hbar)dx}}
182 \end{eqnarray}\)
183 $\square$
185 #+end_quote
189 ** COMMENT Momentum eigenstates
191 In free space, the Hamiltonian is \(H=\frac{1}{2m}P^2\) and the
192 momentum operator $P$ has eigenstates \(p(x) = \exp{(-ipx/\hbar)}\).
194 In the infinitely deep potential well, the Hamiltonian is the same but
195 there is a new condition in order for states to qualify as physically
196 allowed: the states must not exist anywhere outside of well, as it
197 takes an infinite amount of energy to do so.
199 Notice that the momentum eigenstates defined above do /not/ satisfy
200 this condition.
204 * COMMENT
205 For each physical system, there is a Schr\ouml{}dinger equation that
206 describes how a particle's state $|\psi\rangle$ will change over
207 time.
209 \(\begin{eqnarray}
210 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
211 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
213 This is a differential equation; each solution to the
214 Schr\ouml{}dinger equation is a state that is physically allowed for
215 our particle. Here, physically allowed states are
216 those that change in physically allowed ways. However, like any differential
217 equation, the Schr\ouml{}dinger equation can be accompanied by
218 /boundary conditions/\mdash{}conditions that further restrict which
219 states qualify as physically allowed.
224 ** Eigenstates of momentum
229 #In the infinitely deep well potential $V(x)=0$, the Schr\ouml{}dinger
231 #\(i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle\)
239 * COMMENT
241 #* The infinite square well potential
243 A particle exists in a potential that is
244 infinite everywhere except for a region of length \(a\), where the potential is zero. This means that the
245 particle exists in a potential[fn:coords][fn:infinity]
248 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
249 }\;x<0\text{ or }x>a.\end{cases}\)
251 The Schr\ouml{}dinger equation describes how the particle's state
252 \(|\psi\rangle\) will change over time in this system.
254 \(\begin{eqnarray}
255 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
256 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
258 This is a differential equation; each solution to the
259 Schr\ouml{}dinger equation is a state that is physically allowed for
260 our particle. Here, physically allowed states are
261 those that change in physically allowed ways. However, like any differential
262 equation, the Schr\ouml{}dinger equation can be accompanied by
263 /boundary conditions/\mdash{}conditions that further restrict which
264 states qualify as physically allowed.
267 Whenever possible, physicists impose these boundary conditions:
268 - A physically allowed state ought to be a /smoothly-varying function of position./ This means
269 that if a particle in the state is likely to be /at/ a particular location,
270 it is also likely to be /near/ that location.
272 These boundary conditions imply that for the square well potential in
273 this problem,
275 - Physically allowed states must be totally confined to the well,
276 because it takes an infinite amount of energy to exist anywhere
277 outside of the well (and physically allowed states ought to have
278 only finite energy).
279 - Physically allowed states must be increasingly unlikely to find very
280 close to the walls of the well. This is because of two conditions: the above
281 condition says that the particle is /impossible/ to find
282 outside of the well, and the smoothly-varying condition says
283 that if a particle is impossible to find at a particular location,
284 it must be unlikely to be found nearby that location.
286 #; physically allowed states are those that change in physically
287 #allowed ways.
290 #** Boundary conditions
291 Because the potential is infinite everywhere except within the well,
292 a realistic particle must be confined to exist only within the
293 well\mdash{}its wavefunction must be zero everywhere beyond the walls
294 of the well.
297 [fn:coords] I chose my coordinate system so that the well extends from
298 \(0<x<a\). Others choose a coordinate system so that the well extends from
299 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
300 situation, they give different-looking answers.
302 [fn:infinity] Of course, infinite potentials are not
303 realistic. Instead, they are useful approximations to finite
304 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
305 of the well\rdquo{} are close enough for your own practical
306 purposes. Having introduced a physical impossibility into the problem
307 already, we don't expect to get physically realistic solutions; we
308 just expect to get mathematically consistent ones. The forthcoming
309 trouble is that we don't.