|
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 #Bugs in Quantum Mechanics
|
|
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
|
|
rlm@0
|
45 * What I thought I knew
|
|
rlm@0
|
46
|
|
rlm@0
|
47 The following is a list of things I thought were true of quantum
|
|
rlm@0
|
48 mechanics; the catch is that the list contradicts itself.
|
|
rlm@0
|
49
|
|
rlm@0
|
50 1. For any hermitian operator: Eigenstates with different eigenvalues are orthogonal.
|
|
rlm@0
|
51 2. For any hermitian operator: Any physically allowed state can be
|
|
rlm@0
|
52 written as a linear sum of eigenstates of the operator.
|
|
rlm@0
|
53 3. The momentum operator and energy operator are hermitian, because
|
|
rlm@0
|
54 momentum and energy are measureable quantities.
|
|
rlm@0
|
55 4. In the vacuum potential, the momentum and energy operators have these eigenstates:
|
|
rlm@0
|
56 - the momentum operator has an eigenstate
|
|
rlm@0
|
57 \(p(x)=\exp{(ipx/\hbar)}\) for each value of $p$.
|
|
rlm@0
|
58 - the energy operator has an eigenstate \(|E\rangle =
|
|
rlm@0
|
59 \alpha|p\rangle + \beta|-p\rangle\) for any \(\alpha,\beta\) and
|
|
rlm@0
|
60 the particular choice of momentum $p=\sqrt{2mE}$.
|
|
rlm@0
|
61 5. In the infinitely deep potential well, the momentum and energy
|
|
rlm@0
|
62 operators have these eigenstates:
|
|
rlm@0
|
63 - The momentum eigenstates and energy eigenstates have the same form
|
|
rlm@0
|
64 as in the vacuum potential: $p(x) =
|
|
rlm@0
|
65 \exp{(ipx/\hbar)}$ and $|E\rangle = \alpha|p\rangle + \beta|-p\rangle$.
|
|
rlm@0
|
66 - Even so, because of the boundary conditions on the
|
|
rlm@0
|
67 well, we must make the following modifications:
|
|
rlm@0
|
68 + Physically realistic states must be impossible to find outside the well. (Only a state of infinite
|
|
rlm@0
|
69 energy could exist outside the well, and infinite energy is not
|
|
rlm@0
|
70 realistic.) This requirement means, for example, that momentum
|
|
rlm@0
|
71 eigenstates in the infinitely deep well must be
|
|
rlm@0
|
72 \(p(x)
|
|
rlm@0
|
73 = \begin{cases}\exp{(ipx/\hbar)},& \text{for }0\lt{}x\lt{}a;
|
|
rlm@0
|
74 \\0, & \text{for }x<0\text{ or }x>a. \\ \end{cases}\)
|
|
rlm@0
|
75 + Physically realistic states must vary smoothly throughout
|
|
rlm@0
|
76 space. This means that if a particle in some state is very unlikely to be
|
|
rlm@0
|
77 /at/ a particular location, it is also very unlikely be /near/
|
|
rlm@0
|
78 that location. Combining this requirement with the above
|
|
rlm@0
|
79 requirement, we find that the momentum operator no longer has
|
|
rlm@0
|
80 an eigenstate for each value of $p$; instead, only values of
|
|
rlm@0
|
81 $p$ that are integer multiples of $\pi a/\hbar$ are physically
|
|
rlm@0
|
82 realistic. Similarly, the energy operator no longer has an
|
|
rlm@0
|
83 eigenstate for each value of $E$; instead, the only energy
|
|
rlm@0
|
84 eigenstates in the infinitely deep well
|
|
rlm@0
|
85 are $E_n(x)=\sin(n\pi x/ a)$ for positive integers $n$.
|
|
rlm@0
|
86
|
|
rlm@0
|
87 * COMMENT:
|
|
rlm@0
|
88
|
|
rlm@0
|
89 ** Eigenstates with different eigenvalues are orthogonal
|
|
rlm@0
|
90
|
|
rlm@0
|
91 #+begin_quote
|
|
rlm@0
|
92 *Theorem:* Eigenstates with different eigenvalues are orthogonal.
|
|
rlm@0
|
93 #+end_quote
|
|
rlm@0
|
94
|
|
rlm@0
|
95 ** COMMENT :
|
|
rlm@0
|
96 I can prove this: if $\Lambda$ is any linear operator, suppose $|a\rangle$
|
|
rlm@0
|
97 and $|b\rangle$ are eigenstates of $\Lambda$. This means that
|
|
rlm@0
|
98
|
|
rlm@0
|
99
|
|
rlm@0
|
100 \(
|
|
rlm@0
|
101 \begin{eqnarray}
|
|
rlm@0
|
102 \Lambda |a\rangle&=& a|a\rangle,\\
|
|
rlm@0
|
103 \Lambda|b\rangle&=& b|b\rangle.\\
|
|
rlm@0
|
104 \end{eqnarray}
|
|
rlm@0
|
105 \)
|
|
rlm@0
|
106
|
|
rlm@0
|
107 If we take the difference of these eigenstates, we find that
|
|
rlm@0
|
108
|
|
rlm@0
|
109 \(
|
|
rlm@0
|
110 \begin{eqnarray}
|
|
rlm@0
|
111 \Lambda\;\left(|a\rangle-|b\rangle\right) &=& \Lambda |a\rangle - \Lambda |b\rangle
|
|
rlm@0
|
112 \qquad \text{(because $\Lambda$ is linear.)}\\
|
|
rlm@0
|
113 &=& a|a\rangle - b|b\rangle\qquad\text{(because $|a\rangle$ and
|
|
rlm@0
|
114 $|b\rangle$ are eigenstates of $\Lambda$)}
|
|
rlm@0
|
115 \end{eqnarray}\)
|
|
rlm@0
|
116
|
|
rlm@0
|
117
|
|
rlm@0
|
118 which means that $a\neq b$.
|
|
rlm@0
|
119
|
|
rlm@0
|
120 ** Eigenvectors of hermitian operators span the space of solutions
|
|
rlm@0
|
121
|
|
rlm@0
|
122 #+begin_quote
|
|
rlm@0
|
123 *Theorem:* If $\Omega$ is a hermitian operator, then every physically
|
|
rlm@0
|
124 allowed state can be written as a linear sum of eigenstates of
|
|
rlm@0
|
125 $\Omega$.
|
|
rlm@0
|
126 #+end_quote
|
|
rlm@0
|
127
|
|
rlm@0
|
128
|
|
rlm@0
|
129
|
|
rlm@0
|
130 ** Momentum and energy are hermitian operators
|
|
rlm@0
|
131 This ought to be true because hermitian operators correspond to
|
|
rlm@0
|
132 observable quantities. Since we expect momentum and energy to be
|
|
rlm@0
|
133 measureable quantities, we expect that there are hermitian operators
|
|
rlm@0
|
134 to represent them.
|
|
rlm@0
|
135
|
|
rlm@0
|
136
|
|
rlm@0
|
137 ** Momentum and energy eigenstates in vacuum
|
|
rlm@0
|
138 An eigenstate of the momentum operator $P$ would be a state
|
|
rlm@0
|
139 \(|p\rangle\) such that \(P|p\rangle=p|p\rangle\).
|
|
rlm@0
|
140
|
|
rlm@0
|
141 ** Momentum and energy eigenstates in the infinitely deep well
|
|
rlm@0
|
142
|
|
rlm@0
|
143
|
|
rlm@0
|
144
|
|
rlm@0
|
145 * Can you measure momentum in the infinitely deep well?
|
|
rlm@0
|
146 In summary, I thought I knew:
|
|
rlm@0
|
147 1. For any hermitian operator: eigenstates with different eigenvalues
|
|
rlm@0
|
148 are orthogonal.
|
|
rlm@0
|
149 2. For any hermitian operator: any physically realistic state can be
|
|
rlm@0
|
150 written as a linear sum of eigenstates of the operator.
|
|
rlm@0
|
151 3. The momentum operator and energy operator are hermitian, because
|
|
rlm@0
|
152 momentum and energy are observable quantities.
|
|
rlm@0
|
153 4. (The form of the momentum and energy eigenstates in the vacuum potential)
|
|
rlm@0
|
154 5. (The form of the momentum and energy eigenstates in the infinitely deep well potential)
|
|
rlm@0
|
155
|
|
rlm@0
|
156 Additionally, I understood that because the infinitely deep potential
|
|
rlm@0
|
157 well is not realistic, states of such a system are not necessarily
|
|
rlm@0
|
158 physically realistic. Instead, I understood
|
|
rlm@0
|
159 \ldquo{}realistic states\rdquo{} to be those that satisfy the physically
|
|
rlm@0
|
160 unrealistic Schr\ouml{}dinger equation and its boundary conditions.
|
|
rlm@0
|
161
|
|
rlm@0
|
162 With that final caveat, here is the problem:
|
|
rlm@0
|
163
|
|
rlm@0
|
164 According to (5), the momentum eigenstates in the well are
|
|
rlm@0
|
165
|
|
rlm@0
|
166 \(p(x)= \begin{cases}\exp{(ipx/\hbar)},& \text{for }0\lt{}x\lt{}a;\\0, & \text{for }x<0\text{ or }x>a. \\ \end{cases}\)
|
|
rlm@0
|
167
|
|
rlm@0
|
168 However, /these/ states are not orthogonal, which contradicts the
|
|
rlm@0
|
169 assumption that (3) the momentum operator is hermitian and (2)
|
|
rlm@0
|
170 eigenstates of a hermitian are orthogonal if they have different eigenvalues.
|
|
rlm@0
|
171
|
|
rlm@0
|
172 #+begin_quote
|
|
rlm@0
|
173 *Problem 1. The momentum eigenstates of the well are not orthogonal*
|
|
rlm@0
|
174
|
|
rlm@0
|
175 /Proof./ If $p_1\neq p_2$, then
|
|
rlm@0
|
176
|
|
rlm@0
|
177 \(\begin{eqnarray}
|
|
rlm@0
|
178 \langle p_1 | p_2\rangle &=& \int_{\infty}^\infty p_1^*(x)p_2(x)dx\\
|
|
rlm@0
|
179 &=& \int_0^a p_1^*(x)p_2(x)dx\qquad\text{ Since }p_1(x)=p_2(x)=0\text{
|
|
rlm@0
|
180 outside the well.}\\
|
|
rlm@0
|
181 &=& \int_0^a \exp{(-ip_1x/\hbar)\exp{(ip_2x/\hbar)dx}}
|
|
rlm@0
|
182 \end{eqnarray}\)
|
|
rlm@0
|
183 $\square$
|
|
rlm@0
|
184
|
|
rlm@0
|
185 #+end_quote
|
|
rlm@0
|
186
|
|
rlm@0
|
187
|
|
rlm@0
|
188
|
|
rlm@0
|
189 ** COMMENT Momentum eigenstates
|
|
rlm@0
|
190
|
|
rlm@0
|
191 In free space, the Hamiltonian is \(H=\frac{1}{2m}P^2\) and the
|
|
rlm@0
|
192 momentum operator $P$ has eigenstates \(p(x) = \exp{(-ipx/\hbar)}\).
|
|
rlm@0
|
193
|
|
rlm@0
|
194 In the infinitely deep potential well, the Hamiltonian is the same but
|
|
rlm@0
|
195 there is a new condition in order for states to qualify as physically
|
|
rlm@0
|
196 allowed: the states must not exist anywhere outside of well, as it
|
|
rlm@0
|
197 takes an infinite amount of energy to do so.
|
|
rlm@0
|
198
|
|
rlm@0
|
199 Notice that the momentum eigenstates defined above do /not/ satisfy
|
|
rlm@0
|
200 this condition.
|
|
rlm@0
|
201
|
|
rlm@0
|
202
|
|
rlm@0
|
203
|
|
rlm@0
|
204 * COMMENT
|
|
rlm@0
|
205 For each physical system, there is a Schr\ouml{}dinger equation that
|
|
rlm@0
|
206 describes how a particle's state $|\psi\rangle$ will change over
|
|
rlm@0
|
207 time.
|
|
rlm@0
|
208
|
|
rlm@0
|
209 \(\begin{eqnarray}
|
|
rlm@0
|
210 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
|
|
rlm@0
|
211 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
|
|
rlm@0
|
212
|
|
rlm@0
|
213 This is a differential equation; each solution to the
|
|
rlm@0
|
214 Schr\ouml{}dinger equation is a state that is physically allowed for
|
|
rlm@0
|
215 our particle. Here, physically allowed states are
|
|
rlm@0
|
216 those that change in physically allowed ways. However, like any differential
|
|
rlm@0
|
217 equation, the Schr\ouml{}dinger equation can be accompanied by
|
|
rlm@0
|
218 /boundary conditions/\mdash{}conditions that further restrict which
|
|
rlm@0
|
219 states qualify as physically allowed.
|
|
rlm@0
|
220
|
|
rlm@0
|
221
|
|
rlm@0
|
222
|
|
rlm@0
|
223
|
|
rlm@0
|
224 ** Eigenstates of momentum
|
|
rlm@0
|
225
|
|
rlm@0
|
226
|
|
rlm@0
|
227
|
|
rlm@0
|
228
|
|
rlm@0
|
229 #In the infinitely deep well potential $V(x)=0$, the Schr\ouml{}dinger
|
|
rlm@0
|
230
|
|
rlm@0
|
231 #\(i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle\)
|
|
rlm@0
|
232
|
|
rlm@0
|
233
|
|
rlm@0
|
234
|
|
rlm@0
|
235
|
|
rlm@0
|
236
|
|
rlm@0
|
237
|
|
rlm@0
|
238
|
|
rlm@0
|
239 * COMMENT
|
|
rlm@0
|
240
|
|
rlm@0
|
241 #* The infinite square well potential
|
|
rlm@0
|
242
|
|
rlm@0
|
243 A particle exists in a potential that is
|
|
rlm@0
|
244 infinite everywhere except for a region of length \(a\), where the potential is zero. This means that the
|
|
rlm@0
|
245 particle exists in a potential[fn:coords][fn:infinity]
|
|
rlm@0
|
246
|
|
rlm@0
|
247
|
|
rlm@0
|
248 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
|
|
rlm@0
|
249 }\;x<0\text{ or }x>a.\end{cases}\)
|
|
rlm@0
|
250
|
|
rlm@0
|
251 The Schr\ouml{}dinger equation describes how the particle's state
|
|
rlm@0
|
252 \(|\psi\rangle\) will change over time in this system.
|
|
rlm@0
|
253
|
|
rlm@0
|
254 \(\begin{eqnarray}
|
|
rlm@0
|
255 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
|
|
rlm@0
|
256 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
|
|
rlm@0
|
257
|
|
rlm@0
|
258 This is a differential equation; each solution to the
|
|
rlm@0
|
259 Schr\ouml{}dinger equation is a state that is physically allowed for
|
|
rlm@0
|
260 our particle. Here, physically allowed states are
|
|
rlm@0
|
261 those that change in physically allowed ways. However, like any differential
|
|
rlm@0
|
262 equation, the Schr\ouml{}dinger equation can be accompanied by
|
|
rlm@0
|
263 /boundary conditions/\mdash{}conditions that further restrict which
|
|
rlm@0
|
264 states qualify as physically allowed.
|
|
rlm@0
|
265
|
|
rlm@0
|
266
|
|
rlm@0
|
267 Whenever possible, physicists impose these boundary conditions:
|
|
rlm@0
|
268 - A physically allowed state ought to be a /smoothly-varying function of position./ This means
|
|
rlm@0
|
269 that if a particle in the state is likely to be /at/ a particular location,
|
|
rlm@0
|
270 it is also likely to be /near/ that location.
|
|
rlm@0
|
271
|
|
rlm@0
|
272 These boundary conditions imply that for the square well potential in
|
|
rlm@0
|
273 this problem,
|
|
rlm@0
|
274
|
|
rlm@0
|
275 - Physically allowed states must be totally confined to the well,
|
|
rlm@0
|
276 because it takes an infinite amount of energy to exist anywhere
|
|
rlm@0
|
277 outside of the well (and physically allowed states ought to have
|
|
rlm@0
|
278 only finite energy).
|
|
rlm@0
|
279 - Physically allowed states must be increasingly unlikely to find very
|
|
rlm@0
|
280 close to the walls of the well. This is because of two conditions: the above
|
|
rlm@0
|
281 condition says that the particle is /impossible/ to find
|
|
rlm@0
|
282 outside of the well, and the smoothly-varying condition says
|
|
rlm@0
|
283 that if a particle is impossible to find at a particular location,
|
|
rlm@0
|
284 it must be unlikely to be found nearby that location.
|
|
rlm@0
|
285
|
|
rlm@0
|
286 #; physically allowed states are those that change in physically
|
|
rlm@0
|
287 #allowed ways.
|
|
rlm@0
|
288
|
|
rlm@0
|
289
|
|
rlm@0
|
290 #** Boundary conditions
|
|
rlm@0
|
291 Because the potential is infinite everywhere except within the well,
|
|
rlm@0
|
292 a realistic particle must be confined to exist only within the
|
|
rlm@0
|
293 well\mdash{}its wavefunction must be zero everywhere beyond the walls
|
|
rlm@0
|
294 of the well.
|
|
rlm@0
|
295
|
|
rlm@0
|
296
|
|
rlm@0
|
297 [fn:coords] I chose my coordinate system so that the well extends from
|
|
rlm@0
|
298 \(0<x<a\). Others choose a coordinate system so that the well extends from
|
|
rlm@0
|
299 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
|
|
rlm@0
|
300 situation, they give different-looking answers.
|
|
rlm@0
|
301
|
|
rlm@0
|
302 [fn:infinity] Of course, infinite potentials are not
|
|
rlm@0
|
303 realistic. Instead, they are useful approximations to finite
|
|
rlm@0
|
304 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
|
|
rlm@0
|
305 of the well\rdquo{} are close enough for your own practical
|
|
rlm@0
|
306 purposes. Having introduced a physical impossibility into the problem
|
|
rlm@0
|
307 already, we don't expect to get physically realistic solutions; we
|
|
rlm@0
|
308 just expect to get mathematically consistent ones. The forthcoming
|
|
rlm@0
|
309 trouble is that we don't.
|
