Solution Manual to Accompany Volume II of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë E-Book
Guillaume Merle
79,99 €
Mehr erfahren.
- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Provides detailed solutions to all 47 problems in the seminal textbook Quantum Mechanics, Volume II
With its counter-intuitive premises and its radical variations from classical mechanics or electrodynamics, quantum mechanics is among the most important and challenging components of a modern physics education. Students tackling quantum mechanics curricula generally practice by working through increasingly difficult problem sets that demand both a theoretical grounding and a solid understanding of mathematical technique.
Solution Manual to Accompany Volume II of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë is designed to help you grasp the fundamentals of quantum mechanics by doing. This essential set of solutions provides explicit explanations of every step, focusing on the physical theory and formal mathematics needed to solve problems with varying degrees of difficulty.
- Contains in-depth explanations of problems concerning quantum mechanics postulates, mathematical tools, approximation methods, and more
- Covers topics including perturbation theory, addition of angular momenta, electron spin, systems of identical particles, time-dependent problems, and quantum scattering theory
- Guides readers on transferring the solution approaches to comparable problems in quantum mechanics
- Includes numerous figures that demonstrate key steps and clarify key concepts
Solution Manual to Accompany Volume II of Quantum Mechanics by Cohen-Tannoudji, Diu and Laloë is a must-have for students in physics, chemistry, or the materials sciences wanting to master these challenging problems, as well as for instructors looking for pedagogical approaches to the subject.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 368
Veröffentlichungsjahr: 2024
Ähnliche
Table of Contents
Cover
Table of Contents
Title Page
Copyright
8 Solutions to the Exercises of Chapter VIII (Complement C
VIII
, § 3). An Elementary Approach to the Quantum Theory of Scattering by a Potential
8.1 Scattering of the Wave by a Hard Sphere
8.2 “Square Spherical Well”: Bound States and Scattering Resonances
Note
9 Solutions to the Exercises of Chapter IX (Complement B
IX
). Electron Spin
9.1 Spin‐Related Measurements of a Particle of Spin 1/2
9.2 An Operator Coupling Momentum P and Spin S
9.3 The Pauli Hamiltonian
9.4 Polarization of a Neutron Beam by Reflection
Note
10 Solutions to the Exercises of Chapter X (Complement G
X
). Addition of Angular Momenta
10.1 The Deuterium Atom
10.2 The Hydrogen Atom
10.3 Two‐Particle System
10.4 Disintegration of a Particle
10.5 Three‐Particle System
10.6 Two‐Particle System and Collision
10.7 Standard Components of a Vector Operator
10.8 Irreducible Tensor Operators; Wigner–Eckart Theorem
10.9 Irreducible Tensor Operators (Follow‐up Exercise)
10.10 Addition of Three Angular Momenta
References
11 Solutions to the Exercises of Chapter XI (Complement H
XI
). Stationary Perturbation Theory
11.1 Punctual Perturbation in an Infinite One‐Dimensional Well
11.2 Localized Perturbation in an Infinite Two‐Dimensional Well
11.3 Perturbations in a Two‐Dimensional Harmonic Oscillator
11.4 Perturbed Circular Motion of a Quantum System
11.5 Three‐Dimensional Perturbation
11.6 One Electronic and Two Nuclear Spins, Spin‐Spin Interactions
11.7 Interaction Between a Nuclear Spin and an Electric Field via its Electric Quadrupole and Magnetic Dipole Moments
11.8 Linear Perturbation Within an Infinite One‐Dimensional Well and Variational Method
11.9 The Hydrogen Atom and the Variational Method
11.10 Determination of the Energies of a Particle in an Infinite One‐Dimensional Well Using the Variational Method
12 Solutions to the Exercises of Chapter XII. An Application of Perturbation Theory: The Fine and Hyperfine Structure of Hydrogen
13 Solutions to the Exercises of Chapter XIII. Approximation Methods for Time‐Dependent Problems
Part I: Solutions to the Exercises of Complement B
XIII
. Linear and Nonlinear Responses of a Two‐Level System Subjected to a Sinusoidal Perturbation
13.1 Competition Between Pumping and Relaxation in a Two‐Level System
13.2 Nonlinear Response of a Two‐Level System Subjected to a Sinusoidal Perturbation
Part II: Solutions to the Exercises of Complement F
XIII
13.1 One‐Dimensional Harmonic Oscillator Subjected to an Electric Field Pulse
13.2 Spin–Spin Interactions During a Collision
13.3 Two‐Photon Transitions Between Non‐equidistant Levels
13.4 Magnetic Response of a System Placed in an Oscillating Magnetic Field
13.5 The Autler–Townes Effect
13.6 Elastic Scattering by a Particle in a Bound State. Form Factor
13.7 A Simple Model of the Photoelectric Effect
13.8 Disorientation of an Atomic Level due to Collisions with Noble Gas Atoms
13.9 Transition Probability per Unit Time Under the Effect of a Random Perturbation. Simple Relaxation Model
13.10 Absorption of Radiation by a Many‐Particle System Forming a Bound State. The Doppler Effect. Recoil Energy. The Mössbauer Effect
References
14 Solutions to the Exercises of Chapter XIV (Complement D
XIV
). Systems of Identical Particles
14.1 Energy Levels of a System of Three Identical Particles
14.2 Two Identical Bosons in a Central Potential
14.3 Identical Electrons in Hybrid Atomic Orbitals
14.4 Collision Between Two Identical Particles
14.5 Collision Between Two Identical Unpolarized Particles
14.6 Possible Values of the Relative Angular Momentum of Two Identical Particles
14.7 Position Probability Densities for a System of Two Identical Particles
14.8 Symmetrization and Measurements
14.9 One‐ and Two‐Particle Density Functions in an Electron Gas at Absolute Zero
Bibliography
End User License Agreement
List of Tables
Chapter 9
Table 9.1 Stationary states of the particle for a spin parallel to .
Table 9.2 Stationary states of the particle for a spin antiparallel to ....
Chapter 10
Table 10.1 Probabilities of finding particle...
Chapter 14
Table 14.1 Possible spin configurations for the three electrons among the t...
Table 14.2 Energy levels and associated degeneracies for a system of three ...
Table 14.3 Possible configurations of the three bosons among the three ener...
Table 14.4 Energy levels and associated degeneracies of a system of three i...
List of Illustrations
Chapter 8
Figure 8.1 Spherical Bessel function of the first kind.
Figure 8.2 Spherical Neumann function of the first kind.
Figure 8.3 Solution ...
Figure 8.4 Graphical resolution of the equation . is represented by dashe...
Figure 8.5 Representative curve of in terms of .
Chapter 9
Figure 9.1
Chapter 11
Figure 11.1 Different energy levels for the different perturbations consider...
Figure 11.2 Probability density in terms of angle at times (solid line),...
Figure 11.3 Representation of the ethane molecule along the CC bond.
Figure 11.4 Energy diagram of a three‐spin system without coupling.
Figure 11.5 Energy levels of (dashed lines) and of (solid lines). The ve...
Figure 11.6 Shape of the E.P.R. spectrum observed for the three‐spin system....
Figure 11.7 Energy levels of the system with coupling between
M
and for ...
Figure 11.8 Shape of the nuclear magnetic resonance spectrum obtained for a ...
Figure 11.9 Exact (solid line) and approximate (dashed line) wave functions ...
Figure 11.10 Exact (solid line) and approximate (dashed line) wave functions...
Figure 11.11 Exact (solid line) and approximate (dashed line) wave functions...
Figure 11.12 Exact (solid line) and approximate (dashed line) wave functions...
Chapter 13
Figure 13.1 Variations of ...
Figure 13.2 Magnetizations of spins pumped at times ...
Figure 13.3 Evolution of magnetizations when .
Figure 13.4 Evolution of magnetizations when .
Figure 13.5 Illustration of the three processes responsible for the resonanc...
Figure 13.6 Energy diagram representing the transitions between , and .
Figure 13.7 Spectral representation of the doublet for (case a), (case b...
Figure 13.8 Energy diagram of a quasi‐two‐level system in the absence of a f...
Figure 13.9 Illustration of the ‐dependency of the time average .
Figure 13.10
Figure 13.11 Distance traveled by a spin between two collisions.
Chapter 14
Figure 14.1 Variation of ...
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
End User License Agreement
Pages
iv
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
183
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
Solution Manual to Accompany Volume II of Quantum Mechanics by Cohen‐Tannoudji, Diu and Laloë
Guillaume Merle and Oliver J. Harper
Authors
Dr. Guillaume Merle
Beihang Sino-French Engineer School
(École Centrale de Pékin)
Beihang University
37 Xueyuan Road
Teaching building no. 2
Haidian district
100191 Beijing
China
Dr. Oliver J. Harper
Lycée Saint Lambert
7 rue Clavel
75019 Paris
France
Cover Image: © antishock/Getty Images
All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
British Library Cataloguing‐in‐Publication Data A catalogue record for this book is available from the British Library.
Bibliographic information published bythe Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2024 WILEY‐VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐41423‐9
ePDF ISBN: 978‐3‐527‐84294‐0
ePub ISBN: 978‐3‐527‐84293‐3
8Solutions to the Exercises of Chapter VIII (Complement CVIII, § 3). An Elementary Approach to the Quantum Theory of Scattering by a Potential
Chapter VIII studies quantum scattering and deals with the effect of a potential on an incident particle. There are several possible reasons for the existence of such a potential. One can imagine the presence of a second particle (such a system is ideal for studying two‐body collisions) or the potential can represent a crystal lattice. Quantum scattering is therefore very useful in practice for studying systems in interaction. Given that the treatment is quantum, the incident particle is modeled by a wave, the particle wave function; the idea is therefore to deduce the resulting wave function (of the scattered particle) following the interaction between the incident particle and the potential. This short chapter is composed of two exercises. The first builds on Complement , uses the same formalism, and studies a similar system. The difference is that the incident particle here is a wave, whereas the complement deals with an wave. The second aims to revisit the “square potential well” using the quantum scattering formalism; this approach reveals the presence of scattering resonances in some cases.
8.1 Scattering of the Wave by a Hard Sphere
Statement
We wish to study the phase shift produced by a hard sphere on the wave (). In particular, we want to verify that it becomes negligible compared to at low energies.
Write the radial equation for the function for . Show that its general solution is of the form: where and are constants.
Show that the definition of implies that:
Determine the constant from the condition imposed on at .
Show that, as approaches zero, behaves like
1
, which makes it negligible compared to .
Comment
As the title suggests, this exercise focuses on the scattering of a wave by a hard sphere. Phase shifts are excellent parameters for describing and summarizing the effect of a potential on an incident wave. Given that the phase shift contains all the information on the scattering of an wave by the potential, the scattering amplitude is deduced by summing over (see equation (C‐55) of Chapter VIII), which finally leads to the differential scattering cross section, an important parameter since it can be measured experimentally.
Whereas the study in Complement deals with the scattering of the wave () by a hard sphere and the resulting phase shift , this exercise adapts the approach (and the notation) for the wave (). The overall aim is to compare the two phase shifts.
It is important to have in mind the approach and the notation of Complement . Also, it is advised to be familiar with Complement that details the expansion onto the basis of free spherical waves, also used here.
Solution
We wish to study the phase shift produced by a hard sphere on the wave (). In particular, we want to verify that it becomes negligible compared to at low energies.
Write the radial equation for the function for . Show that its general solution is of the form:
where and are constants.
Let us study a hard sphere, impenetrable, of radius , where the potential is infinite inside the sphere and vanishing outside it (set by the reference of the potentials), in other words a central potential of the form:
The radial equation can be written, for , using relation (C‐35) of Chapter VIII:
This is just the Schrödinger equation for the radial function when for a wave vector k and a given value of , which becomes for the wave, for which :
with the additional boundary condition on the surface of the sphere .
Let us set and . Hence:
and
and the preceding equation can be written as
It is the spherical Bessel equation of the first kind, whose two linearly independent solutions are, according to § 2‐c. of Complement : the spherical Bessel function of the first kind
and the spherical Neumann function of the first kind
These two functions are depicted in Figures 8.1 and 8.2, respectively. Note that the divergence of the Neumann function visible in Figure 8.2 is not a problem since the study is for .
Figure 8.1 Spherical Bessel function of the first kind.
Figure 8.2 Spherical Neumann function of the first kind.
The general solution of the radial equation is therefore a linear combination of the two independent solutions of the form:
where and are two constants, or equivalently as requested by the statement:
The solution for is depicted in Figure 8.3.
Figure 8.3 Solution of the radial equation for . The values of for small values of are omitted, only keeping the values for which .
Show that the definition of implies that:
The phase shift is by definition given by the asymptotic behavior of (recall that it describes the phase difference between incident and reflected waves far from the potential), in other words according to relation (C‐43) of Chapter VIII:
However, given the expression obtained in question :
We thus directly deduce by identifying the respective terms:
Hence
Determine the constant from the condition imposed on at .
We know that the wave function is continuous at , so since the sphere is impenetrable, which implies, according to the expression obtained in question :
We can easily see that if , then and therefore according to the results of question . This means that removing the hard sphere cancels any phase shift: it is indeed the presence of the potential which incurs the phase shift.
Show that, as approaches zero, behaves , which makes it negligible compared to .
We can write, according to the expression of obtained in question , when approaches 0 and neglecting all terms of the fourth order or higher in :
by performing a Taylor expansion of and to the third order about 0. therefore behaves like when approaches 0. Since according to expression (20) of Complement , is indeed negligible compared to at low energies (i.e. for small values of ). This indeed confirms that the only nonzero phase shifts are those of the first partial waves ( of lowest ).
8.2 “Square Spherical Well”: Bound States and Scattering Resonances
Statement
Consider a central potential such that:
where is a positive constant. Set:
We shall confine ourselves to the study of the wave ().
Bound states
()
(i)Write the radial equation in the two regions and , as well as the condition at the origin. Show that, if one sets: the function is necessarily of the form:
(ii)Write the matching conditions at . Deduce from them that the only possible values for are those that satisfy the equation:
(iii)Discuss this equation: indicate the number of bound states as a function of the depth of the well (for fixed ) and show, in particular, that there are no bound states if this depth is too small.
Scattering resonances
()
(i) Again write the radial equation, this time setting
Show that is of the form:
(ii)Choose . Show, using the continuity conditions at , that the constant and the phase shift are given by: with:
(iii)Trace the curve representing in terms of . This curve clearly shows resonances, for which is maximum. What are the values of associated with these resonances? What is then the value of ? Show that, if there exists a resonance for a small energy (), the corresponding contribution of the wave to the total cross section is practically maximal.
Relation between bound states and scattering resonances
Assume that is very close to , where is an integer, and set:
(i)Show that, if is positive, there exists a bound state whose binding energy is given by:
(ii)Show that if, on the other hand, is negative, there exists a scattering resonance at energy such that:
(iii)Deduce from this that if the depth of the well is gradually decreased (for fixed ), the bound state which disappears when passes through an odd multiple of gives rise to a low‐energy scattering resonance.
Comment
This exercise reuses a certain number of concepts already studied in Chapter I, with the study of a wave function in various spatial zones, inside and outside the well, in the case of bound states and free states. Although the concepts have been studied before, the aim here is to use the formalism of quantum scattering. We will therefore find the same results as before, but this new approach will reveal more physics than before, in particular scattering resonances: the phenomenon whereby the transmission coefficient, related to the scattering cross section, is maximal (equal to 1 in the case of the transmission coefficient) in its variation with the incident energy. As usual, it is always interesting to compare classical and quantum results, noting their similarities and differences.
Solution
Consider a central potential such that:
where is a positive constant. Set:
We shall confine ourselves to the study of the wave ().
The potential is of spherical symmetry, and its radial representation is a well or step function of depth , hence “square,” which explains the title of the exercise “square spherical well” as chosen by the authors of the original book.
Bound states
()
Our focus is on bound states, which correspond in the classical case to a particle trapped inside the well. Its energy is therefore less than that outside the well. We can write .
(i) Write the radial equation in the two regions and , as well as the condition at the origin. Show that, if one sets:
the function is necessarily of the form:
The study is assumed to be limited to the wave (). The radial equation is thus given by
according to relation (C‐35) of Chapter VIII. Given the expression of the potential, the radial equation simplifies to:
• for :
setting :
and the solutions of the radial equation are thus of the form
• for :
i.e., setting and :
i.e., setting:
and the solutions of the radial equation are thus of the form
However, we firstly note that the function must verify the condition at the origin , which means that
so
Secondly, must be bounded at infinity so necessarily , which means that
Hence, in the end:
Within the well, the wave function is that of a standing wave; this is not surprising given the boundary conditions and is a typical result of wave physics (quantum or classical). As in the exercises of Chapter I, we can see that the particle explores the classically “forbidden” zone outside the well, , with an exponentially decreasing radial probability of presence. The quantum wave therefore does not suddenly stop at the potential boundary, as would a classical particle, but decreases exponentially.
(ii) Write the matching conditions at . Deduce from them that the only possible values for are those that satisfy the equation:
Since the spherical potential well is of finite depth, the function must be continuous at , likewise for its first derivative. We therefore write, according to the results of question (i):
and the only possible values for are those that verify the equation obtained by dividing the first equation by the second (each side respectively):
(iii) Discuss this equation: indicate the number of bound states as a function of the depth of the well (for fixed ) and show, in particular, that there are no bound states if this depth is too small.
The number of bound states is equivalent to, according to the results of question (ii), the number of values (and thus of ) that are solutions to the equation
that is, the number of points of intersection between the functions and . The graphical representations of these two functions are sketched in Figure 8.4 for various values of . Let us now further investigate the different cases. Since the functions are defined for and present a vertical asymptote at , we can deduce that:
• there is no bound state if , which corresponds to case 0 in
Figure 8.4
;
• there is a single bound state if
which corresponds to case 1 in Figure 8.4;
• there are two bound states if
which corresponds to case 2 in Figure 8.4;
• …
• there are bound states if
There is therefore no bound state if the depth of the well is less than or equal to . Note that the limit values of correspond to the eigenenergies of the even eigenstates of a particle of mass in an infinite potential well of width .
Figure 8.4 Graphical resolution of the equation . is represented by dashed and dotted lines for various values of , whereas is represented by solid lines. In case 0, there is no solution. In case 1, there is a single solution. In case 2, there are two solutions, and so on for .
In classical mechanics, the particle cannot leave the well, and there is neither constraint on the depth of the potential well nor on the energy within the well (no quantization). In quantum mechanics, the fact that the particle does explore outside the well sets constraints on the states within the well. The result here is somewhat surprising: if the well is not deep enough, , there is no bound state and the particle escapes from the well. There is no classical equivalent for this case.
In the following questions of this exercise, we will focus on similar effects, but on scattering states.
Scattering resonances
()
In classical terms, the particle can explore all of space since it has enough energy to exit the well.
(i) Again write the radial equation, this time setting
Show that is of the form:
As in question .(i), we write
• for :
i.e., setting :
and the solutions of the radial equation are thus of the form
• for :
i.e., setting and :
which can also be written, setting :
and the solutions of the radial equation are thus of the form
However, we firstly know that the function must verify the condition at the origin , which means that
so
Secondly, we write
which means that
so by definition of the phase shift since . So, in the end:
The reasoning is the same as before, but the end result differs. The form of the solution is now similar in both spatial zones (sinusoidal), inside and outside the well. The particle explores all of space. At this stage, classical and quantum mechanics appear to yield similar results.
(ii) Choose . Show, using the continuity conditions at , that the constant and the phase shift are given by:
with:
If , the expression of obtained in question (i) becomes
which can also be written in terms of progressive waves
which shows that can be interpreted as a transmission coefficient through the potential barrier. The spherical potential well is of finite depth so the function must be continuous at , likewise for its first derivative. Hence:
Multiplying the first equation by , squaring the two resulting equations, and summing them, each side respectively, we find:
so
Moreover, dividing either side of the first equation by either side of the second respectively, we find:
Setting , we do indeed find
and
(iii) Trace the curve representing in terms of . This curve clearly shows resonances, for which is maximum. What are the values of associated with these resonances? What is then the value of ? Show that, if there exists a resonance for a small energy (), the corresponding contribution of the wave to the total cross section is practically maximal.
The representative curve of in terms of is sketched in Figure 8.5. If alone could be interpreted as a transmission coefficient in terms of amplitude, can be interpreted as a transmission coefficient in terms of energy. Figure 8.5 shows resonances for which is maximum equaling 1, that is, a maximum of transmission. This can be understood in terms of interferences, sometimes constructive and sometimes (partially) destructive as with a Fabry–Pérot interferometer. Note that for large , large energies, which makes sense: the effect of the potential well on a highly energetic particle is negligible, the probability for the particle to be transmitted is almost 1.
Figure 8.5 Representative curve of in terms of .
According to the expression obtained in question (ii), is maximum when
This condition, , is the condition for constructive interferences on the transmitted wave (see vertical dashed lines in Figure 8.5). It can also be written
with
Therefore, according to the results of question (ii):
since at resonance, and we deduce
The total cross section is, according to relation (C‐58) of Chapter VIII:
For low energies (), we find, at resonance:
according to the results of question (ii). The term is therefore practically maximal at low energies and the contribution from the wave to the total cross section is practically 1, thus almost maximal.
It is interesting to note that, as in question on bound states, the wave function must be studied in its entirety over all of space. Unlike classical mechanics, where the behavior inside the well only depends on the well, the particle is not localized and its behavior depends on the potential over all of space.
In the last question of this exercise, we will once more focus on bound states, bearing in mind the aforementioned resonances.
Relation between bound states and scattering resonances
Assume that is very close to , where is an integer, and set:
(i) Show that, if is positive, there exists a bound state whose binding energy is given by:
In the case where
with , we write
and there are bound states in this case, according to the results of question .(iii). The bound state, associated with the highest value of , corresponds to the lowest value of and the lowest binding energy value. Its binding energy is given by
according to the relation determined in question .(ii). However, for , we can see graphically in Figure 8.4 that the bound state verifies (see the representative curve of in a dotted line), and thus, for the bound state:
Performing a Taylor expansion of at to the first order in , we find:
We thus deduce:
(ii) Show that if, on the other hand, is negative, there exists a scattering resonance at energy such that:
If we can write
with and , we showed in question .(iii) that there is a scattering resonance at energy such that:
to the first order in .
(iii) Deduce from this that if the depth of the well is gradually decreased (for fixed ), the bound state which disappears when passes through an odd multiple of gives rise to a low‐energy scattering resonance.
If we can write
with , we showed in question (i) that there is a bound state (the ) whose binding energy is given by:
If, for fixed , the depth of the well is gradually decreased, then (and a fortiori) also decreases toward
Now, there are only bound states according to the results of question .(iii), and the previous bound state disappears and, as the depth of the potential continues to decrease toward
with , this gives rise, as studied in question (ii), to a scattering resonance of energy such that:
that is, a low‐energy scattering resonance.
Given the results of question .(iii), decreasing the depth of the well is a way of tuning the potential to match the resonance condition and set . This is an interesting result: a bound state is transformed into a scattering resonance by modifying the shape of the well. This disappearance of the bound state provides another interpretation of the conditions for a scattering resonance. Since the particle in quantum mechanics is not localized, behavior within the well has an impact on the wave function outside the well. The localized nature of classical mechanics is not capable of describing such effects.
Note
1
This result is true in general: for any potential of finite range , the phase shift behaves like at low energies.
9Solutions to the Exercises of Chapter IX (Complement BIX). Electron Spin
The spin operator has many properties in common with the angular momentum operator. The spin of a quantum system is therefore similar in many respects to its angular momentum. However, the physical origin behind spin is very different to angular momentum. Indeed, angular momentum stems from the rotation of the system and is analogous to its classical equivalent. Spin, however, has no classical equivalent; it acts as an “intrinsic” angular momentum, but in no case should it be thought of as the angular momentum of the electron cloud within an atom. The motion of the electron cloud to produce typical spin values would be faster than the speed of light, which is of course impossible.
Until now, we have either put spin considerations aside in order to study the motion of quantum systems or simply considered spin in terms of an angular momentum operator (see Chapter IV in particular). If spin is an intrinsic property of quantum systems, it can still affect the motion of particles in position space through various couplings; such considerations are the object of this chapter.
9.1 Spin‐Related Measurements of a Particle of Spin 1/2
Statement
Consider a spin 1/2 particle. Call its spin S, its orbital angular momentum L, and its state vector . The two functions and are defined by:
Assume that:
where are the coordinates of the particle and is a given function of .
a. What condition must satisfy for to be normalized?
b. is measured with the particle in the state . What results can be found, and with what probabilities? Same question for , then for .
c. A measurement of , with the particle in the state , yielded zero. What state describes the particle just after this measurement? Same question if the measurement of had given .
Comment
This exercise deals with measurements performed on a wave function while explicitly accounting for the particle spin. The techniques are standard, as already studied in Chapters III-VI, but it is important to bear in mind the two possible spin values (up and down) in orbital angular momentum measurements, for example. Revising spherical harmonics could be beneficial for some calculations, although simply knowing that they constitute an orthonormal basis is sufficient for solving this exercise.
No spin‐coupling is introduced here in terms of an operator. There is no mention of a Hamiltonian, but the wave functions depend on both the orbital variables and the spin state. This results in correlations between the two: a certain orbital measurement collapses the wave function onto one of a selection of spin states, each with a given probability, and vice versa for a spin measurement affecting an orbital state.
Solution
Consider a spin 1/2 particle. Call its spin S, its orbital angular momentum L, and its state vector . The two functions and are defined by:
Assume that:
where are the coordinates of the particle and is a given function of .
a. What condition must satisfy for to be normalized?
is normalized if and only if . Using the closure relation in the basis with , we find:
by definition of the functions and . Substituting these two functions by their expressions and switching to spherical coordinates, we thus find:
Then, given that the spherical harmonics constitute an orthonormal basis of the space of square‐integrable functions of and (according to relation (45) of Complement ), we find:
b. is measured with the particle in the state . What results can be found, and with what probabilities? Same question for , then for .
A measurement of can yield the following results:
• with a probability
• with a probability
since spherical harmonics constitute an orthonormal basis of the space of square‐integrable functions of and , and given the result of question a. As for a measurement of , it can yield:
•0 with a probability
• with a probability
• with a probability for the same reasons as before. Finally: and a measurement of can yield the results:
• with a probability
• with a probability for the same reasons as before.
c. A measurement of , with the particle in the state , yielded zero. What state describes the particle just after this measurement? Same question if the measurement of had given .
Since, in state , we can write
if a measurement of yields 0, hence , the state describing the particle immediately after the measurement is the projection of onto the states such that and defined by
and, being normalized, we can write
hence and, setting so that is real and positive:
This means that if a measurement of yields 0, then the spin of the particle is necessarily up.
If a measurement of yields , the state describing the particle immediately after the measurement is defined by
and, being normalized, we can write
hence and, setting so that is real and positive:
In this case, the spin of the particle is a superposition of spin states.
9.2 An Operator Coupling Momentum P and Spin S
Statement
Consider a spin 1/2 particle. P and S designate the observables associated with its momentum and its spin. We choose as the basis of the state space the orthonormal basis of eigenvectors common to , and (whose eigenvalues are, respectively, , and ).
We intend to solve the eigenvalue equation of the operator which is defined by:
a.Is Hermitian?
b. Show that there exists a basis of eigenvectors of which are also eigenvectors of . In the subspace spanned by the kets , where are fixed, what is the matrix representing ?
c.What are the eigenvalues of , and what is their degree of degeneracy? Find a system of eigenvectors common to and .
Comment
This exercise deals with an operator whose action is in orbital state space and spin state space. The exercise is rather guided, as the intermediate step of the overall reasoning is made explicit in question b with the study of the observable restrained to spin state space. The operator produces a coupling between spin and momentum.
Solution
Consider a spin 1/2 particle. P and S designate the observables associated with its momentum and its spin. We choose as the basis of the state space the orthonormal basis of eigenvectors common to , and (whose eigenvalues are, respectively, , and ).
We intend to solve the eigenvalue equation of the operator which is defined by:
a. Is Hermitian?
By definition of , we can write
Hence
since the components of P and S are observables so are a fortiori Hermitian. Furthermore, since orbital and spin observables commute, we deduce that:
and is thus Hermitian. It is thus possible to diagonalize the corresponding matrix and find its eigenvalues as requested by the exercise in question c.
b. Show that there exists a basis of eigenvectors of which are also eigenvectors of . In the subspace spanned by the kets , where are fixed, what is the matrix representing ?
We write
and find similar relations for and . Therefore, commutes with , and , which also mutually commute. Hence, there exists a basis of eigenvectors of that are also eigenvectors of .
Note that
according to relation (1) of Complement , so:
The matrix representing in the subspace spanned by the kets where are fixed can thus be derived directly as follows:
with the three Pauli matrices (see relation (2) of Complement ),
Note that, since are real given that are observables, the matrix representing is Hermitian, which confirms that is indeed Hermitian.
c. What are the eigenvalues of , and what is their degree of degeneracy? Find a system of eigenvectors common to and .
According to the results of the preceding question, we write
Let us denote the direction of vector p, and let us consider the orthonormal basis . In the same way that operator can be written
in the basis , it can be written
in the basis , where the matrices , and are defined in the same way as the Pauli matrices , and , i.e.
where vectors u, v, and w are the unitary vectors associated with the directions , and , respectively. Since the vector p has direction u, we write , hence
Moreover, we write by generalizing relations (5) of Complement , hence
We thus directly deduce that has:
• as an eigenvector associated with eigenvalue ;
• as an eigenvector associated with eigenvalue ;
these eigenvalues are infinitely degenerate since all vectors p on a sphere of constant radius have the same eigenvalue (to within the sign according to the spin projection on ). Since the vectors are eigenvectors of with eigenvalues , we can see that the eigenvectors of are also eigenvectors of with the same eigenvalues such that:
9.3 The Pauli Hamiltonian
Statement
The Hamiltonian of an electron of mass , charge , spin (where are the Pauli matrices), placed in an electromagnetic field described by the vector potential and the scalar potential , is written
The last term represents the interaction between the spin magnetic moment and the magnetic field .
Show, using the properties of the Pauli matrices, that this Hamiltonian can also be written in the following form (the “Pauli Hamiltonian”):
