- Trang chủ >>
- Khoa Học Tự Nhiên >>
- Vật lý
Problems solutions in nonrelativistic quantum mechanics
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (37.69 MB, 507 trang )
(pro 6 Cents
ions
1
i n
/
NONRELATIVISTIC
QUANTUM
MECHANICS
Anton Z. Capri
*
1 i
•
World Scientific
cpfo 6kwis
Soíupons
NONRELATIVISTIC
QUANTUM
MECHANICS
Anton Z. Capri
Department of Physics
University of Alberta, Canada
\
www.pdfgrip.com
2 M
V
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Capri, Anton Z.
Problems & solutions in nonrelativistic quantum mechanics / Anton Z. Capri.
p. cm.
Includes bibliographical references.
ISBN 9810246331 (alk. paper) — ISBN 9810246501 (pbk.: alk. paper)
1. Nonrelativistic quantum mechanics - Problems, exercises, etc. I. Title: Problems and
solutions in nonrelativistic quantum mechanics. II. Title: Nonrelativistic quantum
mechanics. III. Title.
QC174.24.N64 C374 2002
530.12'076--dc21
2002029614
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
This book is printed on acid-free paper.
www.pdfgrip.com
To bkaidrite,
zero".
who knows that physics is simple because "everything equals
www.pdfgrip.com
www.pdfgrip.com
Preface
Soon after the first edition of Nonrelativistic Quantum Mechanics appeared,
I received numerous requests for solutions to the problems in that book. To
remedy this situation I started by writing out solutions to the more difficult
problems, but as I proceeded with the third edition of Nonrelativistic Quantum
Mechanics I also revised some of the problems and added quite a few others.
Since in constructing these new problems I had to solve them, in the first place,
to be sure that they were indeed problems that students could solve, I finally
went on to write out solutions to all the problems. However, I did not simply
want a compendium of solutions of the Schrodinger equation since with programs such as Maple or Mathematica these solutions are accessible to every
student. Instead I wanted to concentrate on problems that teach quantum mechanics. It is with this in mind that I began to collect and solve problems. My
idea was to provide a means for students to learn quantum mechanics by "doing
it". This is why the book begins with extremely simple problems and progresses
to more difficult ones.
Some of the problems extend results that are usually taught in a course on
quantum mechanics. But, by having the students obtain the results themselves
they are more likely to retain the ideas and at the same time gain confidence in
their own abilities.
As usual, I tested most of these problems on my students. Sometimes they
came up with very original ways of looking at old problems. I have learned a lot
from my students. It is this learning process that led me to occasionally introduce more than one way of solving a problem since the solutions are intended
to help students to obtain a better understanding of the techniques involved in
tackling problems in quantum mechanics.
The notation and methods used are those explained in Nonrelativistic Quantum Mechanics and I frequently refer to chapters from that book. The chapter
headings are also the same as in Nonrelativistic Quantum Mechanics. Nevertheless, the present book is independent and should serve as a companion to
any of the numerous excellent books on quantum mechanics. Throughout the
book I have used Gaussian units since these are the units most commonly used
in atomic physics. I also tried to arrange the problems according to increasing
degree of difficulty. This, was not always possible since it would have meant
losing the possibility of arranging them according to topic.
It is a pleasure to thank Professor M. Razavy for his generous help in, not
www.pdfgrip.com
only providing me with some wonderful problems and supplying me with numerous references, but also for his constant moral support.
Of course the students who suffered through the courses in which I subjected
them to all sorts of quantum problems also deserve my heartfelt thanks. To their
credit, the undergraduates seldom complained. On the other hand, there was
many an evening, after I had assigned some more than usually difficult problems
in the graduate course on quantum mechanics, that walking down the hall of
the fourth floor of the physics building I heard my name muttered with less than
flattering epithets. Nevertheless, the graduate students survived and many, after
they completed their degree, even thanked me for what they had learned.
It is my hope that these problems and solutions will be of use to future
generations of physics students. At any rate they should provide more entertainment than solving cross word puzzles.
A.Z.Capri
Edmonton, Alberta
July, 2002.
www.pdfgrip.com
Contents
1
The
1.1
1.2
1.3
1.4
1.5
1.6
1.7
B r e a k d o w n of Classical Mechanics
Quantum Number of the Earth
Thermal Wavelength
Photons in a Beam
Hydrogen Atom and de Broglie
Vibrations in NaCl
Crystal Powder
Einstein Coefficients
2
R e v i e w of Classical Mechanics
2.1 Lagrangian and Hamiltonian for SHO
2.2 Lagrangian and Hamiltonian: Simple Pendulum
2.3 Bohr-Sommerfeld Quantization: SHO
2.4 Bohr-Sommerfeld: Particle in a Box
2.5 Larmor Frequency
2.6 Applicability of Bohr-Sommerfeld Quantization
2.7 Schrodinger and Hamilton-Jacobi
2.8 WKB Approximation
2.9 Dumbbell Molecule: Bohr-Sommerfeld
8
8
9
9
10
11
12
12
13
14
3
Elementary Systems
3.1 Commutator Identities
3.2 Complex Potential
3.3 Group and Phase Velocity
3.4 Linear Operators
3.5 Probability Density
3.6 Angular Momentum Operators
3.7 Beam of Particles
3.8 Time Evolution of Wave Function
3.9 Operator Hamiltonian
3.10 Zero of Energy
3.11 Some Commutators
3.12 Eigenfunction for a Simple Hamiltonian
15
15
16
17
18
19
20
21
22
23
24
25
26
www.pdfgrip.com
1
1
2
2
3
4
4
5
4
One-Dimensional Problems
4.1 Potential Step
4.2 Deep Square Well
4.3 Hydrogenic Wavefunction
4.4 Bound State Wavefunction, Current, Momentum
4.5 Time Evolution for Particle in a Box
4.6 Particle in Box: Energy and Eigenfunctions
4.7 Particle in a Box
4.8 Particles Incident on a Potential
4.9 Two Beams Incident on a Potential
4.10 Ramsauer-Townsend Effect
4.11 Wronskian and Non-degeneracy in 1 Dimension
4.12 Symmetry of Reflection
4.13 Parity and Electric Dipole Moment
4.14 Bound State Degeneracy and Current
4.15 Car Reflected from a Cliff
29
29
30
32
33
34
35
36
36
38
39
40
41
43
44
45
5
More O n e - D i m e n s i o n a l P r o b l e m s
5.1 Motion of a Wavepacket
5.2 Lowest Energy States
5.3 Particle at Rest
5.4 Scattering from Two Delta Functions
5.5 Reflection and Transmission Amplitudes: Phase Shifts
5.6 Oscillator Against a Solid Wall
5.7 Periodic Potential
5.8 Reflection and Transmission Through a Barrier
5.9 Hermite Polynomials: Integral Representation
5.10 Matrix Element Between Degenerate States
5.11 Hellmann-Feynman Theorem
48
48
50
51
51
53
55
56
57
58
60
61
6
M a t h e m a t i c a l Foundations
6.1 Cauchy Sequence in a Finite Vector Space
6.2 Nonuniqueness of Schrodinger Representation
6.3 Degeneracy and Commutator
6.4 von Neumann's Example
6.5 Projection Operator
6.6 Spectral Resolution
6.7 Resolvent Operator
6.8 Deficiency Indices
6.9 Adjoint Operator
6.10 Projection Operator
6.11 Commutator of Lz and
6.13 Domain of Kinetic Energy: Polar Coordinates
6.14 Self-Adjoint Extensions of p 4
63
63
64
64
65
67
68
68
69
71
72
74
74
75
76
www.pdfgrip.com
7
Physical I n t e r p r e t a t i o n
7.1 Tetrahedral Die
7.2 Probabilities, Expectation Values, Evolution
7.3 (Lx) and (Ly) in an Eigenstate of Lz
7.4 Free Particle Propagator
7.5 Minimum Uncertainty Wavefunction
7.6 Spreading of a Wave Packet
7.7 Time-dependent Expectation Values
7.8 Ehrenfest Theorem
7.9 Compatibility Theorem
7.10 Constant of the Motion
7.11 Spreading of a Gaussian Wavepacket
7.12 Incorrect Time Operator
7.13 Probability to Find a Particle
7.14 Sphere Bouncing on Sphere
7.15 Cloud Chamber Tracks
7.16 Spin 1 Measurement in Two Directions
7.17 Particle in a Box: Probabilities and Evolution
7.18 Free Wave Equation: Translation Invariance
7.19 Free Wave Equation: Accelerated Frame
7.20 The Wigner Function
7.21 Properties of the Wigner Function
7.22 Uncertainty Relation and Wigner Function
79
79
79
80
81
82
84
85
87
89
90
90
92
92
93
95
95
97
99
101
103
105
106
8
D i s t r i b u t i o n s a n d Fourier T r a n s f o r m s
8.1 Properties of the Delta Function
8.2 Representation of Delta Function
8.3 Normalization of Scattering Solution
8.4 Tempered Distribution
8.5 Fourier Transform of V
8.6 Tempered Distribution of Fast Decrease
8.7 A Useful Identity
8.8 A Representation of ¿(a:)
8.9 Fourier Transform of ¿ ' " ' ( x )
8.10 Value of x m S < - n \x)
8.11 Distribution Occurring in Fermi's Golden Rule
108
108
110
112
114
114
115
116
118
118
119
119
9
Algebraic M e t h o d s
9.1 An Operator Identity
9.2 Expectation Values: Simple Harmonic Oscillator
9.3 Angular Momentum Matrices
9.4 Displaced Oscillator
9.5 Dipole Matrix Elements
9.6 Scalar Operator
9.7 Probability to Obtain I, m
9.8 Probability to Obtain I, m Along Different Axis
122
122
123
124
127
128
129
129
130
www.pdfgrip.com
9.9
9.10
9.11
9.12
9.13
9.14
9.15
9.16
9.17
9.18
9.19
9.20
9.21
9.22
9.23
9.24
9.25
9.26
9.27
9.28
9.29
9.30
9.31
Commutators of x and p with L
Some Eigenfunctions of Angular Momentum
Expectation Value of Lx
Rotation Invariance of the Hamiltonian
Uncertainty Relation for SHO
Baker-Campbell-Hausdorff Formula
A Useful Commutator
Uncertainty in Lz
Expectation Values of Angular Momentum
Validity of Ehrenfest's Theorem
Wigner Problem: Annihilation and Creation
Wigner Problem: SHO
Identity for Pauli Matrices
Operator Identity - Spin Rotation
An Operator Identity
Commutator with Inverse Operator
Schwinger Method for Angular Momenta
Minimum Uncertainty in Jx
Unsold's Theorem and its Application
Rotation Matrix for j= 1
Algebra and Constants of the Motion
Coherent State and Normal Ordering
Normal Ordering of xn
10 Central Force P r o b l e m s
10.1 Isotropic SHO in Two Dimensions
10.2 Attractive Exponential Potential
10.3 Reduction of the Two-body Problem
10.4 Particle in a Spherical Potential Well
10.5 Particle on Surface of a Cylinder
10.6 Expectation Values: Electron in a H-atom
10.7 Parity in Spherical Coordinates
10.8 Magnetic Moment due to Orbital Motion
10.9 Spherical Square Well
10.10 Binding Energy and Potential
10.11 Generating Function: Laguerre Polynomials
10.12 Normalization of Hydrogen Wavefunction
10.13 Kramers' Relation
10.14 Quantum Mechanical Virial Theorem
10.15 Ehrenfest Theorem for Angular Momentum
10.16 Angular Momentum of a Two-Particle System
10.17 Hulthén Potential: Ground State
10.18 Hydrogenic Atom in Two Dimensions
10.19 Runge-Lenz Vector: Constant of the Motion
10.20 Runge-Lenz Vector: Hydrogen Spectrum
www.pdfgrip.com
132
132
134
136
137
138
139
140
140
142
143
145
147
148
149
150
150
152
153
154
155
158
158
162
162
164
165
166
168
169
170
170
171
173
174
175
177
180
181
182
183
184
186
188
11 Transformation Theory
11.1
Fourier Transform of Hermite Functions
196
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16
11.17
11.18
11.19
11.20
11.21
11.22
11.23
11.24
Schrodinger Equation in Momentum Space
Heisenberg Equation for a Free Particle
Dirac Picture for Displaced SHO
Heisenberg Picture for Displaced SHO
Heisenberg Picture: SHO and Constant Force
Heisenberg Picture: Constant Force
Schrodinger Picture: Constant Force
Dirac Picture: Constant Magnetic Field
Coherent State
Coherent State: Overlap of Two States
Coherent State: Wavefunction
Squeezing Operator
No Eigenstates for Creation Operator
Spin Coherent State: Euler Angles
Minimum Uncertainty Spin Coherent States
Spin Coherent States: Complex Variables
Useful Commutator
Forced SHO
3-D Simple Harmonic Oscillator
Quadrupole Tensor
Eigenfunction of J 2 , L2, and Jz
SHO: A Time-independent Operator
SHO with Time-Dependent Spring
197
198
199
202
203
204
205
206
208
208
209
210
211
212
214
216
219
219
221
221
223
225
226
12 N o n - d e g e n e r a t e P e r t u r b a t i o n T h e o r y
12.1 Expansion of l / | r \ — r^\
12.2 Second Order Correction to State
12.3 1/2 Ax2 Perturbation of SHO
12.4 1/4 Ax4 Perturbation of SHO
12.5 1/4 \x4 - Brillouin-Wigner Perturbation
12.6 Two-level System
12.7 Approximate SHO
12.8 Two-dimensional SHO
12.9 Kuhn-Thomas-Reiche Sum Rule
12.10 Electron in Box Perturbed by Electric Field
12.11 Positronium
12.12 Rigid Rotator in Electric Field
12.13 Electric Dipole Moment Sum Rule
12.14 Another Sum Rule
12.15 Gaussian Perturbation of SHO Bosons
12.16 Gaussian Perturbation of SHO Fermions
12.17 Polarizability: Particle in a Box
12.18 Atomic Isotope Effect
12.19 Relativistic Correction to H atom
www.pdfgrip.com
232
232
234
235
237
238
239
242
243
246
248
249
250
251
253
254
255
256
258
260
12.20 van der Waals' Interaction
260
13 D e g e n e r a t e P e r t u r b a t i o n T h e o r y
13.1 Stark Effect for n = 2 Level in H
13.2 Perturbation of Particle in a Box
13.3 Perturbation of Isotropic Two-dimensional SHO
13.4 Two-dimensional SHO with Off-diagonal Term
13.5 Non-diagonal Two-dimensional SHO
13.6 Particle in a Box Perturbed by Electric Field
13.7 Unusual Particle on Interval
13.8 Rigid Rotator in Magnetic Field
13.9 Axp y Perturbation of SHO
13.10 Paschen-Back Effect
13.11 H Atom: Weak Field Stark Effect
263
263
264
266
267
270
272
273
274
276
277
281
14 Further A p p r o x i m a t i o n M e t h o d s
14.1 Variational Ground State of SHO
14.2 Variational Ground State of H2 Molecule
14.3 Square Barrier: W K B Approximation
14.4 Variational Ground State in Gaussian Potential
14.5 Variational Ground State: Quartic Potential
14.6 WKB: Ball Bouncing on a Floor
14.7 Ground State of H *
14.8 Variational Solution: Particle in a Box
14.9 Hydrogen Atom: Variational Technique
14.10 Bound State in One Dimension
14.11 Field Emission: W K B Approximation
14.12 Deuteron: Variational Principle
14.13 Bouncing Ball: Variational Calculation
14.14 Beta Decay of Tritium
14.15 Anharmonic Oscillator
14.16 Nonlinear SHO: Variational Calculation
14.17 WKB Solution and Parity
284
284
285
287
289
292
294
295
298
299
301
302
303
305
306
307
308
311
15 T i m e - D e p e n d e n t P e r t u r b a t i o n T h e o r y
15.1 Transition Probability: Bound State to Free
15.2 Photo-disintegration of Deuteron
15.3 Excitation of SHO
15.4 Excitation of SHO by Stiffer Spring
15.5 Periodic Perturbation
15.6 Excitation of H-atom
15.7 Expanding Box
15.8 Sudden Displacement of SHO
15.9 Sudden Perturbation of Two-level Atom
15.10 Berry's Phase
15.11 Neutron in Rotating Magnetic Field
314
314
316
319
321
322
323
325
326
327
329
333
www.pdfgrip.com
- -1 • JO Jí ' 11NI I
15.12
15.13
15.14
15.15
15.16
Excitation of Electron by Electric Field
Neutron Magnetic Moment
Electron Passing Through Magnetic Field
SHO: Sudden Transition
Coulomb Excitation
335
336
337
337
338
16 P a r t i c l e i n a U n i f o r m M a g n e t i c F i e l d
16.1 Estimate of Magnetic Energies
16.2 Radii of Landau Levels
16.3 Equation of Continuity
16.4 Gauge Invariance
16.5 Gauge Transformations and Observables
16.6 Spin 1/2 Particle in Magnetic Field
16.7 Spin 1/2 in Magnetic Field: Heisenberg Equations
16.8 Separation of Spin and Space for Spin 1/2
16.9 Spin 1/2 in Time-dependent Magnetic Field
16.10 Spin 1/2 in Rotating Magnetic Field
342
342
343
344
345
346
348
349
352
352
354
17 A n g u l a r M o m e n t u m , E t c .
17.1 Operator to Lower Total J
17.2 Energy Shift Due to a Magnetic Field
17.3 Coupling of Spin 1 to Spin 1/2
17.4 Example of Wigner-Eckart
17.5 Rotations for Spin 1/2 and Spin 1
17.6 Spin 1/2 Coupled to Spin 3 / 2
17.7 Coupling of Spin 1 or 0 and Spin 1/2
17.8 Identity for Constant Magnetic Field
17.9 The State \n,j, m,l)
17.10 Landé ^-factor
17.11 Spin and Space Coordinates
17.12 Clebsch-Gordon for
3/2
17.13 Rigid Rotator in a Step Potential
17.14 Spin Dependent Operators for Two Particles
358
358
359
361
362
364
366
367
368
369
369
372
373
373
375
18 S c a t t e r i n g - T i m e D e p e n d e n t
18.1 Cross-section from Experiment
18.2 Green's Functions for Free Particle States
18.3 Dispersion Relations
18.4 Kalien-Yang-Feldman Equations
18.5 Born Approximation
18.6 Scattering in CM and Laboratory Frame
18.7 Propagator for a Free Particle
18.8 Propagator for Simple Harmonic Oscillator
377
377
378
379
380
381
382
385
386
www.pdfgrip.com
19 Scattering - T i m e I n d e p e n d e n t
19.1 Equations for Spherical Bessel Functions
19.2 Rodrigues Formula: Spherical Bessel Functions
19.3 Wronskian for Spherical Bessel Functions
19.4 Superposition of Yukawa Potentials
19.5 B o r n Approximation for Gaussian Potential
19.6 Born Approximation for Square Well
19.7 Phase Shifts for Delta-function Potential
19.8 Phase Shifts for Yukawa Potential
19.9 Low Energy s-Wave Amplitude
19.10 Spherical Potential Shell
19.11 Expressions for j0{x) and n0(x)
19.12 Effective Range, Scattering Length
19.13 Effective Range, Scattering Length: Yukawa Potential
19.14 Shape-independent Parameters
19.15 Phase Shifts for Hard Sphere
19.16 Resonance for Square Well
19.17 Double Slit
19.18 Born Approximation: Spherically Symmetric Potential
19.19 Scattering from a Separable Potential
19.20 Generalized Optical Potential
19.21 Free Particle Eigenfunctions
19.22 Scattering from an Inverse Square Potential
19.23 Neutron-Proton Scattering: Spin Flip
19.24 Reflectionless Potential in One Dimension
19.25 n-p Scattering: Singlet and Triplet States
19.26 Phase Shift, Scattering Length, Etc
19.27 Scattering off a Diatomic Molecule
19.28 WKB s-Wave Phase Shift: Attractive Potential
19.29 WKB s-Wave Phase Shift: Hulthén Potential
19.30 WKB Approximation for Phase Shifts
19.31 Zero-Range Potential
19.32 Calogero Equation
389
389
392
393
393
394
395
397
399
400
402
403
404
405
406
407
411
412
413
414
415
416
418
419
420
422
423
425
426
427
428
431
435
20 S y s t e m s of Identical Particles
20.1 Periodic Table
20.2 Identical s = 1/2 Particles in I = 0,1 States
20.3 Two Identical s = 0 Particles
20.4 Identical s = 1/2 Particles in Centre of Mass
20.5 Heisenberg Field Operator for Bosons
20.6 Heisenberg Field Operator for Fermions
20.7 Two-body Interaction
20.8 Diagonalization of Boson Hamiltonian
20.9 Formula for exp(—^4)5 exp(>l)
20.10 Bogoliubov Transformation: Fermions
20.11 Diagonalization of Bose Hamiltonian
439
439
439
440
441
442
444
444
445
447
448
450
www.pdfgrip.com
20.12
20.13
20.14
20.15
20.16
20.17
20.18
20.19
20.20
Diagonalization of Quadratic Hamiltonian
Bogoliubov Transformation: Bose Operators
Density Matrix for a Subsystem
Density Matrix and S-Matrix
Zero Energy Bound States of Two Fermions
Bose Number Operator: Constant of Motion
Bose Operator: More Constants of Motion
The Pauli Problem
Atomic Isotope Effect
452
453
456
457
458
460
461
462
464
21 Q u a n t u m S t a t i s t i c a l M e c h a n i c s
21.1 Average Energy of Assembly of SHO's
21.2 Properties of the Density Matrix
21.3 Expectation Values for Spin
21.4 Expectation Value for Number of Particles
21.5 Spin 1/2 Polarization
21.6 Density Matrix for Spin s = l
21.7 Polarization Vector for Spin j
21.8 Composite Density Matrix
21.9 Arbitrariness of Composite Density Matrix
21.10 Two Energy Levels Bose Gas
21.11 Density Matrix: Particles Coupled by Spring
21.12 Particles: Dissimilar, Bose, and Fermi
21.13 A Three-Level Laser
21.14 Integrals from Q u a n t u m Statistical Mechanics
467
467
469
470
471
472
473
475
476
477
479
481
483
484
486
Index
489
www.pdfgrip.com
www.pdfgrip.com
Chapter 1
T h e B r e a k d o w n of Classical
Mechanics
1.1
Quantum N u m b e r of the Earth
Calculate the principle quantum number for the earth in its orbit about the sun.
What is the energy difference between two neighbouring energy levels?
Hint: For large n, E„ ss Eciassicai .
Solution
The classical energy of the earth in its orbit about the sun is
where
m is the mass of the earth = 6 x 10 27 g ,
M is the mass of the sun = 2 x 10 33 g ,
G is the gravitational constant = 6.67 x 1 0 - 8 dyn cm 2 /g 2 and
R is the sun-earth distance = 1.5 x 10 13 cm .
Proceeding as for the hydrogen atom we find that we need only replace e 2 by
GmM. Thus, we find
GmM
(1.1.2)
Solving for n we get
GMm2
= 18 x 10
(1.1.3)
Therefore
n « 4 x 10 73 .
(1.1.4)
www.pdfgrip.com
1
Also we see t h a t since
1 /2tr
\2
An
An
AE — - I —-GmM ) m —3 = E\
2 \ h
J
n
n
,
(1.1.5)
we get AE = 6.4 x 1 0 - 3 4 erg .
1.2
Thermal Wavelength
W h a t is the wavelength associated with gas molecules at a temperature T?
Estimate this wavelength for a typical gas at room temperature and compare it
to visible light.
Solution
The energy U of a molecule at temperature T is given by the equipartition
principle
U = \ k
T = y .
B
(1.2.6)
Therefore, the wavelength is
A = hc/U
.
(1.2.7)
At room temperature T « 300 K. So,
U = 1.5 x 1.38 x 10" 2 3 x 300 J = 6.21 x 1 0 - 2 1 J .
A =
6 63 X
-
6
10
2~
3
^ o 3 _ Q 0 x 1 Q 8 = 3.20 x 1 0 - m .
(1.2.8)
(1.2.9)
This is considerably longer than visible light which has a wavelength of about
5 x 10~ 7 m .
1.3
P h o t o n s in a B e a m
For a monochromatic beam of electromagnetic radiation of wavelength (A «
5000 Á) , intensity 7 = 1 w a t t / m " , calculate the number of photons passing an
area of A = 1 cm 2 normal to the beam in one second.
Solution
The energy of a photon is given by
-^photon —-
—y
•
(1.3.10)
1 he total energy of the beam is E — IA = N E p h o t o n where N is the number of
photons. Therefore,
..
I AX
1 x 10" 4 x 5000 x 10 - 1 0
N = —— =
6.63 x 10 - 34 x 3.00 x 10»
he
=
9 k v i n14
^ X 10" phot°nS "
www.pdfgrip.com
( L 3 1 1 )
i .4.
H r vnulTÜl\
1.4
Aiurn
ó
Hydrogen A t o m and de Broglie
Show that if one assumes that the circumference of a stationary state orbit of an
electron in a hydrogen atom is an integral multiple of the de Broglie wavelength,
one also obtains the correct energy levels.
Solution
The de Broglie wavelength is given by
A = h/p .
The circumference is 2irr. Thus, we write
2nr= n\
.
The energy is given by
it
E
P2
=
e2
to
<L412>
" 7 -
We substitute for p and r in terms of A and introduce the fine structure constant
2ne2
« = -¡¡r
(1.4.13)
to get
h2
2mA2
E =
hca
n\
(1.4.14)
We now equate the Coulomb force with the mass times the centripetal acceleration
mv2
2xe2
( L415 )
— = ^xThis yields
h2
27re2
C-4-16'
S P = 7 T '
We can now solve for A.
1
mac 1
A=—~n-
(1.4.17)
Combining this with our previous result we find
E =
ma2c2
^ 5
ma2c2
1 ma2c2
S 5 - =www.pdfgrip.com
- j — •
, ,
C-418)
1.5
Vibrations in N a C l
The shortest possible wavelength of sound in sodium chloride is twice the lattice
spacing, about 5.8 x 1 0 - 8 cm. The sound velocity is approximately 1.5 x 105
cm/sec.
a) Compute a rough value for the highest sound frequency in the solid.
b) Compute the energy of the corresponding phonons, or quanta of vibrational
energy.
c) Roughly what temperature is required to excite these oscillations appreciably?
Solution
The shortest wavelength is given by
A m , n = 5.8 x 1 0 - 8 cm = 2 x lattice spacing
»o = l , 5 x 105 cm/s .
(1.5.20)
a)
«o
A '
(1.5.21)
Therefore,
vmax
=
Amin
= 2.6 x 10 12 Hz .
(1.5.22)
b)
Emax = hvmax
c
= 1.7 x 1 0 - 2 1 J = 0.011 eV
(1.5.23)
)
u = \kBT
.
Assume u « Emax.
(1.5.24)
Therefore,
T ~ ^Emax
on
3k B - 8 3 K
1.6
(1.5.19)
(1.5.25)
•
Crystal Powder
Estimate the effect on the specific heat of reducing a crystal to a fine powder of
dimensions of about 1 0 - 5 cm.
Hint: Study the Debye model [1.1] of specific heat and realize that the size of
the crystal now also imposes an upper limit on the wavelength of the sound
waves in the crystal.
www.pdfgrip.com
Solution
The effect of grinding up the crystal into a powder is to limit the maximum
wavelength of a standing wave in the crystal to roughly the size of the crystal
particles. This changes the integral in the Debye expression [1.1] for the internal
energy from
u
1 2 t t ( k B T ) 4 r — x3dx
= - i ^ ~ J 0
~ i
(i.e.26)
to
^powder
—
l2n(kBT)4
iXmax
x3 dx
h3y
0
Jxmtn
ex - 1
4
12Tv(k B T) [*""" x3dx
rr
" — V S T J.
— r
f'-"7»
Here, we have introduced
•Emm —
h
^ ^min
k B A
=
h
Do
~j ™ T
m a x
(1.6.28)
where Á m a x = 1 0 - 5 cm « size of the powder particles. If we now estimate x m i n
at room temperature by using that vo ps 500 m / s , we get x m ¡„ « 8 x 1 0 - 4 < < 1.
Therefore, we can approximate the exponential in the last integral by ex ss 1 + a:
and get
rxmtn x3dx
rrm,„
X dX
Jo
e ' - l ™J0
~ ~Z
j
(1.6.29)
X
min-
This means that the internal energy U is reduced by
A u
=
12n-(*Br)4 x /I
iturh3V¡¡
X
hv2 o \ 3
4tt kBT
(£rrr --) = ^ 1^3 kBXmax ) ~ K
(1-6.30)
Therefore, the specific heat per unit volume at constant volume
\_dU_
V dT
is decreased by a constant amount, namely
47rkB _ 4tt x 1.38 x 1 Q - 2 3 J / K
^max
(10- 7 )3m3
1.7
0.173 J / ( K m 3 ) .
(1.6.31)
Einstein Coefficients
For a collection of atoms with energies En , n — 1, 2, 3, . . . submerged in
a background of radiation at a temperature T, the following transitions may
occur:
1) spontaneous from n —>• m En > Em
www.pdfgrip.com
2) induced from n - » m
3) induced from m —» n .
At equilibrium, at a temperature T, the emission and absorption probabilities
are given by
Pmn
=
KNn[Bmnp(v)+Amn]
Pnm
=
KNm[Bnmp(v)]
emission
(1.7.32)
absorption.
(1.7.33)
Here, hu = En — Em, K is a proportionality constant, and Nn, Nm are respectively the number of atoms in the states n and m . The coefficients Amn, Bmn
are known respectively as the "Einstein Coefficients" of spontaneous and induced emission, whereas the coefficient Bnm is known as the "Einstein Coefficient" of induced absorption. [1.2] Use these equations together with Planck's
radiation law for the radiation density p(v) at equilibrium to show that
1) the Einstein coefficients of induced absorption and emission are the same,
that is that Byim — Bum and t h a t
2) the Einstein coefficients of spontaneous and induced emission are related by
Anm ~
có
Bnm
.
Solution
In equilibrium, at a temperature T, if the number of atoms in the state n and
m is given by Nn and Nrn respectively, we have t h a t
Nn=Ne~E"/kBT
,
Nm = Ne~Bm/kBT
(1.7.34)
where N is the total number of atoms. Therefore,
Nm = Nn e(En-Em)/kBT
_
N n ehv/kBT
(1.7.35)
Also at equilibrium the number of transitions from n —»• m equals the number
of transitions from m —> n. Thus, we have
NmPnm = NnPmn
(1.7.36)
or
Nnehv'kBT
Brnnp{v)=Nn[Bnmp{v)
+ Anm}
.
(1.7.37)
Thus, solving for the radiation density p we get
<•<">=^rJBBm\B
c
•
J^nm/
But, at equilibrium the radiation density is given by Planck's Law
www.pdfgrip.com
<17.38)
Therefore, comparing these two equations we see that we have
Bnm
— Bmn
(1.7 .40)
and
871" , 3 n
Aim —
'
(1.7.41)
Bibliography
[1.1] A.Z. Capri, Nonrelativistic Quantum Mechanics 3rd edition, World Scientific Publishing Co. Pte. Ltd., section 1.12, (2002) .
[1.2] F.K. Richtmyer, E.H. Kennard, and J.N. Cooper, Introduction to Modern
Physics, 6th edition, sec 13.12, McGraw-Hill, New York, (1969).
www.pdfgrip.com
Chapter 2
R e v i e w of Classical
Mechanics
2.1
Lagrangian and Hamiltonian for SHO
Find the Lagrangian for a harmonic oscillator. Use the definition of conjugate
momentum p to find it and also the Hamiltonian H .
Solution
For a simple harmonic oscillator we have
T
1
-mv
2
(2.1.1)
Therefore,
L = T — V = \-rnv2 - -kx2
2
2
.
(2.1.2)
Next we compute the canonical momentum
dL
p = -x- = mv .
ov
(2.1.3)
Then, we can substitute p/m for v and get
(2.1.4)
www.pdfgrip.com
8
