Problems in Classical and Quantum Mechanics Extracting the Underlying Concepts

Problems in Classical and Quantum Mechanics
 
Author:
J. Daniel Kelley, Jacob J. Leventhal
Release at: 2017
Pages: 358
Edition:
First Edition
File Size: 4 MB
File Type: pdf
Language: English


Description of Problems in Classical and Quantum Mechanics Extracting the Underlying Concepts


This book is not a textbook. It is a collection of problems intended to aid students in their undergraduate and graduate-level courses in physics. The book was, however, formulated with students who are preparing for the Ph.D. qualifying exam in mind. Thus, the problems that are included by are of the type that could be on this exam or are problems that are meant to elucidate an important principle. There are many compilations of physics problems available to students, so it is reasonable to ask why this one is different. It is different because the aim is to place the problems in the broader context of the subject. The book is meant to facilitate the development of problem-solving skills to aid in understanding physics.
We state the problem and then present the solution in detail. Further, we note and discuss the significance of the problem in the context of the subject under study. e analyze the broader implications of the solution including limiting cases and the elation to other problems. Many of the solutions are accompanied by a tutorial on heir meaning and the route to solution. We stress that the solution to the problem is just the beginning of the learning process. The subtitle infers manipulation of the solution and changing that the associated parameters can provide a great deal of insight.
Our approach is to make the discussions of each problem seem as though the student has come to one of our offices and asked for help solving it. It is our belief that when students come for help, it is not enough to simply show them how to arrive t the solution. We discuss with them the physics that should be learning from the problem. That is, after all, why problems are assigned. We want the students to ask themselves What physics can I learn from the problem? not How can work his problem and go on to the next one as quickly as possible?
Another feature of this book is the inclusion of more mathematical detail in the solutions than is usually provided. We have done this because the book is meant to e an aid in learning physics. Thus, at the risk of lengthening the book, we have attempted, when possible, to relieve the reader of the burden of spending a lot of time on mathematical detail after the plan of attack has been formulated.

Content of Problems in Classical and Quantum Mechanics Extracting the Underlying Concepts


Part I Classical Mechanics
1 Newtonian Physics 3
1 Simple pendulum  3
2 Free fall with drag  5
3 Collisions—2 balls and a brick wall  8
4 Collisions—two blocks and a spring  10
5 Two unequal masses attached to a spring  11
6 Hole dug through a diameter of the earth  12
7 Hole dug through a chord of the earth  14
8 Sphere with a spherical (off center) hole  15
9 Moving inclined plane 18
10 Particle moving in a cosine potential  19
11 Particle moving in inverted Gaussian potential  21
12 Hard sphere scattering—classical  22
2 Lagrangian and Hamiltonian Dynamics  25
1 Brachistochrone  27
2 Lagrange/Newton equivalence  30
3 SHO - Lagrangian and Hamiltonian  31
4 Simple pendulum - Lagrangian  33
5 Simple pendulum with vertically moving pivot point  35
6 Sliding pendulum  37
7 Atwood’s machine I  40
8 Mass on table, pulley, hanging mass - Lagrangian 41
9 Mass on table, pulley, hanging mass - Hamiltonian  43
10 Projectile motion  44
11 Hanging disk - Hamiltonian  47
12 Hanging disk - Lagrangian  49
13 Rod pivoted at the end  49
14 Moving inclined plane  52
15 Rotating massless rod - Lagrangian  54
16 Rotating massless rod - Hamiltonian  55
17 Atwood’s machine II  58
18 Sphere rolling on inclined plane  59
19 Bead on a wire  62
20 Two unequal masses attached to a spring  64
3 Central Forces and Orbits  67
1 Conservation of angular momentum I  68
2 Conservation of angular momentum II  68
3 Kepler’s laws 69
4 Newton’s gravitational law deduced from Kepler’s laws  71
5 Total energy for a central potential  72
6 Equation of the orbit 73
7 Total energy for a circular orbit  74
8 Spiral orbit I  76
9 Spiral orbit II 77
10 Inverse cube force law  79
11 Isotropic oscillator I 81
12 Isotropic oscillator II  82
13 Kepler effective potential  83
14 Kepler orbits  85
15 Runge–Lenz vector I  89
16 Runge–Lenz vector II  92
4 Normal Modes and Coordinates  95
1 Two pendulums coupled by a spring  95
2 Three springs, two masses  102
3 Double pendulum  103
4 Pendulum with oscillating pivot  108
5 Two springs with thin rod attached  111
6 Three particles, two springs (CO2)  114
References 117
Part II Quantum Mechanics
5 Introductory Concepts  121
1 Bohr atom  121
2 Gravitational bohr radius 124
3 deBroglie wavelength of an electron in Bohr orbit  125
4 Uncertainty principle and the H-atom  126
5 Classical radius of the electron  128
6 Magnetic dipole moment of spinning charged sphere  129
7 Uncertainty principle and the SHO  130
8 Measurement, expectation values and probabilities  131
9 Expectation values: SHO 132
10 Expectation values: L-box  135
11 Rectangular barrier 137
12 Transmission and reflection over a square well 141
13 Delta function barrier I 142
14 Delta function barrier II 144
15 Semi-infinite barrier 145
6 Bound States in One Dimension  149
61 Degeneracy  149
62 Parity  150
63 Characteristics of the Eigenfunctions  151
64 Superposition Principle  151
1 Stationary states  152
2 Characteristics of eigenfunctions  153
3 Delta function potential  156
4 Half-well potential 158
5 Sudden approximation I  159
6 Sudden approximation II  162
7 Wave function, probabilities I 164
8 Wave function, probabilities II 166
9 Sudden approximation III  169
7 Ladder Operators for the Harmonic Oscillator  173
1 Commutation of x and p  174
2 General relations 175
3 Expectation values I  179
4 Expectation values II  181
5 Expectation values III  182
6 Matrix elements 185
7 Expectation values IV  186
8 Expectation values V  187
8 Angular Momentum 189
1 Operators and ladder operators  192
2 Angular momentum eigenvalues 195
3 Measurements and expectation values 197
4 Speed of a “spinning hard sphere” electron  199
5 Spinors  200
6 Using Pauli matrices 200
7 Spin ladder operators  202
8 Electron in a B field  203
9 Commutators  205
10 Construct a Clebsch–Gordan table  205
11 More Clebsch–Gordan  207
12 Hyperfine structure of H-atom I  209
13 Hyperfine structure of H-atom II  210
14 Hard sphere scattering - quantum  213
9 Indistinguishable Particles 217
1 Bosons, fermions and the exchange force  218
2 Two fermions in an SHO  219
10 Bound States in Three Dimensions  223
1 3D L-box  224
2 Density of states for a 3D L-box  225
3 Rigid rotor  226
4 Infinitely deep spherical well  227
5 Spherical shell well  228
6 Classically forbidden region: H-atom  230
7 Expectation value for H-atom 231
8 Kramer’s relation  232
9 Bound state for unknown potential  233
10 Relation between H-atom quantum numbers  235
11 H-atom degeneracy 237
12 The quantum defect  239
11 Approximation Methods  243
111 The WKB Approximation  243
1 Energies of an L-box  244
2 Energies of an SHO  245
3 Transmission through a parabolic barrier  246
112 The Variational Method  248
1 Estimate the ground state energy of an SHO  249
2 Estimate the first excited state energy of an SHO 251
3 Linear potential well 253
4 Quartic potential well  256
113 Non-degenerate Perturbation Theory 258
1 Gravitational correction to H-atom energy  259
2 L-box perturbed by a delta-function  260
3 L-box perturbed by a linear function  261
4 Relativistic correction to L-box  263
5 Charged SHO perturbed by constant electric field  266
6 H-atom perturbed by delta function  270
7 Relativistic correction to SHO  271
8 Relativistic correction to H-atom  273
9 Quartic perturbation of SHO  275
10 Matrix eigenvalues  276
114 The Helium Atom  280
1 He-atom ground state energy  281
2 He-atom: perturbation/variation  282
115 Degenerate Perturbation Theory  284
1 2-D a-box with xy perturbation  285
2 Particle on a ring with delta-function perturbation 288
3 2-D SHO with xy perturbation  292
4 Select set for 3x3 matrix  294
116 Time Dependent Perturbation Theory  297
1 Gaussian perturbation applied to particle in an L-box  298
2 E-field pulse applied to to particle in an L-box  300
3 Exponential perturbation applied to SHO  302
4 Pulse applied to SHO 305
5 Exponential E-field applied to H-atom  307
6 Fermi’s Golden Rule  310
References 311
A Greek Alphabet  313
B Acronyms, Descriptors and Coordinates  315
B1 Acronyms and Descriptors 315
B2 Coordinate Systems  315
C Units 317
D Conic Sections in Polar Coordinates  319
E Useful Trigonometric Identities  323
F Useful Vector Relations  325
G Useful Integrals  327
H Useful Series 329
H1 Taylor Series 329
H2 Binomial Expansion 330
I -Functions  331
I1 Integral -Functions  331
I2 Half-Integral -Functions  331
J The Dirac Delta-Function 333
K Hyperbolic Functions  335
K1 Manipulations of Hyperbolic Functions 335
K2 Relationships Between Hyperbolic and Circular Functions 337
L Useful Formulas  339
L1 Classical Mechanics 339
L2 Quantum Mechanics  340
L21 One Dimension  340
L22 Three Dimensions (Central Potentials)  341
M The Infinite Square Well  343
M1 The L-Box  343
M2 The a-Box  344
N Operators, Eigenfunctions, and Commutators 347
N1 Eigenfunctions and Eigenvalues of Operators  347
N2 Operator Algebra; Commutators  347
N3 Commutator Identities  348
N4 Some Quantum Mechanical Commutators 348
O The Quantum Mechanical Harmonic Oscillator  349
P Legendre Polynomials 351
P1 Properties  351
P2 Legendre Series  351
P3 The Function 1= jr1  r2j  352
P4 Polynomials  353
Q Orbital Angular Momentum Operators in Spherical Coordinates  355
R Spherical Harmonics  357
S Clebsch–Gordan Tables 359
T The Hydrogen Atom 361
References 362
Index  363

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