| Author: |
J. Daniel Kelley, Jacob J. Leventhal
|
| Release at: | 2017 |
| Pages: | 358 |
| Edition: |
First Edition (Extracting the Underlying Concepts)
|
| File Size: | 4 MB |
| File Type: | |
| Language: | English |
Description of Problems in Classical and Quantum Mechanics (PDF)
Problems in Classical and Quantum Mechanics written by J. Daniel Kelley, Jacob J. Leventhal is a great Physics book available for (PDF) download. 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 (PDF)
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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