Physical chemistry : quantum physics

This is a new undergraduate textbook on physical chemistry by Horia Metiu published as four separate paperback volumes. These four volumes on physical chemistry combine a clear and thorough presentation of the theoretical and mathematical aspects of the subject with examples and applications drawn from current industrial and academic research. By using the computer to solve problems that include actual experimental data, the author is able to cover the subject matter at a practical level. The books closely integrate the theoretical chemistry being taught with industrial and laboratory practice. This approach enables the student to compare theoretical projections with experimental results, thereby providing a realistic grounding for future practicing chemists and engineers. Each volume of Physical Chemistry includes Mathematica® and Mathcad® Workbooks on CD-ROM.

Metiu's four separate volumes-Thermodynamics, Statistical Mechanics, Kinetics, and Quantum Mechanics-offer built-in flexibility by allowing the subject to be covered in any order.

These textbooks can be used to teach physical chemistry without a computer, but the experience is enriched substantially for those students who do learn how to read and write Mathematica® or Mathcad® programs. A TI-89 scientific calculator can be used to solve most of the exercises and problems.

® Mathematica is a registered trademark of Wolfram Research, Inc.
® Mathcad is a registered trademark of Mathsoft Engineering & Education, Inc.

Author Notes

Horia Metiu is Professor of Chemistry and Physics at the University of California, Santa Barbara

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This four-volume series is primarily designed to be a set of course resources for a full-year program in physical chemistry. The volumes include an introduction to thermodynamics, kinetics, quantum mechanics, and statistical mechanics. Writing conversationally, Metiu (Univ. of California, Santa Barbara) uses first-person techniques to introduce the motivation for particular aspects and approaches to the material. Periodically, the author includes interesting historical context for how ideas, (e.g., the temperature scale) were developed. Each volume includes a CD-ROM that has both Mathematica and MathCAD workbooks for readers to explore the behavior of mathematical expressions. Although the CD-ROM workbooks are useful, readers can use the printed volumes without access to these programs. The author does not limit investigations to ideal cases, as the non-ideal regime is where most practicing physical chemists work. The writing level is such that students with a year of introductory physics and at least a year of calculus can read the book and learn the topics on their own. As with any series, there are compromises in what is covered, but each chapter includes accessible references for further reading. ^BSumming Up: Recommended. Upper-division undergraduates. J. A. Bartz Kalamazoo College

Preface	p. xxi
How to use the workbooks, exercises, and problems	p. xxvii
Chapter 1 Why quantum mechanics?	p. 1
Molecular vibrations	p. 1
Radiation by a hot body	p. 2
The stability of atoms	p. 3
Chapter 2 Dynamic variables and operators	p. 5
Operators: the definition	p. 5
Examples of operators	p. 6
Operations with operators	p. 7
Operator addition	p. 7
Operator multiplication	p. 8
Powers of operators	p. 9
The commutator of two operators	p. 9
Linear operators	p. 11
Dynamical variables as operators	p. 11
Position and potential energy operators	p. 12
Potential energy operator	p. 13
The momentum operator	p. 14
The kinetic energy operator	p. 14
The total energy (the Hamiltonian)	p. 16
Angular momentum	p. 17
Supplement 2.1 Review of complex numbers	p. 21
Complex functions of real variables	p. 24
Chapter 3 The eigenvalue problem	p. 25
The eigenvalue problem: definition and examples	p. 25
The eigenvalue problem for p[subscript x]	p. 26
p[subscript x] has an infinite number of eigenvalues	p. 26
The eigenvalue problem for angular momentum	p. 27
If two operators commute, they have the same eigenfunctions	p. 29
Degenerate eigenvalues	p. 30
Which eigenfunctions have a meaning in physics?	p. 32
Not all eigenfunctions are physically meaningful	p. 32
Normalization	p. 32
A relaxed condition	p. 33
An example: the eigenfunctions of kinetic energy	p. 34
One-dimensional motion in the force-free, unbounded space	p. 34
The eigenfunctions of the kinetic energy operator	p. 35
Which of these eigenfunctions can be normalized?	p. 37
Boundary conditions: the particle in a box	p. 38
Forcing the eigenfunctions to satisfy the boundary conditions	p. 38
Quantization	p. 40
The particle cannot have zero kinetic energy	p. 40
Imposing the boundary conditions removes the trouble with normalization	p. 42
Some properties of eigenvalues and eigenfunctions	p. 42
Chapter 4 What do we measure when we study quantum systems?	p. 45
Introduction	p. 45
The preparation of the initial state	p. 46
Not all quantum systems have "well-defined" energy	p. 46
Three examples of energy eigenstates	p. 48
A particle in a one-dimensional box	p. 49
The vibrational energy of a diatomic molecule	p. 50
The energy eigenstates of the hydrogen atom	p. 51
Energy measurements by electron scattering	p. 52
Electron scattering from a gas of diatomic molecules	p. 53
Energy measurements by photon emission	p. 57
Photon emission	p. 57
Applications	p. 58
Chapter 5 Some results are certain, most are just probable	p. 61
Introduction	p. 61
What is the outcome of an electron-scattering experiment?	p. 62
Classical interpretation of the experiment	p. 63
The quantum description of the experiment	p. 64
Probabilities	p. 65
A discussion of photon absorption measurements	p. 68
Why the outcome of most absorption experiments is certain	p. 69
A discussion of photon emission measurements	p. 69
A one-photon, one-molecule experiment	p. 70
The probabilities of different events	p. 72
Chapter 6 The physical interpretation of the wave function	p. 73
Application to a vibrating diatomic molecule	p. 75
Data for HCl	p. 76
Interpretation	p. 79
Average values	p. 79
The effect of position uncertainty on a diffraction experiment	p. 80
The effect of position uncertainty in an ESDAID experiment	p. 81
Chapter 7 Tunneling	p. 85
Classically forbidden region	p. 85
How large is the accessible region?	p. 85
The classically allowed region for an oscillator	p. 86
Tunneling depends on mass and energy	p. 89
Tunneling junctions	p. 90
Scanning tunneling microscopy	p. 91
Chapter 8 Particle in a box	p. 95
Define the system	p. 95
The classical Hamiltonian	p. 96
Quantizing the system	p. 97
The boundary conditions	p. 98
Solving the eigenvalue problem (the Schrodinger equation) for the particle in a box	p. 100
Separation of variables	p. 100
Boundary conditions	p. 101
The behavior of a particle in a box	p. 104
The ground state energy is not zero	p. 105
Degeneracy	p. 107
Degeneracy is related to the symmetry of the system	p. 109
The eigenfunctions are normalized	p. 111
Orthogonality	p. 112
The position of a particle in a given state	p. 113
Chapter 9 Light emission and absorption: the phenomena	p. 117
Introduction	p. 117
Light absorption and emission: the phenomena	p. 118
An absorption experiment	p. 118
Characterization of an absorption spectrum	p. 119
Why the transmitted intensity is low at certain frequencies	p. 120
Emission spectroscopy	p. 122
Units	p. 126
How to convert from one unit to another	p. 128
Why we need to know the laws of light absorption and emission	p. 130
Chapter 10 Light emission and absorption: Einstein's phenomenological theory	p. 135
Introduction	p. 135
Photon absorption and emission: the model	p. 136
Photon energy and energy conservation	p. 136
How to reconcile this energy conservation with the existence of a line-width	p. 137
The model	p. 137
Photon absorption and emission: the rate equations	p. 138
The rate of photon absorption	p. 138
The rate of spontaneous photon emission	p. 139
The rate of stimulated emission	p. 140
The total rate of change of N[subscript 0] (or N[subscript 1])	p. 140
Photon absorption and emission: the detailed balance	p. 141
Molecules in thermal equilibrium with radiation	p. 141
The detailed balance	p. 143
The solution of the rate equations	p. 144
Using the detailed balance results to simplify the rate equation	p. 145
The initial conditions	p. 146
The ground state population when the molecules are continuously exposed to light	p. 146
Analysis of the result: saturation	p. 147
Why Einstein introduced stimulated emission	p. 149
Simulated emission and the laser	p. 150
The rate of population relaxation	p. 151
Chapter 11 Light absorption: the quantum theory	p. 153
Introduction	p. 153
Quantum theory of light emission and absorption	p. 154
The absorption probability	p. 154
Electrodynamic quantities: light pulses	p. 154
Electromagnetic quantities: the polarization of light	p. 156
Electromagnetic properties: the direction of propagation	p. 157
Electromagnetic quantities: the energy and the intensity of the pulse	p. 157
The properties of the molecule: the transition dipole moment	p. 158
The properties of the molecule: the line shape	p. 159
How the transition probability formula is used	p. 162
The probability of stimulated emission	p. 163
Validity conditions	p. 163
The connection to Einstein's B coefficient	p. 163
Single molecule spectroscopy and the spectroscopy of an ensemble of molecules	p. 165
Chapter 12 Light emission and absorption by a particle in a box and a harmonic oscillator	p. 169
Introduction	p. 169
Light absorption by a particle in a box	p. 170
Quantum dots	p. 170
Photon absorption probability	p. 172
The amount of light and its frequency	p. 173
The energies and absorption frequencies	p. 173
The transition dipole	p. 174
The energy eigenfunctions for a particle in a box	p. 174
The dipole operator m	p. 175
The role of light polarization	p. 176
The evaluation of the transition moment for a particle in a box	p. 176
The first selection rule	p. 178
Physical interpretation	p. 178
The second selection rule	p. 178
A calculation of the spectrum in arbitrary units	p. 180
Light absorption by a harmonic oscillator	p. 181
The eigenstates and eigenvalues of a harmonic oscillator	p. 183
The transition dipole	p. 183
The molecular orientation and polarization	p. 184
Chapter 13 Two-particle systems	p. 187
Introduction	p. 187
The Schrodinger equation for the internal motion	p. 189
The laboratory coordinate system	p. 189
The energy	p. 190
Because there is no external force, the system moves with constant velocity	p. 191
A coordinate system with the origin in the center of mass	p. 194
The total energy in terms of momentum	p. 195
The Hamiltonian operator	p. 196
The concept of quasi-particle	p. 197
The Hamiltonian of the quasi-particle in spherical coordinates	p. 199
The role of angular momentum in the motion of the two-particle system	p. 201
Angular momentum in classical mechanics	p. 201
The properties of angular momentum	p. 201
The Schrodinger equation of the quasi-particle and the square of the angular momentum	p. 205
Supplement 13.1 The role of angular momentum in the motion of the two-particle system	p. 207
The evolution of the angular momentum	p. 207
A polar coordinate system	p. 208
The angular momentum in polar coordinates	p. 210
The energy in polar coordinates	p. 210
Chapter 14 Angular momentum in quantum mechanics	p. 213
Introduction	p. 213
The operators representing the angular momentum in quantum mechanics	p. 215
Angular momentum in classical mechanics	p. 215
The angular momentum operator in quantum mechanics	p. 216
Angular momentum in spherical coordinates	p. 217
The operator L[superscript 2]	p. 218
The commutation relations between L[superscript 2] and L[subscript x], L[subscript y], L[subscript z]	p. 218
The eigenvalue equations for L[superscript 2] and L[subscript z]	p. 220
The eigenvalue problem for L[superscript 2]	p. 221
The eigenvalue problem for L[subscript z]	p. 222
Spherical harmonics	p. 222
The physical interpretation of these eigenstates	p. 225
Supplement 14.1 A brief explanation of the procedure for changing coordinates	p. 226
Chapter 15 Two particle systems: the radial and angular Schrodinger equations	p. 231
Introduction	p. 231
The Schrodinger equation in terms of L[superscript 2]	p. 232
The Hamiltonian of a two-particle system in terms of L[superscript 2]	p. 232
The radial and the angular Schrodinger equations	p. 233
The separation of variables in the Schrodinger equation	p. 233
Additional physical conditions	p. 234
The integral over the angles	p. 235
The radial normalization	p. 235
Chapter 16 The energy eigenstates of a diatomic molecule	p. 239
Introduction	p. 239
The harmonic approximation	p. 240
The form of the function V(r)	p. 240
The physics described by V(r)	p. 241
When the approximation is correct	p. 244
The harmonic approximation to V(r)	p. 245
The rigid-rotor approximation	p. 247
Physical interpretation	p. 247
The rigid-rotor approximation decouples vibrational and rotational motion	p. 248
The eigenstates and eigenvalues of the radial Schrodinger equation in the harmonic and rigid-rotor approximation	p. 249
The eigenvalues	p. 249
An improved formula for the energy	p. 250
The radial eigenfunctions	p. 252
The physical interpretation of the eigenfunctions	p. 254
The probability of having a given interatomic distance and orientation	p. 255
The probability of having a given bond length	p. 256
Turning point and tunneling	p. 258
The average values of r - r[subscript 0] and (r - r[subscript 0])[superscript 2]	p. 260
The energy eigenstates and eigenvalues of a diatomic molecule	p. 262
Chapter 17 Diatomic molecule: its spectroscopy	p. 265
Introduction	p. 265
Collect the necessary equations	p. 269
The frequencies of the absorbed photons	p. 269
Photon absorption and emission probabilities	p. 271
The wave function for the nuclear motion	p. 272
The electronic wave function	p. 273
The dipole moment of the molecule	p. 274
The separation of the transition dipole matrix element into a rotational and vibrational contribution	p. 275
The harmonic approximation for the dipole moment	p. 277
Physical interpretation	p. 279
The integral	p. 280
Vibrational and rotational excitation by absorption of an infrared photon	p. 281
The molecules in a gas have a variety of states	p. 285
The infrared absorption spectrum of a gas at a fixed temperature	p. 286
The probability of [characters not reproducible] (v, l, m; T)	p. 288
The probability of various states in a gas: a numerical study	p. 289
Back to the spectrum: the relative intensity of the absorption peaks	p. 291
Numerical analysis	p. 291
Chapter 18 The hydrogen atom	p. 295
Introduction	p. 295
The Schrodinger equation for a one-electron atom	p. 296
Why the properties of a one-electron atom are so different from those of a diatomic molecule	p. 297
The solution of the Schrodinger equation for a one-electron atom	p. 299
The eigenvalues	p. 300
The eigenfunctions	p. 301
The energy	p. 302
The magnitude of the energies	p. 302
The energy scale [epsilon] and the length scale a	p. 304
The radial wave functions R[subscript n,l] (r) and the mean values of various physical quantities	p. 306
The probability of finding the electron at a certain distance from the nucleus	p. 308
The functions R[subscript n,l] (r)	p. 310
Plots of R[subscript n,l] (r)	p. 311
The mean values of r, r[superscript 2], Coulomb energy, centrifugal energy, and radial kinetic energy	p. 312
The angular dependence of the wave function	p. 317
Some nomenclature	p. 319
The s-states	p. 319
The np orbitals	p. 320
The nd orbitals	p. 323
Hydrogen atom: absorption and emission spectroscopy	p. 325
The transition dipole	p. 327
The selection rules	p. 328
The radial integrals	p. 330
Chapter 19 The spin of the electron and its role in spectroscopy	p. 331
Introduction	p. 331
The spin operators	p. 334
Spin eigenstates and eigenvalues	p. 335
The scalar product	p. 337
The emission spectrum of a hydrogen atom in a magnetic field: the normal Zeeman effect	p. 339
The experiment	p. 340
A modern (but oversimplified) version of the Lorentz model	p. 342
The energy of the hydrogen atom in a magnetic field	p. 343
The emission frequencies	p. 345
The spectrum	p. 347
The role of spin in light emission by a hydrogen atom: the anomalous Zeeman effect	p. 348
The interaction between spin and a magnetic field	p. 348
The energies of the electron in the hydrogen atom: the contribution of spin	p. 348
Comments and warnings	p. 350
Chapter 20 The electronic structure of molecules: The H[subscript 2] molecule	p. 353
Introduction	p. 353
The Born-Oppenheimer approximation	p. 355
The electronic energies E[subscript n] (R) are the potential energies for the nuclear motion	p. 358
How to use the variational principle	p. 363
Application to the harmonic oscillator	p. 363
An application of the variational principle that uses a basis set	p. 366
The many-body wave function as a product of orbitals	p. 369
The curse of multi-dimensionality	p. 369
The wave function as a product of orbitals	p. 370
How to choose good orbitals	p. 372
The electron wave function must be antisymmetric	p. 372
Indistinguishable particles	p. 372
The antisymmetrization of a product of orbitals	p. 375
How to generalize to more than two electrons	p. 377
The Pauli principle	p. 379
Which electrons should be antisymmetrized	p. 380
The molecular orbitals in a minimal basis set: [sigma subscript u] and [sigma subscript g]	p. 383
The MO-LCAO method	p. 383
The minimal basis set	p. 384
Determine the molecular orbitals by using symmetry	p. 384
The molecular orbitals are normalized	p. 385
The symmetry of [sigma subscript g] and [sigma subscript u]	p. 387
The antisymmetrized products used in the configuration interaction wave functions must be eigenfunctions of S[superscript 2] and S[subscript z]	p. 387
The strategy for constructing the functions [Phi subscript i](1, 2)	p. 389
The spin states	p. 389
The singlet state [vertical bar] 0, 0>	p. 390
The triplet states [vertical bar] 1, m[subscript s]>	p. 391
Pairing up the spin and the orbital functions to create antisymmetric configurations	p. 391
The states [Phi subscript 1], [Phi subscript 2], and [Phi subscript 6]	p. 391
The states [Phi subscript 3], [Phi subscript 4], and [Phi subscript 5]	p. 392
The notations [superscript 1 Sigma subscript g+], [superscript 1 Sigma subscript g-], [superscript 3 Sigma subscript u,-1], [superscript 3 Sigma subscript u,0], [superscript 3 Sigma subscript u,1], [superscript 1 Sigma subscript u,0]	p. 392
The configurations in terms of atomic orbitals: physical interpretation	p. 393
The integrals required by the configuration interaction method	p. 394
The overlap matrix is diagonal	p. 395
Only the off-diagonal matrix elements and differ from zero	p. 395
The Hamiltonian matrix	p. 396
The Hamiltonian in atomic units	p. 396
Atomic units	p. 397
The matrix elements in terms of atomic orbitals	p. 397
Expression for the matrix elements	p. 398
The overlap integral S(R)	p. 399
The integral J(R)	p. 399
The integral K(R)	p. 401
The integral J'(R)	p. 401
The integral K'(R)	p. 402
The integral L(R)	p. 404
The ground and excited state energies given by perturbation theory	p. 405
Behavior of H[subscript 11] (R) at large R	p. 407
The configuration interaction method	p. 409
The variational eigenvalue problem: a summary	p. 409
The eigenvalues and the eigen vectors of matrix H	p. 410
The coupling between configurations	p. 411
When perturbation theory is accurate	p. 412
The configuration interaction energies	p. 412
The configuration interaction wave function of the ground state	p. 414
Summary	p. 416
Chapter 21 Nuclear magnetic resonance and electron spin resonance	p. 421
Introduction	p. 421
More information about spin operators and spin states	p. 424
The NMR spectrum of a system with one independent spin	p. 428
The energy of the spin states for non-interacting spins	p. 428
The energy levels	p. 429
The rate of energy absorption	p. 430
NMR notation and units	p. 430
The order of magnitude of various quantities	p. 431
Hot bands	p. 432
The chemical shift	p. 434
The magnetic field acting on a nucleus depends on environment	p. 434
The NMR spectrum of a system of two non-interacting nuclei having spin 1/2	p. 436
The Hamiltonian and the states of a system of two non-interacting spin 1/2 particles	p. 436
The order of magnitude of these energies	p. 438
The selection rules	p. 438
The frequencies of the allowed transitions	p. 440
The spin-spin interaction	p. 441
Spin-spin coupling	p. 441
The interaction between nuclear spins	p. 442
The spectrum of two distinguishable, interacting nuclei	p. 443
The energies of the spin states	p. 443
The Hamiltonian	p. 444
The states of the interacting spins	p. 444
The Galerkin method: how to turn an operator equation into a matrix equation	p. 446
The matrix elements H[subscript nm] =	p. 447
Perturbation theory	p. 449
The off-diagonal elements	p. 450
The lowest energy E[subscript 1] and eigenvector [vertical bar phi subscript 1]>	p. 451
The eigenvalue E[subscript 2] and the spin state [vertical bar phi subscript 2]>	p. 452
Why [vertical bar phi subscript 1]> and [vertical bar phi subscript 2]> are so different	p. 453
The physical meaning of [vertical bar phi subscript 2]>	p. 453
The states [vertical bar phi subscript 3]> and [vertical bar phi subscript 4]> and the energies E[subscript 3] and E[subscript 4]	p. 455
The orders of magnitude	p. 456
The NMR spectrum of two weakly interacting, indistinguishable nuclei having spin 1/2	p. 457
The states that satisfy the symmetry requirements	p. 457
Perturbation theory	p. 459
The selection rules	p. 461
Appendices	p. 463
A1 Values of some physical constants	p. 463
A2 Energy conversion factors	p. 464
Further Reading	p. 465
Index	p. 467

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