i The Physics of Quantum Mechanics Daniel F. Styer ii The Physics of Quantum Mechanics Daniel F. Styer Schiffer Professor of Physics, Oberlin College This book is in draft form | it is not polished or complete. It needs more problems. I appreciate your comments. copyright c 19 August 2021 Daniel F. Styer The copyright holder grants the freedom to copy, modify, convey, adapt, and/or redistribute this work under the terms of the Creative Commons Attribution Share Alike 4.0 International License. A copy of that license is available at http://creativecommons.org/licenses/by-sa/4.0/legalcode. You may freely download this book in pdf format from http://www.oberlin.edu/physics/dstyer/ThePhysicsOfQM: It is formatted to print nicely on either A4 or U.S. Letter paper. The author receives no monetary gain from your download: it is reward enough for him that you want to explore quantum mechanics. Instructions for living a life: Pay attention. Be astonished. Tell about it. | Mary Oliver, Sometimes iii Contents Welcome 1 1. What is Quantum Mechanics About? 5 1.1 Quantization . .5 1.2 Interference . 22 1.3 Aharonov-Bohm effect . 31 1.4 Light on the atoms . 33 1.5 Entanglement . 35 1.6 Quantum cryptography . 48 1.7 What is a qubit? . 52 2. Forging Mathematical Tools 55 2.1 What is a quantal state? . 55 2.2 Amplitude . 57 2.3 Reversal-conjugation relation . 64 2.4 Establishing a phase convention . 66 2.5 How can I specify a quantal state? . 68 2.6 States for entangled systems . 77 2.7 Are states \real"? . 82 2.8 What is a qubit? . 82 v vi Contents 3. Refining Mathematical Tools 85 3.1 Extras . 85 3.2 Outer products, operators, measurement . 90 3.3 Photon polarization . 96 3.4 Lightning linear algebra . 100 3.5 Problems . 109 4. Formalism 113 4.1 The role of formalism . 113 4.2 The density matrix . 117 5. Time Evolution 119 5.1 Operator for time evolution . 119 5.2 Working with the Schr¨odingerequation . 123 5.3 Formal properties of time evolution; Conservation laws . 135 5.4 Magnetic moment in a uniform magnetic field . 139 5.5 The neutral K meson . 139 6. The Quantum Mechanics of Position 143 6.1 Describing states in continuum systems . 143 6.2 How does position amplitude change with time? . 150 6.3 What is wavefunction? . 158 6.4 Operators and their representations; The momentum basis 159 6.5 Time evolution of average quantities . 169 7. The Free Particle 173 7.1 Problems . 173 Contents vii 8. Square Wells 177 8.1 What does an electron look like? . 178 9. The Simple Harmonic Oscillator 181 9.1 Resume of energy eigenproblem . 181 9.2 Solution of the energy eigenproblem: Differential equation approach . 181 9.3 Solution of the energy eigenproblem: Operator factoriza- tion approach . 186 9.4 Problems . 190 10. Qualitative Solution of Energy Eigenproblems 193 11. Perturbation Theory 195 11.1 The O notation . 195 11.2 Perturbation theory for cubic equations . 198 11.3 Derivation of perturbation theory for the energy eigenprob- lem............................... 201 11.4 Perturbation theory for the energy eigenproblem: Sum- mary of results . 205 11.5 Problems . 206 12. Quantum Mechanics in Two and Three Dimensions 211 12.1 More degrees of freedom . 211 12.2 Vector operators . 214 12.3 Multiple particles . 215 12.4 The phenomena of quantum mechanics . 216 viii Contents 13. Angular Momentum 219 13.1 Solution of the angular momentum eigenproblem . 219 13.2 Summary of the angular momentum eigenproblem . 223 (j) 13.3 Ordinary differential equations for the dm;m0 (θ)...... 223 13.4 Problems . 224 14. Central Force Motion 227 14.1 Energy eigenproblem in two dimensions . 227 14.2 Energy eigenproblem in three dimensions . 234 14.3 Bound state energy eigenproblem for Coulombic potentials 239 14.4 Summary of the bound state energy eigenproblem for a Coulombic potential . 243 14.5 Problems . 244 15. Identical Particles 247 15.1 Many-particle systems in quantum mechanics . 247 15.2 An antisymmetric basis for the helium problem . 261 16. Breather 271 16.1 Scaled variables . 272 16.2 Variational method for finding the ground state energy . 275 16.3 Problems . 277 17. Hydrogen 281 17.1 The Stark effect . 281 18. Helium 291 18.1 Ground state energy of helium . 291 Contents ix 19. Atoms 297 19.1 Addition of angular momenta . 297 19.2 Hartree-Fock approximation . 304 19.3 Atomic ground states . 305 20. Molecules 307 20.1 The hydrogen molecule ion . 307 20.2 Problems . 313 20.3 The hydrogen molecule . 314 20.4 Can we do better? . 315 21. WKB: The Quasiclassical Approximation 317 21.1 The connection region . 319 21.2 Why is WKB the \quasiclassical" approximation? . 320 21.3 The \power law" potential . 320 22. The Interaction of Matter and Radiation 329 22.1 Perturbation Theory for the Time Development Problem . 329 22.2 Setup . 330 22.3 Light absorption . 334 22.4 Absorbing incoherent light . 340 22.5 Absorbing and emitting light . 341 22.6 Problems . 346 23. The Territory Ahead 349 x Contents Appendix A Tutorial on Matrix Diagonalization 351 A.1 What's in a name? . 351 A.2 Vectors in two dimensions . 352 A.3 Tensors in two dimensions . 355 A.4 Tensors in three dimensions . 359 A.5 Tensors in d dimensions . 360 A.6 Linear transformations in two dimensions . 361 A.7 What does \eigen" mean? . 363 A.8 How to diagonalize a symmetric matrix . 364 A.9 A glance at computer algorithms . 370 A.10 A glance at non-symmetric matrices and the Jordan form 371 Appendix B The Spherical Harmonics 377 Appendix C Radial Wavefunctions for the Coulomb Problem 379 Appendix D Quantum Mechanics Cheat Sheet 381 Index 385 Welcome Why would anyone want to study a book titled The Physics of Quantum Mechanics? Starting in the year 1900, physicists exploring the newly discovered atom found that the atomic world of electrons and protons is not just smaller than our familiar world of trees, balls, and automobiles, it is also fundamentally different in character. Objects in the atomic world obey different rules from those obeyed by a tossed ball or an orbiting planet. These atomic rules are so different from the familiar rules of everyday physics, so counterintuitive and unexpected, that it took more than 25 years of intense research to uncover them. But it is really only since the year 1990 that physicists have come to appreciate that the rules of the atomic world (now called \quantum mechan- ics") are not just different from the everyday rules (now called \classical mechanics"). The atomic rules are also far richer. The atomic rules provide for phenomena like particle interference and entanglement that are simply absent from the everyday world. Every phenomenon of classical mechanics is also present in quantum mechanics, but the quantum world provides for many additional phenomena. Here's an analogy: Some films are in black-and-white and some are in color. It does not malign any black-and-white film to say that a color film has more possibilities, more richness. In fact, black-and-white films are simply one category of color films, because black and white are both colors. Anyone moving from the world of only black-and-white to the world of color is opening up the door to a new world | a world ripe with new possibilities and new expression | without closing the door to the old world. 1 2 Welcome This same flood of richness and freshness comes from entering the quan- tum world. It is a difficult world to enter, because we humans have no expe- rience, no intuition, no expectations about this world. Even our language, invented by people living in the everyday world, has no words for the new quantal phenomena | just as a language among a race of the color-blind would have no word for \red". Reading this book is not easy: it is like a color-blind student learning about color from a color-blind teacher. The book is just one long argument, building up the structure of a world that we can explore not through touch or through sight or through scent, but only through logic. Those willing to follow and to challenge the logic, to open their minds to a new world, will find themselves richly rewarded. The place of quantum mechanics in nature Quantum mechanics is the framework for describing and analyzing small things, like atoms and nuclei. Quantum mechanics also applies to big things, like baseballs and galaxies, but when applied to big things, cer- tain approximations become legitimate: taken together, these are called the classical approximation to quantum mechanics, and the result is the familiar classical mechanics. Quantum mechanics is not only less familiar and less intuitive than classical mechanics; it is also harder than classical mechanics. So whenever the classical approximation is sufficiently accurate, we would be foolish not to use it. This leads some to develop the misimpression that quantum mechanics applies to small things, while classical mechanics applies to big things. No. Quantum mechanics applies to all sizes, but classical mechanics is a good approximation to quantum mechanics when it is applied to big things. For what size is the classical approximation good enough? That depends on the accuracy desired. The higher the accuracy demanded, the more situ- ations will require full quantal treatment rather than approximate classical treatment. But as a rule of thumb, something as big as a DNA strand is almost always treated classically, not quantum mechanically.