i Invitation to Quantum Mechanics Daniel F. Styer ii Invitation to Quantum Mechanics Daniel F. Styer Schiffer Professor of Physics, Oberlin College copyright c 31 May 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/InvitationToQM: It is formatted to print nicely on either A4 or U.S. Letter paper. You may also purchase a printed and bound copy from World Scientific Publishing Company. In neither case does the author receive monetary gain from your download/purchase: it is reward enough for him that you want to explore quantum mechanics. Love all God's creation, the whole and every grain of sand in it. Love the stars, the trees, the thunderstorms, the atoms. The more you love, the more you will grow curious. The more you grow curious, the more you will question. The more you question, the more you will uncover. The more you uncover, the more you will love. And so at last you will come to love the entire universe with an agile and resilient love founded upon facts and understanding. | This improvisation by Dan Styer was inspired by the first sentence, which appears in Fyodor Dostoyevsky's The Brothers Karamazov. iii iv Dedicated to Linda Ong Styer, adventurer Contents Synoptic Contents 1 Welcome 3 1. \Something Isn't Quite Right" 9 1.1 Light in thermal equilibrium: Blackbody radiation . 10 1.2 Photoelectric effect . 23 1.3 Wave character of electrons . 30 1.4 How does an electron behave? . 35 1.5 Quantization of atomic energies . 36 1.6 Quantization of magnetic moment . 42 2. What Is Quantum Mechanics About? 47 2.1 Quantization . 47 2.2 Interference . 59 2.3 Aharonov-Bohm effect . 67 2.4 Light on the atoms . 70 2.5 Entanglement . 72 2.6 Quantum cryptography . 84 2.7 What is a qubit? . 88 v vi Contents 3. Forging Mathematical Tools 89 3.1 What is a quantal state? . 89 3.2 Amplitude . 91 3.3 Reversal-conjugation relation . 99 3.4 Establishing a phase convention . 101 3.5 How can I specify a quantal state? . 103 3.6 States for entangled systems . 112 3.7 Are states \real"? . 116 3.8 What is a qubit? . 116 4. The Quantum Mechanics of Position 119 4.1 Probability and probability density: One particle in one dimension . 119 4.2 Probability amplitude . 124 4.3 How does wavefunction change with time? . 126 4.4 Wavefunction: Two particles . 127 4.5 Solving the Schr¨odingertime evolution equation for the infinite square well . 131 4.6 What did we learn by solving the Schr¨odinger time evolu- tion equation for the infinite square well? . 139 4.7 Other potentials . 149 4.8 Energy loss . 152 4.9 Mean values . 153 4.10 The classical limit of quantum mechanics . 157 4.11 Transitions induced by light . 166 4.12 Position plus spin . 172 Contents vii 5. Solving the Energy Eigenproblem 177 5.1 Sketching energy eigenfunctions . 178 5.2 Scaled quantities . 198 5.3 Numerical solution of the energy eigenproblem . 201 6. Identical Particles 205 6.1 Two or three identical particles . 205 6.2 Symmetrization and antisymmetrization . 208 6.3 Consequences of the Pauli principle . 214 6.4 Consequences of the Pauli principle for product states . 217 6.5 Energy states for two identical, noninteracting particles . 217 6.6 Spin plus space, two electrons . 219 6.7 Spin plus space, three electrons, ground state . 225 7. Atoms 229 7.1 Central potentials in two dimensions . 229 7.2 Central potentials in three dimensions . 236 7.3 The hydrogen atom . 238 7.4 The helium atom . 247 7.5 The lithium atom . 252 7.6 All other atoms . 253 7.7 The periodic table . 256 8. The Vistas Open to Us 259 Appendix A Significant Figures 263 Appendix B Dimensions 269 viii Contents Appendix C Complex Arithmetic 277 Appendix D Problem-Solving Tips and Techniques 279 Appendix E Catalog of Misconceptions 283 Index 285 Synoptic Contents Welcome What is quantum mechanics and why should I care about it? 1. \Something Isn't Quite Right" Historical experiments show that classical mechanics is flawed. 2. What Is Quantum Mechanics About? If classical mechanics is wrong, then what is right? We explore, in the context of modern experiments with qubits, the atomic phenomena that quantum mechanics needs to explain. 3. Forging Mathematical Tools We build a framework for the quantum mechanics of qubits, using a mathematical tool called \amplitude". 4. The Quantum Mechanics of Position The framework, built to treat qubits, extends to treat continuum position as well. Energy plays a central role here. 1 2 Synoptic Contents 5. Solving the Energy Eigenproblem Since energy plays a central role, we devote a chapter to solving such problems. We find that solving particular problems strengthens our conceptual understanding, and that conceptual understanding strengthens our skill in solving particular problems. 6. Identical Particles This surprisingly subtle topic deserves a chapter of its own. 7. Atoms We apply our new knowledge to physical (rather than model) systems. 8. The Vistas Open to Us This book is an invitation. Where might you and quantum mechanics travel together? Welcome Why would anyone want to study 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. 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- 3 4 Welcome 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. This situation is analogous to the relationship between relativistic me- chanics and classical mechanics. Relativity applies always, but classical mechanics is a good approximation to relativistic mechanics when applied to slow things (that is, with speeds much less than light speed c). The speed at which the classical approximation becomes legitimate depends upon the Welcome 5 accuracy demanded, but as a rule of thumb particles moving less than a quarter of light speed are treated classically.