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Introductory Quantum Theory School of Mathematics and Natural Sciences Physics Department Bergische Universitat¨ Wuppertal A course on Introductory Quantum Theory t x by Reinhard Hentschke Contents 1 Early developments in quantum mechanics1 1.1 Quantization conditions.....................................1 1.2 The black body problem.....................................4 1.3 Building a general formalism..................................7 1.4 The hydrogen atom........................................ 18 1.5 Angular momentum algebra................................... 27 2 Formal quantum mechanics 33 2.1 Operator averages........................................ 36 2.2 The uncertainty principle.................................... 38 2.3 From wave mechanics to matrix mechanics........................... 40 2.4 The time evolution operator................................... 41 2.5 The density operator....................................... 43 2.6 Path integration and the density operator........................... 45 3 Scattering theory 51 3.1 The Lippmann-Schwinger equation............................... 51 3.2 The 1st Born approximation................................... 54 3.3 Partial wave expansion...................................... 55 3.4 Some remarks on formal scattering theory........................... 56 4 Solution methods in quantum mechanics 59 4.1 Time-independent perturbations................................ 59 4.2 Variation method......................................... 64 4.3 The quasi classical approximation................................ 66 4.4 Time-dependent perturbation theory.............................. 67 4.5 Interaction of charges with electromagnetic fields....................... 75 5 Many particle systems 77 5.1 Two kinds of statistics...................................... 77 5.2 Constructing N-particle wave functions............................ 78 5.3 Electron spin........................................... 81 5.4 Simplified theory for helium................................... 86 5.5 Simplified theory for the hydrogen molecule.......................... 88 5.6 Quantum mechanics in chemistry................................ 90 iii 6 The quantized radiation field and second quantization 101 6.1 Quantum theory of radiation.................................. 101 6.2 Second quantization....................................... 104 6.3 Interacting quantum fields:................................... 106 6.4 Outlook.............................................. 106 A Constants and units 109 B Mathematical appendix 111 B.1 Conversion to generalized coordinates............................. 111 B.2 Spherical harmonics....................................... 112 B.3 δ-function............................................. 113 B.4 Fourier series and Fourier transforms.............................. 114 C The Casimir effect 115 D Problems 117 Preface mal basis for quantum mechanics treating bound state problems; chapter three deals with scatter- I recall a physics student expressing his opinion ing; chapter four introduces perturbation theory to- that "real physics" begins with quantum mechan- gether with selected numerical techniques; in chap- ics. This opinion, to which I cannot agree, proba- ter five we turn to many particle systems and ex- bly is due to the "strangeness" in quantum theory, ploit the fact that in nature there are two types which is summarized in an often cited quote by the of particles-fermions and bosons; the final chap- late Richard Feynman 1 expressing his believe that ter, chapter six, presents a short glimpse at the "nobody really understands quantum mechanics". quantized radiation field and second quantization. These lecture notes cover introductory quantum Chapters five and six mark the departure of physics theory to an extend that can be presented in a from chemistry as far as university education is con- one semester course. The subject is approached by cerned. Atomic or molecular systems with more looking first at some of the pressing questions by than two electrons are usually reserved for students the end of the 19th century, when classical physics, of quantum chemistry. Solids as well as advanced in the eyes of many, had come close to explaining quantum theory, which leads to the physics of el- all known physical phenomena. We will focus on ementary particles, appear to be the domain of a special question (e.g. the black body problem), physics students. Here, in a limited sense of course, then introduce an idea or concept to answer this I attempt to satisfy both tastes. Because of this the question in simple terms (e.g. energy quantization), lecture does not follow one particular text. How- relate the quantum theoretical answer to classical ever, in preparing these notes I have been guided theory or experiment, and finally progress deeper by the references [5,6,7,8]. Additional textbooks into the mathematical formalism if it provides a on quantum mechanics containing most of the ma- general basis for answering the next question. In terial discussed here include [9, 10, 11, 12, 13]. this spirit we develop quantum theory by adding in The initial version of these notes was prepared a step by step procedure postulates and abstract for a course on introductory quantum mechanics concepts, testing the theory as we go along, i.e. we during the fall/winter term 2002/2003. Subse- will accept abstract and maybe sometimes counter- quently these notes were used and extended during intuitive concepts as long as they lead to verifiable the summers of 2004, 2006, 2007, as well as during predictions. the fall/winter terms 2008/2009 and 2015/16. I Chapter one is devoted to the early developments owe thanks to many students during these years in quantum theory 2; chapter two provides a for- for their valuable comments. 1 Feynman, Richard Phillips, american physicist, *New Wuppertal, February 2016 (revised version of York 11.5.1918, yLos Angeles 15.2.1988; he made many sem- inal contributions to theoretical physics including the theory January 18, 2021) of superfluid helium and, in particular, to the development of quantum electrodynamics. He introduced the Feynman Reinhard Hentschke diagram technique and the concept of path integration. He not only was a brilliant scientist but also famous for his way of teaching physics. In 1965 he was awarded the Nobel prize Bergische Universit¨at in physics together with Julian S. Schwinger and Sin-Itiro School of Mathematics and Natural Sciences Tomonaga. Gauss-Str. 20 2The student should be aware that these lecture notes, for the most part, cover only a few years in the development 42097 Wuppertal of quantum theory, i.e. "early" is appropriate in a limited e-mail: [email protected] sense only. In 1900 Planck presented his treatment of black http://constanze.materials.uni-wuppertal.de body radiation; Einstein's interpretation of the photoelec- tric effect appeared in 1905/06; Bohr's seminal work "On the constitution of atoms and molecules" was published in 1913. But after 1925/26, when a formal framework (matrix lecture notes, as almost all texts aimed at beginners, do mechanics/wave mechanics) had been suggested to do cal- (partially!) cover this exciting period only. A good histor- culations, the development became extremely rapid! Maybe ical source on important steps in physics in the 19th and not surprising it took only about five years and a number of early 20th century leading up to the developments discussed textbooks on quantum mechanics had appeared (e.g, Weyl, here is A. Sommerfeld's "Atombau und Spektrallinien" [4], 1930 [1]; Dirac, 1930 [2]; von Neumann, 1932 [3]). These which first appeared in 1919! Chapter 1 Early developments in quantum mechanics 1.1 Quantization conditions h = 6:6256 ± 0:0005 · 10−34Js : The year 1900 marks the beginning of what we call "Quantum Theory". Until then various physicists Discrete entities were not new in science at that of reputation had failed to derive a valid formula time. Chemists had long accepted the notion that describing the frequency dependence of radiation matter consists of atoms. This idea was formulated intensity emitted from a black body. A black body by Dalton 3 in his book A new system of chemical can be a metal cavity with walls kept at a fixed philosophy, which appeared in two volumes in 1808 temperature T . A small hole drilled into the walls and 1810. Also, Thomson 4 in 1897 had discovered of this oven will allow radiation to escape - the the electron. However, energy was firmly believed black body radiation. This may sound academic. to be continuous. E.g., we all are used to chang- But what makes this problem especially interest- ing the speed of our car and thus its kinetic energy ing is, that all prior calculations were based on continuously. However, like matter, which macro- apparently well founded theories like Maxwell's 1 scopically appears continuous but microscopically equations and classical statistical mechanics. These must be 'chemically' discrete in order to explain pillars of physics, however, as we shall see below, chemical reactions and their mass ratios, energy produced nonsense. In order to obtain a correct also reveals its discrete nature on the microscopic formula describing black body radiation the physi- scale. That classical physics had worked to every- cist Max Planck 2 had to introduce a revolutionary body's satisfaction had to do with the experimental concept. The energy of the radiation field inside inability of that time to probe the microscopic (or the black body is not continuous. It is the sum of atomic) scale.
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