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From Quantum to Molecule Contents Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 From Quantum to Molecule Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ [email protected] Lecture notes, current version: January 31, 2020 Contents Page 1 HC1: Introduction to Quantum Theory I 4 1.1 Wave-particleduality......................................... 5 1.2 Thequantumwave-function..................................... 7 1.3 TheSchroedingerequation ..................................... 11 1.4 Heisenberg’suncertaintyprinciple. 13 2 HC2: Introduction to Quantum Theory II 16 2.1 Observables and operators in quantum theory . 16 2.2 The quantum superposition principle and the free particle revisited . 21 2.3 Hermiticity and orthogonality. 23 2.4 Expectation values of physical quantities . 25 2.5 Radioactivity ............................................. 27 3 HC3: The quantum mechanics of simple systems 30 3.1 The particle in a box and energy quantization . 30 3.2 Particleinatwo-dimensionalbox. 34 3.3 The quantum harmonic oscillator . 35 3.4 Quantumtunneling.......................................... 40 4 HC4: Rotational motion in quantum mechanics 46 4.1 Polar and spherical coordinates . 47 4.2 Quantummechanicsofaparticleinaring. 49 4.3 Angular momentum in quantum mechanics . 52 4.4 Quantum mechanics of a particle in spherical motion . 54 4.5 Probabilistic interpretation in spherical coordinates . 59 Page 1 of 177 Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 5 HC5: The hydrogen atom 62 5.1 The Schroedinger equation for the hydrogen atom . 62 5.2 The radial wave functions of the hydrogen atom . 67 5.3 The orbital structure of the hydrogen atom . 72 5.4 Radial probability distributions . 73 5.5 Hydrogen-like atoms . 77 5.6 General properties of the p- and d-typeorbitals.......................... 78 6 HC6: Multi-electron atoms and the electron spin 82 6.1 Theheliumatom........................................... 82 6.2 Spininquantummechanics ..................................... 85 6.3 Fermions and bosons: spin and statistics . 87 6.4 ThePauliexclusionprincipleandtwo-electronsystems . 88 6.5 Multi-electronatomsandthebuilding-upprinciple . 91 7 HC7: Chemical bonds and molecules I 96 7.1 Chemical bonding in valence Bond theory . 98 7.2 Molecular bonds in diatomic molecules . 100 7.3 Orbitalhybridisation......................................... 103 7.4 MolecularOrbitaltheory....................................... 106 8 HC8: Chemical bounds and molecules II 114 8.1 Chemical bonding in homo-nuclear diatomic molecules . 114 8.2 Heteronuclear diatomic molecules . 119 8.3 The variational principle for diatomic molecules . 122 9 HC9: Chemical bounds and molecules III 128 9.1 Molecular orbitals for poly-atomic molecules with the Huckel method . 128 9.2 Matrix formulation of Huckel theory . 130 9.3 Huckel theory for poly-atomic orbitals . 132 9.4 Hydrogen bonds . 135 10 HC10: Molecular and vibrational spectroscopy 139 10.1 Basics of light-matter interaction: emission and absorption . 140 10.2 Vibrational modes and light interaction in diatomic molecules . 146 10.3 Vibrational modes and light interaction in poly-atomic molecules . 149 11 HC11: Molecular spectroscopy and electronic transitions 153 11.1 Electronic transitions in molecules . 153 11.2 TheFranck-Condonprinciple .................................... 157 11.3 Fluorescence and phosphorescence . 159 11.4Lasers ................................................. 162 11.5 Photodynamic Therapy (PDT) (*) . 164 Page 2 of 177 Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 12 HC12: Magnetic Nuclear Resonance 168 12.1Nuclearspin.............................................. 168 12.2Thechemicalshift .......................................... 171 12.3NMRspectroscopy .......................................... 171 12.4 Magnetic resonance imaging (MRI) . 175 Page 3 of 177 Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 From Quantum to Molecule Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ [email protected] 1 HC1: Introduction to Quantum Theory I In this lecture we present a first introduction to the mathematical framework of quantum theory. We intro- duce the main postulates of quantum mechanics and study the main physical consequences. We introduce the quantum wave-function and study the equation that determine its behaviour: the Schroedinger equation. We also introduce a number of mathematical tools which are required to compute phenomena in the quan- tum formalism. We also discuss Heisenberg’s uncertainty principle, and show that it entails a fundamental limitation about the physical knowledge that we can have about quantum systems. Learning goals of this lecture: • Assess the consequences of the wave-particle duality posited by quantum theory. • Understand that the state of a quantum system is fully described by the quantum wave-function, which is a solution of the time-independent Schroedinger equation. • Become familiar the probabilistic interpretation associated to the quantum wave-function and compute the normalisation constants in simple cases. • Apply Heisenberg’s uncertainty principle to pairs of correlated physical observables. Introduction. In this first two lectures of the course (HC1 and HC2), we will present a concise overview of the basic principles of quantum theory, the mathematical language relevant for the description of small objects such as atoms, molecules, and electrons. We will state these central principles of quantum theory as established facts, avoiding (due to lack of time) the historical discussion of how these principles were developed or what is the experimental evidence that validates them. The interested student can find a brief introduction to the historical developments that lead to the formation of quantum theory on the slides that I have posted in Canvas: https://canvas.vu.nl/courses/35399/files/folder/Historical%20Introduction%20to%20Quantum%20Theory Page 4 of 177 Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 In the following, we present the central concepts of quantum theory, focusing on those which are of special relevance for this course. When required, we will also briefly review the corresponding mathematical methods that are needed to deal with the idea of quantum theory. 1.1 Wave-particle duality Every object, from a plane to a proton or an electron, will under specific conditions behave as a wave: this concept is known as wave-particle duality. Therefore, this object will experience a behaviour usually associated with waves, such as di↵raction when crossing a narrow slit and the formation of interference patters. This of course sounds quite anti-intuitive: we don’t see every day cars interfering between them or di↵racting as they cross a tunnel. The reason that the wave-like character of everyday objects is that quantum theory tells us that it should becomes apparent only when the length scales involved (say the width of an aperture) are of the same order or smaller than their so-called De Broglie wavelength, defined as h h λ = , (1.1) ⌘ p mv 36 2 where h =6.26 10− m kg/s is a numerical constant known as Planck’s constant, ubiquitous in the ⇥ quantum theory, and p = mv is the linear momentum of the object with m being its mass and v its velocity. Planck’s constant is one of the defining features of quantum theory, and in particular it determines when genuinely quantum e↵ects will be important. Let us illustrate under which conditions the wave-like nature of two very di↵erent objects will become relevant, an electron and a car. First of all let us consider an electron that travels at 1% of the speed of light. This electron will 24 have associated a linear momentum of p = m v =2.73 10− kg m/s, and therefore its De Broglie · ⇥ wavelength, that is, the distance scale for which an electron should exhibit a wave-like character, is h 10 λ = 0.5 10− m , (1.2) p ' ⇥ which is the typical separation between atoms in an crystal. Therefore one expects to observe the wave character of electrons when crossing a crystalline lattice. On the other hand, for a car of mass m = 1000 kg moving at a velocity of v = 100 km/h, such that its momentum is p =2.8 104 kg m/s, we can associate a De Broglie wavelength of ⇥ h 38 λ = 2 10− m , (1.3) p ' ⇥ implying that the wave character of a car is unobservable: a car would di↵ract only when crossing a tunnel of width λ, much smaller than an atom itself. ⇠ Therefore, in quantum theory, when particles interact with objects of length similar or smaller than their de Broglie wavelength, their wave-like character will become apparent. In general particles such as protons and electrons will exhibit phenomena that we classically associate with either particles or waves depending on the specific circumstances. The remarkable wave-particle behaviour of quantum theory is illustrated nicely by the famous double slit Page 5 of 177 Dr Juan Rojo From Quantum to Molecule: Lecture Notes January 31, 2020 Classical waves Classical particles Quantum particles Figure 1.1: In classical theory, the behaviour of waves and particles in the double slit experiment is very di↵erent: waves lead to an interference pattern, while particles lead to just two blobs in the detector screen (upper diagrams). In quantum particle, the situation is very di↵erent (lower diagrams): if the number of incident particles is large enough, a clear interference patter also appears, just as if the particles behaved as waves. experiment, schematically represented in Fig. 1.1.
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