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Lecture notes in Atomic Physics

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Atomic physics and Quantum mechanics acompanying notes Univ.-Prof. Dr. Enrico Arrigoni TU Graz - Austria Version from: November 13, 2010 Students of the course 515.300 “Atomphysik und Quantenmechanik fu¨r ET/MB” can access a more complete version of the lecture notes (access password restricted) interested people can ask me by e-mail: arrigonitugraz.at PDF presentation using LaTeX and the Beamer Class http://latex-beamer.sourceforge.net Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 1 / 193Content of this lecture Content of this lecture 5 The wave function and Schr¨odinger equation Euristic derivation of Schr¨odinger equation Time-independent Schr¨odinger equation Interpretation of the wave function Summary: Schr¨odinger equation 6 Quantum mechanics of some simple systems Free particle Particle in a box Generalisations of the particle in a box Tunnel effect Three-dimensional box 7 Principles and Postulates of Quantum mechanics Postulates of Quantum Mechanics Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 3 / 193Content of this lecture Content of this lecture 8 Angular momentum and electron spin First step: “particle on a ring” Second step: “particle on the surface of a sphere” Electron spin 9 The Hydrogen atom Schr¨odinger equation and separation of variables Qualitative solution Classification of atomic orbitals 10 Many-electron atoms and the periodic table Pauli principle Building-up principle Hund’s rule Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 4 / 193Suggested literature Suggested literature I 1 S. M. Blinder, Introduction to Quantum Mechanics in chemistry, Material Science, and Biology see also http://www.umich.edu/ chem461/ the class essentially based on this book 2 L. van Dommelen Fundamental Quantum Mechanics for Engineers notes available at http://www.eng.fsu.edu/ dommelen/quantum 3 J. E. House, Fundamentals of Quantum Chemistry some mathematical aspects are treated in more detail here 4 P. W. Atkins, Physical Chemistry Chap. 2 also a good book, many details and examples, many physical aspects discussed. 5 P. A. Tipler and R. A. Llewellyn, Moderne Physik simpler treatment Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 5 / 193Suggested literature Suggested literature II 6 J.J. Sakurai, Modern Quantum Mechanics High level book 7 D. Ferry Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineer More advanced, special topics of interest for material physicists. Device physics, transport theory. 8 Applets http://www.quantum-physics.polytechnique.fr/en/index.html Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 6 / 193Failures of classical physics Blackbody radiation 4000K I(ω) 3000K 2000K 2πc frequency ω = λ 2πc Energy intensity I(ω) versus frequency (ω = 2πν = ) of blackbody λ radiation at different temperatures: The energy intensity I(ω) vanishes at small and large ω, there is a maximum in between. The maximum frequency ω (“color”) of the distribution obeys the max law (Wien’s law) ω = const. T max Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 12 / 193Failures of classical physics Blackbody radiation An idealized description is the so-called blackbody model, which describes a perfect absorber and emitter of radiation. One single electromagnetic wave is characterised by a wavevector k which indicates the propagation direction and is related to the frequency and 2π ω wavelength byk = = . λ c In a blackbody, electromagnetic waves of all wavevectors k are present and distributed in equilibrium. One can consider an electromagnetic wave with wavevector k as an independent oscillator (“mode”). Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 13 / 193Failures of classical physics Blackbody radiation For a given frequency ω (= 2πν), there are many oscillators k having that frequency. Sinceω = c k the number (density) n(ω) of oscillators with frequencyω is proportional to the surface of a sphere with radiusω/c, i. e. 2 n(ω)∝ω (4.1) The energy equipartition law of statistical physics tells us that at temperature T each mode is excited (on average) to the same energy K T. B Therefore, at temperature T the energy density u(ω,T) of all oscillators with a certain frequency ω would be given by 2 u(ω,T)∝ K T ω (4.2) B (Rayleigh hypothesis). Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 14 / 193Failures of classical physics Blackbody radiation For a given frequency ω (= 2πν), there are many oscillators k having that frequency. Sinceω = c k the number (density) n(ω) of oscillators with frequencyω is proportional to the surface of a sphere with radiusω/c, i. e. 2 n(ω)∝ω (4.1) The energy equipartition law of statistical physics tells us that at temperature T each mode is excited (on average) to the same energy K T. B Therefore, at temperature T the energy density u(ω,T) of all oscillators with a certain frequency ω would be given by 2 u(ω,T)∝ K T ω (4.2) B (Rayleigh hypothesis). Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 14 / 193Failures of classical physics Blackbody radiation For a given frequency ω (= 2πν), there are many oscillators k having that frequency. Sinceω = c k the number (density) n(ω) of oscillators with frequencyω is proportional to the surface of a sphere with radiusω/c, i. e. 2 n(ω)∝ω (4.1) The energy equipartition law of statistical physics tells us that at temperature T each mode is excited (on average) to the same energy K T. B Therefore, at temperature T the energy density u(ω,T) of all oscillators with a certain frequency ω would be given by 2 u(ω,T)∝ K T ω (4.2) B (Rayleigh hypothesis). Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 14 / 193Failures of classical physics Blackbody radiation Planck’s hypothesis: The “oscillators” (electromagnetic waves), cannot have a continuous of energies. Their energies come in “packets” (quanta) of size h ν = ω. −34 h h≈ 6.6×10 Joules sec ( = ) Planck’s constant. 2π At small frequencies, as long as K T ≫ ω, this effect is irrelevant. B The effect will start to be important at K T ∼ ω: here u(ω,T) will start B to decrease. And in fact, Wien’s empiric observation is that u(ω,T) displays a maximum at ω∝ K T. B Eventually, for K T ≪ ω the oscillators are not excited at all, their B energy is vanishingly small. A more elaborate theoretical treatment gives the correct functional form. Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 16 / 193Failures of classical physics Blackbody radiation Planck’s hypothesis: The “oscillators” (electromagnetic waves), cannot have a continuous of energies. Their energies come in “packets” (quanta) of size h ν = ω. −34 h h≈ 6.6×10 Joules sec ( = ) Planck’s constant. 2π At small frequencies, as long as K T ≫ ω, this effect is irrelevant. B The effect will start to be important at K T ∼ ω: here u(ω,T) will start B to decrease. And in fact, Wien’s empiric observation is that u(ω,T) displays a maximum at ω∝ K T. B Eventually, for K T ≪ ω the oscillators are not excited at all, their B energy is vanishingly small. A more elaborate theoretical treatment gives the correct functional form. Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 16 / 193Failures of classical physics Blackbody radiation Planck’s hypothesis: The “oscillators” (electromagnetic waves), cannot have a continuous of energies. Their energies come in “packets” (quanta) of size h ν = ω. −34 h h≈ 6.6×10 Joules sec ( = ) Planck’s constant. 2π At small frequencies, as long as K T ≫ ω, this effect is irrelevant. B The effect will start to be important at K T ∼ ω: here u(ω,T) will start B to decrease. And in fact, Wien’s empiric observation is that u(ω,T) displays a maximum at ω∝ K T. B Eventually, for K T ≪ ω the oscillators are not excited at all, their B energy is vanishingly small. A more elaborate theoretical treatment gives the correct functional form. Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 16 / 193Failures of classical physics Photoelectric effect Photoelectric effect Electrons in a metal are confined by an energy barrier (work function) φ. One way to extract them is to shine light onto a metallic plate. Light transfers an energy E to the electrons. light The rest of the energy E −φ goes into the kinetic energy of the light 1 2 electron E = m v . kin 2 By measuring E , one can get E . kin light −−−−−−−−−−−−− anode V =E /e kin,min E kin ν φ Metal catode ++++++++++ Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 19 / 193Failures of classical physics Photoelectric effect examples: Photoelectric effect Classicaly, we would espect the total energy transferred to an electron E =φ+E to be proportional to the radiation intensity. The light kin experimental results give a different picture: while the current (i. e. the number of electrons per second expelled from the metal) is proportional to the radiation intensity, E is proportional light E = h ν (4.4) light to the frequency of light: E light E E kin φ φ ν = min ν h Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 20 / 193Failures of classical physics Photoelectric effect Summary: Planck’s energy quantum The explanation of Blackbody radiation and of the Photoelectric effect are explained by Planck’s idea that light carries energy only in “quanta” of size E = hν (4.5) This means that light is not continuous object, but rather its constituent are discrete: the photons. Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 21 / 193Failures of classical physics Photoelectric effect Summary: Planck’s energy quantum The explanation of Blackbody radiation and of the Photoelectric effect are explained by Planck’s idea that light carries energy only in “quanta” of size E = hν (4.5) This means that light is not continuous object, but rather its constituent are discrete: the photons. Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 21 / 193Wave and Particle duality Light carries momentum: Compton scattering This result can be understood if one assumes that the particles constituting electromagnetic waves (photons) have a momentum h p = (5.3) λ and due to the kinematics part of the momentum is transferred to the electron. This is consistent with Planck’s energy formula for photons and with relativity, assuming that photons velocity is c: 2 5.4 E = m c = hν ⇒ p = m c = E/c =hν/c = h/λ ( ) Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 33 / 193Wave and Particle duality Light carries momentum: Compton scattering This result can be understood if one assumes that the particles constituting electromagnetic waves (photons) have a momentum h p = (5.3) λ and due to the kinematics part of the momentum is transferred to the electron. This is consistent with Planck’s energy formula for photons and with relativity, assuming that photons velocity is c: 2 5.4 E = m c = hν ⇒ p = m c = E/c =hν/c = h/λ ( ) Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 33 / 193Wave and Particle duality Matter (Electrons) as waves examples: “Double slit” experiment with crystals Diffraction crystal For x-rays the natural “slit” system consists of an arrangement of atoms in a crystalline structure the distance between atoms is of the order of the wavelength of x-rays Surprisingly, an interference pattern was observed for electrons as well. Based on these ideas, de Broglie suggested that matter (electrons) might Enrico Arrigoni (TU Graz) Atomic Physics and Quantum Mechanics WS 2009 34 / 193 also behave as waves.