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Transcript
PHY206: Atomic Spectra Lecturer: Dr Stathes Paganis Office: D29, Hicks Building Phone: 222 4352 Email: [email protected] Text: A. C. Phillips, ‘Introduction to QM’ http://www.shef.ac.uk/physics/teaching/phy206 Marks: Final 70%, Homework 2x10%, Problems Class 10% Course Outline (1) Lecture 1 : Bohr Theory Lecture 2 : Angular Momentum (1) Introduction Bohr Theory (the first QM picture of the atom) Quantum Mechanics Orbital Angular Momentum (1) Magnetic Moments Lecture 3 : Angular Momentum (2) Stern-Gerlach experiment: the Spin Orbital Angular Momentum (2) Examples Operators of orbital angular momentum Lecture 4 : Angular Momentum (3) Orbital Angular Momentum (3) Angular Shapes of particle Wavefunctions Spherical Harmonics Examples PHY206: Spring Semester Atomic Spectra 2 Course Outline (2) Lecture 5 : The Hydrogen Atom (1) Central Potentials QM of the Hydrogen Atom (1) The Schrodinger Equation for the Coulomb Potential Lecture 6 : The Hydrogen Atom (2) QM of the Hydrogen Atom (2) Classical and QM central potentials Energy levels and Eigenfunctions Sizes and Shapes of the H-atom Quantum States Lecture 7 : The Hydrogen Atom (3) The Reduced Mass Effect Relativistic Effects PHY206: Spring Semester Atomic Spectra 3 Course Outline (3) Lecture 8 : Identical Particles (1) Lecture 9 : Identical Particles (2) Particle Exchange Symmetry and its Physical Consequences Exchange Symmetry with Spin Bosons and Fermions Lecture 10 : Atomic Spectra (1) Atomic Quantum States Lecture 11 : Atomic Spectra (2) Central Field Approximation and Corrections The Periodic Table Lecture 12 : Review Lecture PHY206: Spring Semester Atomic Spectra 4 Atoms, Protons, Quarks and Gluons Atomic Nucleus Atom Proton Proton PHY206: Spring Semester gluons Atomic Spectra 5 Atomic Structure PHY206: Spring Semester Atomic Spectra 6 Early Models of the Atom Rutherford’s model Planetary model Based on results of thin foil experiments (1907) Positive charge is concentrated in the center of the atom, called the nucleus Electrons orbit the nucleus like planets orbit the sun PHY206: Spring Semester Atomic Spectra 7 Classical Physics: atoms should collapse Classical Electrodynamics: charged particles This means an electron should fall radiate EM energy (photons) when their into the nucleus. velocity vector changes (e.g. they accelerate). PHY206: Spring Semester Atomic Spectra 8 Light: the big puzzle in the 1800s Light from the sun or a light bulb has a continuous frequency spectrum Light from Hydrogen gas has a discrete frequency spectrum PHY206: Spring Semester Atomic Spectra 9 Emission lines of some elements (all quantized!) PHY206: Spring Semester Atomic Spectra 10 Emission spectrum of Hydrogen “Quantized” spectrum DE DE “Continuous” spectrum Any DE is possible Only certain DE are allowed Relaxation from one energy level to another by emitting a photon, with DE = hc/l If l = 440 nm, DE = 4.5 x 10-19 J PHY206: Spring Semester Atomic Spectra 11 Emission spectrum of Hydrogen The goal: use the emission spectrum to determine the energy levels for the hydrogen atom (H-atomic spectrum) PHY206: Spring Semester Atomic Spectra 12 Balmer model (1885) Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by: 1 1 2 2 2 n n = 3, 4, 5, ….. The above equation predicts that as n increases, the frequencies become more closely spaced. PHY206: Spring Semester Atomic Spectra 13 Rydberg Model Johann Rydberg extended the Balmer model by finding more emission lines outside the visible region of the spectrum: 1 1 Ry 2 2 n1 n 2 n1 = 1, 2, 3, ….. n2 = n1+1, n1+2, … Ry = 3.29 x 1015 1/s this model the energy levels of the H atom are proportional In to 1/n2 PHY206: Spring Semester Atomic Spectra 14 The Bohr Model (1) Bohr’s Postulates (1913) Bohr set down postulates to account for (1) the stability of the hydrogen atom and (2) the line spectrum of the atom. 1. Energy level postulate An electron can have only specific energy levels in an atom. – Electrons move in orbits restricted by the requirement that the angular momentum be an integral multiple of h/2p, which means that for circular orbits of radius r the z component of the angular momentum L is quantized: L mvr n 2. Transitions between energy levels An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another. PHY206: Spring Semester Atomic Spectra 15 The Bohr Model (2) Bohr derived the following formula for the energy levels of the electron in the hydrogen atom. Bohr model for the H atom is capable of reproducing the energy levels given by the empirical formulas of Balmer and Rydberg. 2 18 Z E 2.178 x10 2 n • Ry x h = -2.178 x 10-18 J PHY206: Spring Semester Energy in Joules Z = atomic number (1 for H) n is an integer (1, 2, ….) The Bohr constant is the same as the Rydberg multiplied by Planck’s constant! Atomic Spectra 16 The Bohr Model (3) 2 Z 18 E 2.178 x10 2 n • Energy levels get closer together as n increases • at n = infinity, E = 0 PHY206: Spring Semester Atomic Spectra 17 Prediction of energy spectra • We can use the Bohr model to predict what DE is for any two energy levels DE E final E initial 1 1 18 18 DE 2.178x10 J 2 (2.178x10 J) 2 ninitial n final 1 1 DE 2.178x1018 J 2 2 n final ninitial PHY206: Spring Semester Atomic Spectra 18 Example calculation (1) • Example: At what wavelength will an emission from n = 4 to n = 1 for the H atom be observed? 1 1 DE 2.178x1018 J 2 2 n final ninitial 1 4 1 DE 2.178x10 J1 2.04x1018 J 16 18 18 DE 2.04x10 J PHY206: Spring Semester hc l l 9.74 x108 m 97.4nm Atomic Spectra 19 Example calculation (2) • Example: What is the longest wavelength of light that will result in removal of the e- from H? 1 1 DE 2.178x10 J 2 2 n final ninitial 1 18 DE 2.178x1018 J0 1 2.178x1018 J 18 DE 2.178x10 J PHY206: Spring Semester hc l l 9.13x108 m 91.3nm Atomic Spectra 20 Bohr model extedned to higher Z • The Bohr model can be extended to any single electron system….must keep track of Z (atomic number). 2 Z 18 E 2.178 x10 2 n Z = atomic number n = integer (1, 2, ….) • Examples: He+ (Z = 2), Li+2 (Z = 3), etc. PHY206: Spring Semester Atomic Spectra 21 Example calculation (3) • Example: At what wavelength will emission from n = 4 to n = 1 for the He+ atom be observed? 1 1 DE 2.178x1018 JZ 2 2 2 n final ninitial 2 1 4 1 DE 2.178x10 J41 8.16x1018 J 16 hc 18 l 2.43x108 m 24.3nm DE 8.16x10 J l l H l He 18 PHY206: Spring Semester Atomic Spectra 22 Problems with the Bohr model Why electrons do not collapse to the nucleus? How is it possible to have only certain fixed orbits available for the electrons? Where is the wave-like nature of the electrons? First clue towards the correct theory: De Broglie relation (1923) E h hc / l wher e c l E mc2 Einstein h h l mc p PHY206: Spring Semester De Broglie relation: particles with certain momentum, oscillate with frequency hv. Atomic Spectra 23 Quantum Mechanics Particles in quantum mechanics are expressed by wavefunctions Wavefunctions are defined in spacetime (x,t) They could extend to infinity (electrons) They could occupy a region in space (quarks/gluons inside proton) In QM we are talking about the probability to find a particle inside a volume at (x,t) 2 3 r , t d r So the wavefunction modulus is a Probability Density (probablity per unit volume) In QM, quantities (like Energy) become eigenvalues of operators acting on the wavefunctions PHY206: Spring Semester Atomic Spectra 24 QM: we can only talk about the probability to find the electron around the atom – there is no orbit! PHY206: Spring Semester Atomic Spectra 25