PHY206: Atomic Spectra  Lecturer: Dr Stathes Paganis  Office: D29, Hicks Building  Phone:   Text: A. C. Phillips, ‘Introduction to QM’   Marks: Final 70%, Homework 2x10%, Problems Class 10%

PHY206: Spring SemesterAtomic Spectra2 Course Outline (1)  Lecture 1 : Bohr Theory Introduction Bohr Theory (the first QM picture of the atom) Quantum Mechanics  Lecture 2 : Angular Momentum (1) Orbital Angular Momentum (1) Magnetic Moments  Lecture 3 : Angular Momentum (2) Stern-Gerlach experiment: the Spin Examples Orbital Angular Momentum (2) Operators of orbital angular momentum  Lecture 4 : Angular Momentum (3) Orbital Angular Momentum (3) Angular Shapes of particle Wavefunctions Spherical Harmonics Examples

PHY206: Spring SemesterAtomic Spectra3 Course Outline (2)  Lecture 5 : The Hydrogen Atom (1) Central Potentials Classical and QM 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) 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 SemesterAtomic Spectra4 Course Outline (3)  Lecture 8 : Identical Particles (1) Particle Exchange Symmetry and its Physical Consequences  Lecture 9 : Identical Particles (2) Exchange Symmetry with Spin Bosons and Fermions  Lecture 10 : Atomic Spectra (1) Atomic Quantum States Central Field Approximation and Corrections  Lecture 11 : Atomic Spectra (2) The Periodic Table  Lecture 12 : Review Lecture

PHY206: Spring SemesterAtomic Spectra5 Atoms, Protons, Quarks and Gluons Atomic Nucleus Proton gluons Atom Proton

PHY206: Spring SemesterAtomic Spectra6 Atomic Structure

PHY206: Spring SemesterAtomic Spectra7 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 SemesterAtomic Spectra8 atoms should collapse Classical Electrodynamics: charged particles radiate EM energy (photons) when their velocity vector changes (e.g. they accelerate). This means an electron should fall into the nucleus. Classical Physics:

PHY206: Spring SemesterAtomic Spectra9 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 SemesterAtomic Spectra10 Emission lines of some elements (all quantized!)

PHY206: Spring SemesterAtomic Spectra11 Emission spectrum of Hydrogen “Continuous” spectrum “Continuous” spectrum “Quantized” spectrum Any  E is possible Only certain  E are allowed EE EE  Relaxation from one energy level to another by emitting a photon, with  E = hc/  Relaxation from one energy level to another by emitting a photon, with  E = hc/  If = 440 nm,  = 4.5 x J

PHY206: Spring SemesterAtomic Spectra12 The goal: use the emission spectrum to determine the energy levels for the hydrogen atom (H-atomic spectrum) Emission spectrum of Hydrogen

PHY206: Spring SemesterAtomic Spectra13  Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by: n = 3, 4, 5, …..  The above equation predicts that as n increases, the frequencies become more closely spaced. Balmer model (1885)

PHY206: Spring SemesterAtomic Spectra14 Rydberg Model  Johann Rydberg extended the Balmer model by finding more emission lines outside the visible region of the spectrum: n 1 = 1, 2, 3, …..  In this model the energy levels of the H atom are proportional to 1/n 2 n 2 = n 1 +1, n 1 +2, … R y = 3.29 x /s

PHY206: Spring SemesterAtomic Spectra15 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/2 , which means that for circular orbits of radius r the z component of the angular momentum L is quantized: 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 SemesterAtomic Spectra16 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. Energy in Joules Z = atomic number (1 for H) n is an integer (1, 2, ….) R y x h = x J The Bohr constant is the same as the Rydberg multiplied by Planck’s constant!

PHY206: Spring SemesterAtomic Spectra17 Energy levels get closer together as n increases at n = infinity, E = 0 The Bohr Model (3)

PHY206: Spring SemesterAtomic Spectra18 We can use the Bohr model to predict what  E is for any two energy levels Prediction of energy spectra

PHY206: Spring SemesterAtomic Spectra19 Example: At what wavelength will an emission from n = 4 to n = 1 for the H atom be observed? 14 Example calculation (1)

PHY206: Spring SemesterAtomic Spectra20 Example: What is the longest wavelength of light that will result in removal of the e - from H?  1 Example calculation (2)

PHY206: Spring SemesterAtomic Spectra21 The Bohr model can be extended to any single electron system….must keep track of Z (atomic number). Examples: He + (Z = 2), Li +2 (Z = 3), etc. Z = atomic number n = integer (1, 2, ….) Bohr model extedned to higher Z

PHY206: Spring SemesterAtomic Spectra22 Example: At what wavelength will emission from n = 4 to n = 1 for the He + atom be observed? 2 14 Example calculation (3)

PHY206: Spring SemesterAtomic Spectra23 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) Einstein De Broglie relation: particles with certain momentum, oscillate with frequency hv.

PHY206: Spring SemesterAtomic Spectra24 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)  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 SemesterAtomic Spectra25 QM: we can only talk about the probability to find the electron around the atom – there is no orbit!