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CHM2S1 A Introduction to Quantum MechanicsDr R L JohnstonTHE UNIVERSITYOF BIRMINGHAMI Foundations of Quantum Mechanics.1 Classical Mechanics1 1 Features of classical mechanics 1 2 Some relevant equations in classical mechanics 1 3 Example The 1 Dimensional Harmonic Oscillator1 4 Experimental evidence for the breakdown of classical mechanics .1 5 The Bohr model of the atom 2 Wave Particle Duality2 1 Waves behaving as particles 2 2 Particles behaving as waves 2 3 The De Broglie Relationship . 3 Wavefunctions3 1 Definitions 3 2 Interpretation of the wavefunction 3 3 Normalization of the wavefunction 3 4 Quantization of the wavefunction.3 5 Heisenberg s Uncertainty Principle 4 Wave Mechanics4 1 Operators and observables 4 2 The Schr dinger equation 4 3 Particle in a 1 dimensional box .4 4 Further examples Learning Objectives To appreciate the differences between Classical CM and QuantumMechanics QM To know the failures in CM that led to the development of QM . To know how to interpret the wavefunction and how to normalize it To appreciate the origins and implications of quantization and theuncertainty principle To understand wave particle duality and know the relationships betweenmomentum frequency wavelength and energy for particles and waves . To be able to write down the Schr dinger equation for particles in a 1 Dbox in 1 and 2 electron atoms in 1 and 2 electron molecules To know the origins and allowed values of atomic quantum numbers andhow the energies and angular momenta of hydrogen atomic orbitalsdepend on them . To be able to sketch the angular and radial nodal properties of atomic To appreciate the origins of sheilding and its effect on the ordering oforbital energies in many electron atoms To use the Aufbau Principle the Pauli Principle and Hund s Rule topredict the lowest energy electron configuration for many electron. To appreciate how the Born Oppenheimer approximation can be usedto separate electronic and nuclear motion in molecules To understand how molecular orbitals MOs can be generated aslinear combinations of atomic orbitals and the difference betweenbonding and antibonding orbitals . To be able to sketch MOs and their corresponding electron densities To construct MO diagrams for homonuclear and heteronuclear diatomicmolecules To predict the electron configurations for diatomic molecules calculatebond orders and relate these to bond lengths strengths and vibrational.frequencies ReferencesFundamentals P W Atkins J de Paula Atkins Physical Chemistry 7th edn OUP Oxford 2001 . D O Hayward Quantum Mechanics for Chemists RSCTutorial Chemistry Texts 14 Royal Society of Chemistry 2002 W G Richards and P R Scott Energy Levels in Atoms andMolecules Oxford Chemistry Primers 26 OUP Oxford 1994 More Advanced. P W Atkins and R S Friedman Molecular QuantumMechanics 3rd edn OUP Oxford 1997 P A Cox Introduction to Quantum Theory and AtomicStructure Oxford Chemistry Primers 37 OUP Oxford 1996 1 Classical Mechanics. Do the electrons in atoms and molecules obey Newton sclassical laws of motion We shall see that the answer to this question is No This has led to the development of Quantum Mechanics wewill contrast classical and quantum mechanics . 1 1 Features of Classical Mechanics CM 1 CM predicts a precise trajectory for a particle velocity vposition r x y z The exact position r and velocity v and hence the momentum p. mv of a particle mass m can be known simultaneously ateach point in time Note position r velocity v and momentum p are vectors having magnitude and direction v vx vy vz 2 Any type of motion translation vibration rotation can have any.value of energy associated with it i e there is a continuum of energy states 3 Particles and waves are distinguishable phenomena with different characteristic properties and behaviour Property Behaviour.mass momentumParticles position collisionsWaves wavelength diffractionfrequency interference 1 2 Revision of Some Relevant Equations in CM.Total energy of particle E Kinetic Energy KE Potential Energy PE T depends on v V depends on rV depends on the systemE mv2 V e g positional electrostatic PE. E p2 2m V p mv Note strictly E T V and r v p are all defined at a particular time t E t Consider a 1 dimensional system straight line translational motion of a particleunder the influence of a potential acting parallel to the direction of motion Define position r x.velocity v dx dtmomentum p mv m dx dt forceF dV dx Newton s 2nd Law of MotionF ma m dv dt m d2x dt2 . Therefore if we know the forces acting on a particle we can solve a differentialequation to determine it s trajectory x t p t acceleration 1 3 Example The 1 Dimensional Harmonic OscillatorF NB assuming no friction or.k other forces act on the particlem except F The particle experiences a restoring force F proportional to itsdisplacement x from its equilibrium position x 0 Hooke s Law F kx.k is the stiffness of the spring or stretching force constant of the bondif considering molecular vibrations Substituting F into Newton s 2nd Law we get m d2x dt2 kx a second order differential equation Solution .position x t Asin t mof particlefrequency 2 m of oscillation Note frequency depends only on characteristics of the system. m k not the amplitude A x time period 1 Assuming that the potential energy V 0 at x 0 it can be shownthat the total energy of the harmonic oscillator is given by As the amplitude A can take any value this means that the energy. E can also take any value i e energy is continuous At any time t the position x t and velocity v t can be determinedexactly i e the particle trajectory can be specified precisely We shall see that these ideas of classical mechanics fail when we goto the atomic regime where E and m are very small then we need.to consider Quantum Mechanics CM also fails when velocity is very large as v c due to relativistic 1 4 Experimental Evidence for the Breakdown of Classical Mechanics By the early 20th century there were a number of experimentalresults and phenomena that could not be explained by classical.mechanics a Black Body Radiation Planck 1900 UV Catastrophe Classical MechanicsEnergy 2000 K Rayleigh Jeans .0 2000 4000 6000 nm Planck s Quantum Theory Planck 1900 proposed that the light energy emitted by the blackbody is quantized in units of h frequency of light E nh n 1 2 3 . High frequency light only emitted if thermal energy kT h h a quantum of energy Planck s constant h 6 626 10 34 Js If h 0 we regain classical mechanics Conclusions . Energy is quantized not continuous Energy can only change by well defined amounts b Heat Capacities Einstein Debye 1905 06 Heat capacity relates rise in energy of a material with its rise intemperature .CV dU dT V Classical physics CV m 3R for all T Experiment CV m 3R CV as T At low T heat capacity of solids determined byvibrations of solid . Einstein and Debye adopted Planck s hypothesis Conclusion vibrational energy in solids is quantized vibrational frequencies of solids canonly have certain values vibrational energy can only change.by integer multiples of h c Photoelectric Effect Einstein 1905 h Photoelectrons ejected withkinetic energy e Peh o t e l e c t r o n s.Ek h Metal surfacework function Ideas of Planck applied to electromagnetic radiation No electrons are ejected regardless of light intensity unless. exceeds a threshold value characteristic of the metal Ek independent of light intensity but linearly dependent on Even if light intensity is low electrons are ejected if is above thethreshold Number of electrons ejected increases with lightintensity . Conclusion Light consists of discrete packets quanta ofenergy photons Lewis 1922 d Atomic and Molecular Spectroscopy It was found that atoms and molecules absorb and emit light only atspecific discrete frequencies spectral lines not.continuously e g Hydrogen atom emission spectrum Balmer 1885 n1 1 Lymann1 2 Balmern1 3 Paschen.n1 4 Brackettn1 5 Pfund 1 R H 2 2 c n n 1 2 . Empirical fit to spectral lines Rydberg Ritz n1 n2 n1 integers Rydberg constant RH 109 737 3 cm 1 but can also be expressed Revision Electromagnetic RadiationA Amplitude wavelength frequency c x or c . wavenumber c 1 c velocity of light in vacuum 2 9979 x 108 m s 1 1 5 The Bohr Model of the Atom The H atom emission spectrum was rationalized by Bohr Energies of H atom are restricted to certain discrete values. i e electron is restricted to well defined circular orbits labelled by quantum number n Energy light absorbed in discrete amounts quanta photons corresponding to differences between thesee restricted values .n2 E E2 E1 h Absorption Emission Conclusion Spectroscopy provides direct evidence for quantization ofenergies electronic vibrational rotational etc of atoms and molecules Limitations of Bohr Model Rydberg Ritz Equation. The model only works for hydrogen and other one electronions ignores e e repulsion Does not explain fine structure of spectral lines Note The Bohr model assuming circular electron orbits isfundamentally incorrect . 2 Wave Particle Duality Remember Classically particles and waves are distinct Particles characterised by position mass velocity Waves characterised by wavelength frequency By the 1920s however it was becoming apparent that.sometimes matter classically particles can behave like wavesand radiation classically waves can behave like particles 2 1 Waves Behaving as Particlesa The Photoelectric EffectElectromagnetic radiation of frequency can be thought of as.being made up of particles photons each with energy E h This is the basis of Photoelectron Spectroscopy PES b SpectroscopyDiscrete spectral lines of atoms and molecules correspond tothe absorption or emission of a photon of energy h causing.the atom molecule to change between energy levels E h Many different types of spectroscopy are possible c The Compton Effect 1923 Experiment A monochromatic beam of X rays i incident ona graphite block . Observation Some of the X rays passing through the block arefound to have longer wavelengths s Explanation The scattered X rays undergo elastic collisions withelectrons in the graphite Momentum and energy transferred from X rays to electrons . Conclusion Light electromagnetic radiation possesses momentum Momentum of photon p h Energy of photon E h hc p h s Applying the laws of conservation of energy and momentum we get . s i 1 cos m e c 2 2 Particles Behaving as WavesElectron Diffraction Davisson and Germer 1925 Davisson and Germer showed that.a beam of electrons could be diffractedfrom the surface of a nickel crystal Diffraction is a wave property arisesdue to interference between scatteredThis forms the basis of electron.diffraction an analytical technique fordetermining the structures of molecules solids and surfaces e g LEED NB Other particles e g neutrons protons He atoms can also be.diffracted by crystals 2 3 The De Broglie Relationship 1924 In 1924 i e one year before Davisson and Germer sexperiment De Broglie predicted that all matter has wave likeproperties . A particle of mass m travelling at velocity v has linearmomentum p mv By analogy with photons the associated wavelength of theparticle is given by 3 Wavefunctions. A particle trajectory is a classical concept In Quantum Mechanics a particle e g an electron does notfollow a definite trajectory r t p t but rather it is best describedTo be able to write down the SchrÃ¶dinger equation for particles: in a 1-D box; in 1- and 2-electron atoms; in 1- and 2-electron molecules. To know the origins and allowed values of atomic quantum numbers and how the energies and angular momenta of hydrogen atomic orbitals depend on them.