Quantum Mechanics Exam 3 Hour Exam 3 Wednesday November 29th Quantization of light In class Quantum Physics and Nuclear Physics Twenty multiple choice questions Will cover Chapters 13 14 15 and 16 Lecture material You should bring 1 page notes written single sided 2 Pencil and a Calculator Review test will be available online Light comes in discrete clumps photons Light shows both particle and wave like properties Photon energy E hf hc Evidence for particle properties the photoelectric effect Matter waves Matter shows both particle and wave like properties deBroglie wavelength Planck s constant h momentum p Evidence for wave properties is interference and diffraction 1 Phy107 Fall 2006 Photon or electron interference Photoelectric effect summary Light is made up of photons individual hc 1240 eV nm particles each with energy E hf One photon collides with one electron knocks it out of metal Do an interference experiment again But turn down the intensity until only ONE photon at a time is between slits and screen Only one photon present here If photon doesn t have enough energy cannot knock electron out Intensity photons sec doesn t change this Photon greater than a minimum frequency less than a maximum wavelength required to eject electron 3 Wavelengths of massive objects Using Quantum Mechanics The wavefunction of a quantum state The ground state and excited states Probabilistic interpretation of the wavefunction Heisenberg uncertainty principle Same constant as before h hc p 2 mc 2 E kinetic 4 Models of the hydrogen atom Absorption and emission of light line spectra h mv p mv for a nonrelativistic v c particle with mass Needed the idea of probabilities of an outcome happening to explain the wavelike and particle Phy107 Fall 2006 like results of interference experiments Quantum states in a hydrogen atom h p deBroglie wavelength Is there still interference Phy107 Fall 2006 2 Phy107 Fall 2006 Position and momentum cannot be know simultaneously Consequency of wave properties kinetic energy rest energy Phy107 Fall 2006 5 Phy107 Fall 2006 6 1 Electron standing waves on an atom Energy levels Instead of drawing orbits we can just indicate the energy an electron would have if it were in that orbit Wave representing electron Electron wave extends around circumference of orbit Only integer number of wavelengths around orbit allowed Wave representing electron 7 Phy107 Fall 2006 Emitting and absorbing light E3 13 6 eV 32 n 2 E2 13 6 eV 22 Photon emitted hf E2 E1 n 1 n 4 n 3 E3 13 6 eV 32 n 2 E2 13 6 eV 22 13 6 eV 12 n 1 13 6 E1 2 eV 1 Absorbing a photon of correct energy makes electron jump to higher quantum state Photon is emitted when electron drops from one quantum state to another Phy107 Fall 2006 9 Particle in a box Wavefunctions Wavefunction 13 6 eV 32 n 2 E2 13 6 eV 22 n 1 E1 13 6 eV 12 8 Phy107 Fall 2006 Particle can exist in different quantum states having Different energy Different momentum Different wavelength The quantum wavefunction describes wave nature of particle Photon absorbed hf E2 E1 E1 E3 Topic The wavefunction Zero energy n 4 n 3 n 4 n 3 Energy axis Zero energy Probability Wavefunction 2 Square of the wavefunction gives probability of finding particle Zero s in probability arise from interference of the particle wave with itself 10 Phy107 Fall 2006 Particle and wave Every particle has a wavelength h p However particles are at approximately one position not very wavelike Works if the particles is a superposition nearby of wavelengths rather than one definite wavelength Ground state wavefunction and probability Height of probability curve represents likelihood of finding particle at that point Heisenberg uncertainty principle Phy107 Fall 2006 11 440 439 438 437 436 Hz Hz Hz Hz Hz x p h 2 However particle is still spread out over small volume in addition to being spread out over several wavelengths Phy107 Fall 2006 12 2 Uncertainty in Quantum Mechanics Position uncertainty L Particle in a box or a sphere Simple in 1D or 2 3D box Since 2L L h h h h Momentum ranges from to range 2 L More complex in the hydrogen atom 2L One halfwavelength Box the force that keeps the electron near the nucleus is the coulomb force Coulomb force is spherically symmetric the same in any direction Still 3 quantum numbers L Reducing the box size reduces position uncertainty but the momentum uncertainty goes up The product is constant x p is always greater than h 4 13 Phy107 Fall 2006 Hydrogen Quantum Numbers Phy107 Fall 2006 14 Additional Lecture Material Quantum numbers n l ml n how charge is distributed radially around the nucleus Average radial distance This determines the energy Spin An additional quantum property of a particle Indistinguishability and symmetry l how spherical the charge distribution Fermions and Bosons Pauli exclusion principle l 0 spherical l 1 less spherical ml rotation of the charge around the z axis Rotation clockwise or counterclockwise and how fast Small energy differences for l and ml states Fit n half wavelengths in the box Physics of solids Energy bands in a solid Metals insulators and semiconductors Superconductors n 1 l 0 ml 0 n 2 l 1 ml 1 Phy107 Fall 2006 15 Free electron by itself in space not only has a charge but also acts like a bar magnet with a N and S pole Several important conceptual aspects of quantum mechanics N Indistinguishability Since electron has charge could explain this if the electron is spinning particles are absolutely identical Leads to Pauli exclusion principle one Fermion quantum state Then resulting current loops would produce magnetic field just like a bar magnet Electron in NOT spinning As far as we know electron is a point particle Phy107 Fall 2006 16 Topic Indistinguishability symmetry Topic spin But Phy107 Fall 2006 Symmetry Characterizes the wavefunctions S Leads to different energy levels 17 Phy107 Fall 2006 18 3 Other elements Fermions and bosons Boson symmetric wavefunction spin 1 1 2 2 More electrons requires next higher energy states Lithium three electrons n 2 states 1 higher energy Fermion antisymmetric wavefunction spin 1 2 Can t try to put two Fermions in the same quantum state for instance both in the s state 1 2 Other states empty n 1 states 2 0 1 lowest energy fill first 19 Phy107 Fall 2006 Phy107 Fall 2006 Elements with more electrons have more complex states occupied 20 More than one atom Wavefunctions Energy levels Elements in same column have similar chemical properties 21 Phy107 Fall 2006 n and p type semiconductors
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