Chapter 4 Introduc1on to Quantum Mechanics 139 4 1 Preliminaries Wave Mo1on and Light 141 4 2 Evidence for Energy Quan1za1on in Atoms 145 4 3 The Bohr Model Predic1ng Discrete Energy Levels in Atoms 153 4 4 Evidence for Wave Par1cle Duality 157 4 5 The Schr dinger Equa1on 167 4 6 Quantum Mechanics of Par1cle in a Box Models 172 All great truths begin as blasphemies George Bernard Shaw CLASSICAL QUANTUM End of 19TH Century Par1cles Newton s equa1ons of mo1on Light James Clerk Maxell theory of electromagne1sm Physicist believed all was known but they were in for some nasty surprises Michelson Many instances might be cited but these will su ce to jus8fy the statement that our future discoveries must be looked for in the sixth place of decimals could be Kelvin or Maxwell Quantum Dots seeing quan1za1on Nanotechnology can control the color by the size of the par1cle under UV illumina1on CdSe range from 2 to 10 nanometers in diameter about the width of 50 atoms Fig 4 CO p 139 Table 4 1 p 141 Basics of Waves Waves Characterized by Wavelength m Frequency s 1 Hertz Amplitude m Calcula1ng Frequency from propaga1ng waves 10 water waves in 30 s Frequency 10 30 s 1 0 3333 s 1 0 3333 Hz Speed of wave distance travelled 1me elapsed speed m s 1 m 1 s Electromagne1c waves Maxwell 1865 E x t Emax cos 2 x t Light travels in vacuum c 3 0x 108 m s 1 Fig 4 2 p 143 The electromagne1c spectrum Fig 4 3 p 143 Laser light White light monochroma1c dispersed into colors Fig 4 5 p 144 Wave par1cle duality photons have momentum and yet has no mass E mc2 E h c E hc energy of a photon must be equivalent to a quan1ty of mass Einstein rela1vity momentum p mv h Superposi1on of waves Construc1ve destruc1ve Note If waves are slightly di erent frequencies bea1ng panerns emerge Fig 4 6 p 144 Superposi1on of monochroma1c waves Di rac1on Panerns Superposi1on of monochroma1c waves Di rac1on Panerns hnp www daviddarling info images interference water waves jpg Fe atoms on Cu STM of electron standing waves Don Eigler IBM Quan za on of energy Maner is quan1zed into atoms Dalton 1802 Energy is quan1zed Planck 1900 For an electromagne1c wave of a given frequency he postulated that the minimum energy that such a wave could carry was equal to h Implies a minimum unit of light called a photon One photon equals one quanta of light Plank was commissioned by electric companies to create maximum light from light bulbs with minimum energy and worked on the physics of incandescent lights Black Body An object that is a perfect eminer and a perfect absorber of radia1on object does not have to appear black Sun and earth s surface behave approximately as black bodies Black body calibra1on source for pyrometers Spectral intensity distribu1on at T 3000 6000 K 0 Kelvin 273 150C Ultraviolet Catastrophe Intensity diverges at high frequencies kB R NA For single par1cles Boltzmann constant kB 1 3806503 10 23 m2 kg s 2 K 1 Experimental data The harmonic oscillator quan1zed Planck proposes quan1za1on of energy to mathema1cally explain the ultraviolet catastrophe Loved music played piano and organ Used a harmonic oscillator as star1ng point Classical Harmonic oscillator F k x x0 K spring constant N m 1 V x k x x0 2 Equa1ons assume x0 0 Force F Posi1on x Frequency f Poten1al Energy U K spring constant N m 1 Classical and quan1zed approaches E kBT E h mm Quan1zed model Planck s constant h fundimental to many equa1ons T 1646 K Fit of theory to experimental data gives value of h Linking Planck s equa1on to Classical model At high temperatures discovery of energy quanta Max Planck was awarded the 1918 Nobel Prize for Physics in recogni1on of the services he rendered to the advancement of Physics by his discovery of energy quanta Atomic spectra Transi1ons between discrete states Ne Ar Hg Electrical discharge tubes Fig 4 10 p 150 Spectrum of Emission and Absorp1on of light Fig 4 11 p 150 EMISSION Fig 4 11a p 150 ABSORPTION Fig 4 11b p 150 Atomic emission from H Hg and Ne Atoms absorb and emit at discrete frequencies Fig 4 12 p 151 1885 J J Balmer H atoms n 3 4 5 red green blue Empirical t Franck Gustav Hertz Experiments on energy levels in atoms Looked at energy lost to atoms by electrons eVthreshold indicates excita1on of rst excited state in atom also con rmed with light emission threshold At higher voltages saw new emission lines of light p 152 Example Hg Neils Bohr 1913 Bohr Model Proposed light is emined or absorbed from transi1ons between discrete states To avoid ve frequencies used E 0 emission E 0 absorp1on The Bohr Model Bohr s atomic model was introduced in 1913 The model s key success lay in explaining the Rydberg formula used in atomic physics p 152 We start using classical mechanics Total E poten1al plus kine1c energy Kine1cally and electrosta1cally E is The associated forces are balaned momentum Classical approach predicts electron losses energy and crashes into nucleus p 154 Postulate angulat momemntum is quan1zed mev r n h 2 V r unknown The Bohr radius a0 0 529 rn 2 n Z a 0 Predicts radius increases as the square of n Decreases with 1 Z Fig 4 14 p 155 levels En vn n h 2 me rn Z e2 2 0nh En Z2 e4 me 8 02 n2 h2 E 2 18 x 10 18 J Z2 n2 n Rydberg units Ioniza1on energy transi1on from n 1 to n in nity IE E nal Einit x NA 1 31 x 106 J mol 1 1310 kJ mol 1 Agrees with experimental IE for Hydrogen Theory explains the energies of the series of emission lines 1926 replaced by QM Fig 4 15 p 157 Interpreta1on of atomic spectra nf ni nf Ei Ef h Wave Par1cle duality Photoelectric e ect Di rac1on Xray di rac1on electron di rac1on And how electrons are also waves De Broglie s equa1ons beauty The photoelectric e ect Electrons are ejected by incident photons of su cient energy Fig 4 16 p 158 Threshold in h to generate photoelectrons not h intensity Fig 4 17 p 158 Einstein photoelectric e ect Fix h in this case Emax eVmax Adjust voltage on collector to retard electrons measure curren phi is called the work func1on like IE for a metal p 159 Einstein s photoelectric e ect phi is called the work func on like IE for a metal Fig 4 18 p 159 Louis Victor Pierre Raymond 7th duc 2 Broglie FRS mc h The fundamental idea of my 1924 thesis was the following The fact that following Einstein s introduc1on of photons in light waves one knew that light contains par1cles which are concentra1ons of energy incorporated into the wave suggests that all par1cles like the electron must be transported by a wave into which it is incorporated My essen1al idea was …