de Broglie atom harmonics Fig 4 20 p 161 Quiz IE EA and electronega8vity Ionic bonding coulomb stabiliza8on Covalent and polar covalent boding Lewis dots VSEPR steric number geometries Dipole moments Wavelength frequency and energy De Broglie Bohr de Broglie implica8ons od the electron as a wave as well as a par8cle Di rac8on Typical laMce spacing in picometers pm 10 12 m micro 6 n nano 9 p pico 12 f femto 15 1914 X ray Di rac8on 8 keV photon 0 154 nm 154 pm 20 keV photon 0 062 nm 62 pm 1 488 keV photon 0 833 nm 830 pm p 163 Calculate the wavelength of electron and see if its in range of laMce spacing h p E mv 2 p2 2m p 2m eV h 2 me eV Electron energy ev electron wavelength nm Electron wavelength in pm 20 0 274 274 50 0 173 173 100 0 1226 122 6 200 0 0867 86 7 Fig 4 21 p 163 Low energy electron di rac8on wavelets interfering Davisson and Germer 1927 Path A B C D E G E F a laMce spacing For construc8ve interference E F n a sin n a sin a sin n a a Di rac8on in 2 dimensions laMce with distances a and b at right angles interfere to form spots na a sin a nb b sin b Fig 4 22 p 164 LOW ENERGY ELECTRON DIFFRACTION LEED Primary electron beam in range of 20 200 eV Fig 4 23 p 164 LEED pajern of Si 111 7x7 Pajern is in reciprocal or k space Electrons behave as par8cles and waves so do photons and He atoms STM real space image Fig 4 24 p 165 1927 The Heisenberg uncertainty principle Uncertainty Indeterminacy Principle h p Wave is spread out so posi8on is indeterminate no de nite value General feature of QM Photon does not change the trajectory of the baseball Photon does change the electron trajectory Fig 4 25 p 166 Lets look at the par8cle in a box the simple way rst Schr dinger wave equa8ons Schr dinger expressed Broglie s hypothesis concerning the wave behavior of majer in a mathema8cal form Associated wave equa8ons with probability of par8cle loca8on Wave func8on squared describe par8cle probability Classical wave equa8ons relate second deriva8ve of the amplitude wrt distance to to the second deriva8ve wrt 8me x both sides by Schr dinger s equa8on External elds V x Rela8onship of second deriva8ve of wavefun8on and energy Lets look at the par8cle in a box the Schr dinger way Total probability 1 between x 0 L Normaliza8on condi8on is par8cle in box Schrodinger equa8on rearranged is wavefunc8on is At x 0 the amplitude is zero so sin 0 0 cos 0 1 so Second boundary condi8on Take this so rearrange and For each n Now n 1 2 3 4 n 0 would give zero probability To determine A total probability from x 0 to L 1 Energy level diagram together with some of the wavefunc8ons and their associated probability densi8es zero point energy En is eigen value of H nodes 0 1 2 non zero E Fig 4 27c p 175 As n increases it approaches a con8nuum classical mechanics Measuring a quantum system changes its 8me evolu8on means that the experimenter is now completely coupled to the quantum system In classical mechanics this coupling does not exist A classical system will evolve according to Newton s laws of mo8on independent of whether or not we observe it This is not true for quantum systems The very act of observing the system changes how it evolves in 8me Put another way by simply observing a system we change it Fig 4 28 p 175 Light emission form STM equivalent of par8cle in box 6nm long in Pt atoms Fig 4 27 p 175 3 dimensional par8cle in box brie y cubic box of equal lengths in x y z here states can be degenerate equal energy 3 quantum numbers Degenerate states g degeneracy Fig 4 29 p 177 Light emission and absorp8on Wavefunc8ons in 2 dimensions Representa8on of wavefunc8ons in 2 dimensions 11 ground state nx 1 ny 1 21 rst excited state Nodal line at x 0 5 22 Fig 4 30 p 179 123 3 dimensional Wavefunc8on As contour plot at speci c cuts 123 123 Fig 4 31 p 181 Fig 4 31a p 181 Fig 4 31b p 181 Fig 4 31c p 181 123 isosurface 123 0 8 and 0 2 Fig 4 32 p 181 222 contours at z 0 75 Isosurfaces 123 0 98 and 0 3 Its not possible to visualize the wavefunc8on in 3D Fig 4 33 p 182 p 187