Chapter 5 wed class Fig 5 CO p 193 Chapter 5 Quantum Mechanics and Atomic structure The hydrogen atom n is the principal quantum number Polar spherical coordinates r Schrodinger soluDon for 1 e atom IdenDcal to Bohr theory Fig 5 1 p 195 More quantum numbers Schr dinger also square of quanDzes angular momentum L2 and Lz the projecDon of L in z axis l is the angular quantum number l 0 to n 1 angular momentum projecDon in z is called magne c quantum number m l 0 l Doing the quantum numbers there are n2 sets for now n 1 l 0 m 0 n 2 l 1 0 m 1 0 1 for l 1 and m 0 for l 0 n 3 l 2 1 0 m 2 1 0 1 2 for l 2 etc For a given n with single electron atom states are degenerate L 0 1 2 3 are assigned leWers s p d f sharp principal di use and fundamental from pre QM spectral characterisDcs Naming quantum states n 1 s state 1s n 2 l 0 m 0 2s state l 1 m 1 0 1 gives three 2p states n 3 3s three 3p states and ve 3 p states Learn how to do this Table 5 1 p 196 Energy level diagram of H atom from QM At each n there are n2 degenerate quantum states the states have values of l 0 1 n and m l 0 l En proportional to 1 n2 Fig 5 2 p 197 For each n l m soluDon there is a wavefuncDon spherical harmonics Wave funcDon is Radial part angular part Probability of locaDng an electron at a point is Fig 5 3 p 198 THE DIFFERENTIAL VOLUME IN SPHERICAL POLAR COORDINATES 0 to 2 0 to radians Wave funcDon of one electron atom is called an orbital Not Bohr orbital but short for when electron has quantum numbers n l m the probability density to nd it at is Orbitals are called 1s 2s 2p 3s 3p 3d Radial wave functions are in dimensionless units Zr a0 a0 Bohr radius 0 529 10 10m 529 pm Z 1 1 H 1s Table 5 2 p 199 Representation of H 1s orbital on X Y plane as 3 D plot and contours 100 r 10 30 50 70 90 Fig 5 4a p 200 Representation of H 1s orbital as distance from nucleus spherically symettric 100 r Fig 5 4a p 200 Probability density for H1s orbital R100 r 2 vs r 3 d representation R 4 2 a0 encloses 99 of total density Fig 5 6 p 201 orbital 1s radial nodes 0 2s 1 3s 2 Electron probability density 90 Radial wave function vs r n 1 radial nodes Electron probability vs r 2 S orbitals penetrate nucleus Fig 5 7 p 201 Radial probability density Radial probability density 2 r2 r2 Rn0 r 2 dr Probability of nding an electron within a shell of thickness dr at distance r over all angles and 0 to 2 0 to Contour plots wavefunction and Radial probability density 2 r2 Fig 5 8 p 203 1s 2s 3s orbitals The s orbitals have a nite probability to be at nucleus Consequence of no angular momentum l 0 Size increases with n They have n 1 radial nodes What s the size of an orbital Its in nite since wave funcDon extends to in nity Take a balloon skin at 90 of the probability density 1s 141 pm 2s 483 pm 3s 1029 pm P orbitals l 1 Y10 l 1 m 0 is called pz orbital Ypz proporDonal to cos Has nodal plane or angular node Wave funcDon 0 and changes sign Electron probability 0 at nodal plane P orbitals l 1 Y11 l 1 m 1 1 2 orbitals no so simple similar form but run along x and y axis nodes on y z and x z planes Radial wave funcDons Rnl for p orbitals radial probability densiDes r2 Rnl2 Fig 5 10 p 205 Anatomy of pz Contour plot for amplitude of pz orbital in X Z plane Phase Contours flatten and closer spaced Rapid decrease in amplitude near nucleus Nodal plane Fig 5 11 p 206 Radial wave funcDons Rnl for p orbitals radial probability densiDes Rnl2 r2 R21 has no radial nodes R31 has 1 radial node R41 has 2 radial nodes Rnl n l 1 radial nodes But has one nodal plane Total nodes n 1 R21 0 at nucleus Except ns wave funcDon i e l 0 np nd nf wave funcDons have zero probability to be at the nucleus Electrons with angular momentum l 0 move around it Fig 5 10 p 205 Isosurfaces at 0 2 of the maximum amplitude Blue is nodal plane 2pz Fig 5 12a p 206 Fig 5 12b p 206 Fig 5 12c p 206 2px 2py 2pz Three orthogonal orbitals Isosurfaces at 0 2 of the maximum amplitude Light blue is nodal planes Fig 5 12 p 206 d orbitals l 2 has 5 projecDons of angular momentum along z axis m 2 1 0 1 2 gives 4 orbitals with same shape but di erent cartesian orientaDons dxy dyz dxz and dx2 dy2 and one m 0 dz2 d orbital facts 5 orbitals 2 angular nodes 2 nodal planes Rn2 r has n 3 radial nodes TOTAL NODES n 3 2 n 1 Fig 5 13 p 207 Fig 5 13a p 207 Fig 5 13b p 207 Fig 5 13c p 207 Fig 5 13d p 207 Fig 5 13e p 207 Fig 5 13f p 207 summary d 1s Average value of distance of electron from nucleus is via QM 2s 2p 3s First term is nth Bohr orbit radial probability densities Rnl2 r2 3p 3d Fig 5 14 p 208