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# class43 - Announcement Final exam This Saturday May 1st...

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Announcement Final exam: This Saturday, May 1 st , 7:30pm – 10pm in this room (DUAN G1B20) Allowed materials: Three letter paper-sized, handwritten cheat-sheets and a calculator. (No cell-phones, PCs, textbooks, BBQs etc.) Exam format: Similar to Exam 2 and 3 (all bubble; emphasis on understanding concepts) Current score distribution Class average: 80 Standard deviation: 11 The Hydrogen atom b 3D-Schrödinger with V(r) = ke 2 /r 2 2 2 2 2 2 2 2 1 sin 1 1 ψ θ + + = r ) , , ( ) , , ( ) , , ( 2 2 2 2 φ r E r r ke r m = + h with: ) ( ) ( ) ( ) , , ( g f r R r = h / iEt e ) ( ) ( ) ( ) , , , ( g f r R t r = Ψ sin sin r r r r r Use separation of variables: Once we have ψ (x) we also know the time-dependent solution: Done! Infinite Square Well: (1D) • V(x) = 0 if 0<x<L otherwise H Atom: (3D) • V(r) = -ke 2 /r Energy eigenstates: r 0 L x Energy eigenstates: • Wave functions: • Wave functions: 2 2 2 2 2 mL n E n h π = h / ) ( ) , ( t iE n n n e x t x = Ψ ) sin( ) ( 2 L x n L n x = ) ( ) ( ) ( ) , , ( m lm nl nlm g f r R r = h / ) , , ( ) , , , ( t iE nlm nlm n e r t r = Ψ 2 2 4 2 2 n e mk E n h = : generalized Laguerre polynomials : spherical harmonic functions What do the wave functions look like? ) ( ) ( ) ( ) , , ( m lm nl nlm g f r R r = l (restricted to 0, 1, 2 ± n-1) m (restricted to – l to l ) n = 1, 2, 3, e n=1 Much harder to draw in 3D than 1D. Indicate amplitude of ψ with brightness. Increasing n: more nodes in radial direction s ( l =0) p ( l =1) d ( l =2) See simulation: falstad.com/qmatom m = -l . . +l changes angular distribution n=2 n=3 Increasing l: less nodes in radial direction; More nodes in azimuthal direction Shapes of hydrogen wave functions: ) ( ) ( ) ( ) , , ( m lm nl nlm g f r R r = Look at s-orbitals (l=0) : no angular dependence n=1 n=2

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n=1 l =0 n=2 Higher n b average r bigger b more spherical shells stacked within each other b more nodes as function of r Probability finding electron at specific radius r (Note: This is not just | Ψ | 2 But rather 4 π r 2 | Ψ | 2 ) 1s 2s l =0 n=3 l =0 Radius (units of Bohr radius, a 0 ) a B 0.05nm 3s Shapes of hydrogen wave functions: ) ( ) ( ) ( ) , , ( φ θ ψ m lm nl nlm g f r R r = l =1, called p-orbitals : angular dependence (n>1) l=1, m=0: p z = dumbbell shaped. l=1, m=-1: bagel shaped around z-axis (traveling wave) l=1, m=+1 = p x =superposition (addition of m=-1 and m=+1) p y =superposition (subtraction of m=-1 and m=+1) Superposition applies: Dumbbells (chemistry) = → = = = = → = = = π i a r a r e e a
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class43 - Announcement Final exam This Saturday May 1st...

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