VIEWS: 100 PAGES: 3 POSTED ON: 4/29/2010
The End 4 Quantum Numbers for the • All homework solutions are now available Hydrogen Atom • Check CULearn for all scores (HW, Clickers, Exams) For hydrogenic atoms (one electron), energy levels only depend on n and we find the same formula as Bohr: En = −Z 2 ER / n2 – Any requested changes are due today For multielectron atoms the energy also depends on ℓ. • Final exam is Tuesday, December 15, 2009 There is a shorthand for giving the n and ℓ values. 4:30 pm – 7:00 pm in G125 (this room) 2p Different letters correspond to different values of ℓ • Bring Calculator! Arrive on time, check seating rows. s p d f g h… n=2 ℓ=1 • Additional review help (in the Physics Help Room), l= 0 1 2 3 4 5 today Friday 1:30 – 4:00 pm The electrons also have another quantum number for • I have to travel to Brookhaven National Lab next week their spin (ms = +/- ½). and so the exam will be proctored. Clicker question Hydrogen energy levels n = 1, 2, 3, … = Principal Quantum Number En = −Z ER / n 2 2 ℓ=0 ℓ=1 ℓ=2 ℓ = 0, 1, 2, … n-1 = angular momentum quantum number (s) (p) (d) = s, p, d, f, … L = l(l + 1)h n=3 E3 = − ER / 32 = −1.5 eV 3s 3p 3d m = 0, ±1, ±2, … ±ℓ is the z-component of Lz = mh angular momentum n=2 E2 = − ER / 22 = −3.4 eV 2s 2p ms = ±1/2 from the spin contribution A hydrogen atom electron is excited to an energy of −13.6/9 eV. How many different quantum states could the electron be in? E = −13.6/9 eV means n2 = 9 so n = 3 A. 1 B. 4 For n = 3, ℓ = 0,1,2 and thus C. 9 l = 0, m = 0 x 2 for spin! D. 18 l = 1, m = -1, 0, +1 n=1 1s E1 = − ER = −13.6 eV E. more than 18 l = 2, m = -2, -1, 0, +1, +2 Probability versus radius: P(r) = |Rnl(r)|2r 2 Higher n average r bigger more spherical shells stacked within each other In spherical coordinates, more nodes as function of r the volume element has an r2 term so probability n=1 increases with r2. l=0 Probability finding Most probable radius for electron as function of r the n = 1 state is at the n=2 Bohr radius aB l=0 Most probable radius for all ℓ=n-1 states (those with only n=3 one peak) is at the radius l=0 predicted by Bohr (n2 aB). 0.05nm Radius (units of Bohr radius, a0) 1 Shapes of hydrogen wave functions: Shapes of hydrogen wave functions: ψ nlm (r ,θ , φ ) = Rnl (r ) f lm (θ ) g m (φ ) ψ nlm (r ,θ , φ ) = Rnl (r ) f lm (θ ) g m (φ ) l=1, called p-orbitals: angular dependence (n=2) Look at s-orbitals (l=0): no angular dependence l=1, m=0: pz = dumbbell shaped. n=1 n=2 l=1, m=-1: bagel shaped around z-axis (traveling wave) l=1, m=+1 1 r − r / 2 a0 ⎛ 3 ⎞ n = 2, l = 1, m = 0 ⎯ ψ 211 = ⎯→ e ⎜− cos θ ⎟ 3 ⎜ 4π ⎟ 2 6 a0 a 0 ⎝ ⎠ 1 r − r / 2 a0 ⎛ 3 ⎞ w/time dependence n = 2, l = 1, m = 1 ⎯ ψ 211 = ⎯→ e ⎜− sin θe iφ ⎟ eimφ+iΕt/h 3 ⎜ 8π ⎟ 2 6a a 00 ⎝ ⎠ Superposition applies: px=superposition (addition of m=-1 and m=+1) Dumbbells py=superposition (subtraction of m=-1 and m=+1) (chemistry) Schrodinger finds quantization of energy and angular momentum: Physics vs Chemistry view of orbits: n=1, 2, 3 … l=0, 1, 2, 3 (restricted to 0, 1, 2 … n-1) 2p wave functions Dumbbell Orbits En = − E1 / n 2 | L |= l (l + 1) h (Physics view) (chemistry) (n=2, l=1) How does Schrodinger compare to what Bohr thought? same I. The energy of the ground state solution is ________ II. The angular momentum of the ground state solution is different _______ px III. The location of the electron is different _______ pz py m=1 m=-1 m=0 px=superposition a. same, same, same (addition of m=-1 and m=+1) b. same, same, different Bohr got energy right, py=superposition c. same, different, different but he said angular (subtraction of m=-1 and m=+1) d. different, same, different momentum L=nh, and e. different, different, different thought the electron was a point particle orbiting around nucleus. Can Schrodinger make sense of the periodic table? Schrödinger’s Cat A radioactive sample has a 50% chance of emitting an alpha particle. If it decays, a Geiger detector triggers the release of poison killing a cat in the box. Before opening the box, the cat is in a superposition of wave functions: ψ = 1ψ + 1ψ 2 dead 2 alive When does the wave function collapse to either dead or alive? No clear agreement. Interesting physics/philosophical question. Schrodinger’s solution for multi-electron atoms Need to account for all the interactions among the electrons Must solve for all electrons at once! (use matrices) V (for q1) = kqnucleus*q1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + …. 2 Good luck on the final exam! Thank you for all your attention and hard work this semester. 3