1. Tues. Dec. 8, 2009Physics 208, Lecture 271 Last Time… 3-dimensional quantum states and wave functions Decreasing particle size Quantum dots (particle in box) Course evaluations Thursday, Dec. 10 in class Last HW assignment available : for practice Covers material of last several lectures in course.

2. Tues. Dec. 8, 2009Physics 208, Lecture 272 3-D particle in box: summary Three quantum numbers (n x,n y,n z ) label each state n x,y,z =1, 2, 3 … (integers starting at 1) Each state has different motion in x, y, z Quantum numbers determine Momentum in each direction: e.g. Energy: Some quantum states have same energy

3. Tues. Dec. 8, 2009Physics 208, Lecture 273 3-dimensional Hydrogen atom Bohr model: Considered only circular orbits Found 1 quantum number n Energy, orbit radius From 3-D particle in box, expect that H atom should have more quantum numbers Expect different types of motion w/ same energy

4. Tues. Dec. 8, 2009Physics 208, Lecture 274 Modified Bohr model Different orbit shapes Big angular momentum Small angular momentum These orbits have same energy, but different angular momenta: Rank the angular momenta from largest to smallest: A B C a)A, B, C b)C, B, A c)B, C, A d)B, A, C e)C, A, B

5. Tues. Dec. 8, 2009Physics 208, Lecture 275 Angular momentum is quantized: orbital quantum number ℓ Angular momentum quantized, ℓ is the orbital quantum number For a particular n, ℓ has values 0, 1, 2, … n-1 ℓ=0, most elliptical ℓ=n-1, most circular For hydrogen atom, all have same energy

6. Tues. Dec. 8, 2009Physics 208, Lecture 276 Orbital magnetic dipole moment Angular momentum L Perpendicular to orbital plane Orbiting electron produces a current loop Current loop produces magnetic dipole moment  along L I

7. Tues. Dec. 8, 2009Physics 208, Lecture 277 3D Hydrogen atom so far… Each orbit has Same energy: Different orbit shape (angular momentum): Different magnetic moment:

8. Tues. Dec. 8, 2009Physics 208, Lecture 278 Orbital mag. quantum number m ℓ Directions of ‘orbital bar magnet’ quantized. Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values Determines z-component of L: Also determines angle of L For example: ℓ=1 gives 3 states:

9. Tues. Dec. 8, 2009Physics 208, Lecture 279 Question For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there? A. 1 B. 2 C. 3 D. 4 E. 5

10. Tues. Dec. 8, 2009Physics 208, Lecture 2710 Summary of quantum numbers n : describes energy of orbit ℓ describes the magnitude of orbital angular momentum m ℓ describes the angle of the orbital angular momentum For hydrogen atom:

11. Tues. Dec. 8, 2009Physics 208, Lecture 2711 Hydrogen wavefunctions Radial probability Angular not shown For given n, probability peaks at ~ same place Idea of “atomic shell” Notation: s: ℓ=0 p:ℓ=1 d:ℓ=2 f:ℓ=3 g:ℓ=4

12. Tues. Dec. 8, 2009Physics 208, Lecture 2712 Full hydrogen wave functions: Surface of constant probability Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability 1s-state

13. Tues. Dec. 8, 2009Physics 208, Lecture 2713 n=2: next highest energy 2s-state 2p-state Same energy, but different probabilities

14. Tues. Dec. 8, 2009Physics 208, Lecture 2714 3s-state 3p-state n=3: two s-states, six p-states and…

15. Tues. Dec. 8, 2009Physics 208, Lecture 2715 …ten d-states 3d-state

16. Tues. Dec. 8, 2009Physics 208, Lecture 2716 Electron spin New electron property: Electron acts like a bar magnet with N and S pole. Magnetic moment fixed… …but 2 possible orientations of magnet: up and down Spin down N S Described by spin quantum number m s N S Spin up z-component of spin angular momentum

17. Tues. Dec. 8, 2009Physics 208, Lecture 2717 Include spin Quantum state specified by four quantum numbers: Three spatial quantum numbers (3-dimensional) One spin quantum number

18. Tues. Dec. 8, 2009Physics 208, Lecture 2718 Quantum Number Question How many different quantum states exist with n=2? A. 1 B. 2 C. 4 D. 8 ℓ = 0 : m l = 0: m s = 1/2, -1/2 2 states ℓ = 1 : m l = +1: m s = 1/2, -1/2 2 states m l = 0: m s = 1/2, -1/22 states m l = -1: m s = 1/2, -1/2 2 states 2s 2 2p 6 There are a total of 8 states with n=2

19. Tues. Dec. 8, 2009Physics 208, Lecture 2719 Question How many different quantum states are in a 5g (n=5, ℓ =4) sub-shell of an atom? A. 22 B. 20 C. 18 D. 16 E. 14 ℓ =4, so 2(2 ℓ +1)=18. In detail, m l = -4, -3, -2, -1, 0, 1, 2, 3, 4 and m s =+1/2 or -1/2 for each. 18 available quantum states for electrons

20. Tues. Dec. 8, 2009Physics 208, Lecture 2720 Putting electrons on atom Electrons obey Pauli exclusion principle Only one electron per quantum state (n, ℓ, m ℓ, m s ) Hydrogen: 1 electron one quantum state occupied occupied unoccupied n=1 states Helium: 2 electrons two quantum states occupied n=1 states

21. Tues. Dec. 8, 2009Physics 208, Lecture 2721 Other elements: Li has 3 electrons n=1 states, 2 total, 2 occupied one spin up, one spin down n=2 states, 8 total, 1 occupied

22. Tues. Dec. 8, 2009Physics 208, Lecture 2722 Multi-electron atoms Electrons interact with nucleus (like hydrogen) Also with other electrons Causes energy to depend on ℓ Energy depends only on n Energy depends on n and ℓ 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d States fill in order of energy:

23. Tues. Dec. 8, 2009Physics 208, Lecture 2723 Atom Configuration H1s 1 He1s 2 Li1s 2 2s 1 Be1s 2 2s 2 B1s 2 2s 2 2p 1 Ne1s 2 2s 2 2p 6 1s shell filled 2s shell filled 2p shell filled etc (n=1 shell filled - noble gas) (n=2 shell filled - noble gas) Electron Configurations

24. Tues. Dec. 8, 2009Physics 208, Lecture 2724 The periodic table Atoms in same column have ‘similar’ chemical properties. Quantum mechanical explanation: similar ‘outer’ electron configurations. Be 2s 2 Li 2s 1 N2p3N2p3 C2p2C2p2 B2p1B2p1 Ne 2p 6 F2p5F2p5 O2p4O2p4 Mg 3s 2 Na 3s 1 P3p3P3p3 Si 3p 2 Al 3p 1 Ar 3p 6 Cl 3p 5 S3p4S3p4 H1s1H1s1 He 1s 2 CaKAs 4p 3 Ge 4p 2 Ga 4p 1 Kr 4p 6 Br 4p 5 Se 4p 4 Sc 3d 1 Y3d2Y3d2 8 more transition metals Ca 4s 2 K4s1K4s1 Na 3s 1

25. Tues. Dec. 8, 2009Physics 208, Lecture 2725 Excited states of Sodium Na level structure 11 electrons Ne core = 1s 2 2s 2 2p 6 (closed shell) 1 electron outside closed shell Na = [Ne]3s 1 Outside (11 th ) electron easily excited to other states.

26. Tues. Dec. 8, 2009Physics 208, Lecture 2726 Optical spectrum Optical spectrum of sodium Transitions from high to low energy states Relatively simple 1 electron outside closed shell Na 589 nm, 3p -> 3s