1. 10 February 2010Modern Physics II Lecture 51 University of San Francisco Modern Physics for Frommies II The Universe of Schrödinger’s Cat Lecture 5

2. 10 February 2010Modern Physics II Lecture 52 Agenda Administrative Matters The Hydrogen Atom in Wave Mechanics Schrödinger’s Equation in Spherical Coordinates The Angular Solution, Spherical Harmonics The Radial Solution Quantum Numbers and Probability Distribution Atoms with Multiple Electrons Quantum Statistics The Pauli Exclusion Principle and the Periodic Table

3. 10 February 2010Modern Physics II Lecture 53 First Physics and Astronomy Colloquium: Today, Wednesday, 10 February 2010 at 4 PM Professor Richard Muller, Department of Physics, UC Berkeley Refreshments at 3:30 PM Harney Science Center Room 127 Administrative Matters

4. 10 February 2010Modern Physics II Lecture 54

5. 10 February 2010Modern Physics II Lecture 55 Simple Harmonic Oscillator Some systems may be approximated by a quantum mechanical 1-D oscillator, e.g. a vibrating diatomic molecule. Any system in a potential minimum behaves approximately like a SHO.

6. 10 February 2010Modern Physics II Lecture 56 Difficult to solve directly so try some guesses.

7. 10 February 2010Modern Physics II Lecture 57 This solution corresponds to the ground state of the oscillator. Note: again we have a zero-point energy

8. 10 February 2010Modern Physics II Lecture 58 Ground state is n=0 not n=1 and  zero-point energy

9. 10 February 2010Modern Physics II Lecture 59 Classically forbidden

10. 10 February 2010Modern Physics II Lecture 510

11. 10 February 2010Modern Physics II Lecture 511 Note penetration of  into classically forbidden regions

12. 10 February 2010Modern Physics II Lecture 512 Quantum Mechanical View of Atoms Bohr model discarded as an accurate description of nature Certain aspects have however been retained e.g. Electrons in an atom exist only in discrete states of definite energy, the stationary states Transitions between these states require the emission (or absorption of a photon. According to wave mechanics, electrons do not travel in well defined circular orbits ala Bohr. The electron, because of its wave nature, is better thought of as spread out in space as a “cloud”. The size and shape of the electron cloud can be found by solving the Schrödinger equation for the atom and forming the probability distribution, |  | 2.

13. 10 February 2010Modern Physics II Lecture 513 Ground state of hydrogen

14. 10 February 2010Modern Physics II Lecture 514 Schrödinger’s Equation in Spherical Coordinates x y zIn Cartesian coordinates

15. 10 February 2010Modern Physics II Lecture 515 Want to do the same thing with spherical symmetry

16. 10 February 2010Modern Physics II Lecture 516 Spherical Time Independent Schrödinger Equation

17. 10 February 2010Modern Physics II Lecture 517

18. 10 February 2010Modern Physics II Lecture 518

19. 10 February 2010Modern Physics II Lecture 519 Quantum Numbers Recall last week’s discussion of the 1-D and 3-D potential well or rigid box. The solutions were characterized by a single quantum number (n) in the 1-D case and by three numbers (n x, n y and n z ) in 3-D. These quantum numbers arise from the imposition of boundary conditions on the solutions. We might expect that in the 3-D problem of the hydrogen atom the solutions will be characterized by numbers corresponding to Boundary conditions applied in 3-D. Restrictions on the values of these quantum numbers arise from the mathematics of the spherical harmonics. Actually, we need a fourth number. There is an additional degree of freedom which I will treat in a few minutes.

20. 10 February 2010Modern Physics II Lecture 520 Results from boundary conditions on solution of the R part of the separated Schrödinger eqn. Bohr result R part contains the potential energy n alone determines the energy levels (actually there is a slight deviation from this) Consequence of central inverse square force.

21. 10 February 2010Modern Physics II Lecture 521 The semiclassical planetary model with electrons in orbits is not a good one

22. 10 February 2010Modern Physics II Lecture 522 Note: All these transitions have  l =  1

23. 10 February 2010Modern Physics II Lecture 523 spdfghetc. l=012345etc. Notation for states: nl, e.g. 4d is n=4, l=2

24. 10 February 2010Modern Physics II Lecture 524 Aside on Angular Momentum r v  Particle of mass m moving with circular speed v around an axis at radius r.

25. 10 February 2010Modern Physics II Lecture 525 A C  To the plane of AB Right Hand Rule Direction of advance of a right hand screw

26. 10 February 2010Modern Physics II Lecture 526 Note that the vector product is not commutative Again look at Right Hand Rule ‛ A‛ A C A × B B × A

27. 10 February 2010Modern Physics II Lecture 527 Space quantization Note choice of z axis is arbitrary.

28. 10 February 2010Modern Physics II Lecture 528 Energy is dependent solely on n. Presence of multiple less and malls for a given n  states are degenerate This degeneracy is removed if directional symmetry is broken by say a B or E field. What about L x and L y ?

29. 10 February 2010Modern Physics II Lecture 529

30. 10 February 2010Modern Physics II Lecture 530 x y z LyLy LxLx Liz L

31. 10 February 2010Modern Physics II Lecture 531 Magnetic effects Normal Zeeman effect: Transition between 1s and 2p Spectral lines broaden and split into 3 lines as B is applied and increased. 3 lines = “normal Zeeman effect” Consider the electron “orbit” to be a current loop with  IA

32. 10 February 2010Modern Physics II Lecture 532 Apply external magnetic field

33. 10 February 2010Modern Physics II Lecture 533 Each degenerate energy level, l, is split into 2l+1 separate energy levels, m l. B has specified a direction in space (z axis) and the symmetry responsible for the degeneracy has been broken.

34. 10 February 2010Modern Physics II Lecture 534 The Stern-Gerlach experiment: If B is inhomogeneous there will be a net force as well as torque on the atom

35. 10 February 2010Modern Physics II Lecture 535 For l  0 the states should separate according to m l 2 lines seen instead of the expected 3 (or 2l+1 = odd) Haven’t seen the whole picture yet.

36. 10 February 2010Modern Physics II Lecture 536 Electron Spin Wolfgang Pauli: Relativity  besides n, l, m l need 4 th quantum number G. Hollenbeck and S. Goudsmit: Propose intrinsic “spin” angular momentum for the electron s = ½ħ Another magnetic quantum number: m s =  ½ 1928, P. A. M. Dirac justifies this from relativity.

37. 10 February 2010Modern Physics II Lecture 537 Gives magnetic effects like orbital angular momentum. Intrinsic spin → intrinsic magnetic dipole moment New magnetic quantum numbers: m s = ± 1/2 Doubles number of states for a given n States are degenerate unless a spatial direction is specified, e.g. external E or B field Quantum state now specified by {n, l, m l, m s }

38. 10 February 2010Modern Physics II Lecture 538 Return to the Stern-Gerlach experiment l = 0 state will give 2 lines for m s = ± 1/2 Fine structure: Even in the absence of external fields, very high resolution spectroscopy reveals splitting of spectral lines. Rest frame of electron: nucleus orbits and appears as a current loop. Interacts with spin magnetic moment and breaks degeneracy Line separation is about 5 x 10 -5 eV compared to the 2p  1s transition energy of 10.2 eV Hyperfine structure: Arises from the spin angular momentum and consequent spin magnetism of the nucleon(s)

39. 10 February 2010Modern Physics II Lecture 539 s-wave states are spherically symmetric, not so for l  0

40. 10 February 2010Modern Physics II Lecture 540 Quantum Statistics Consider a system of 2 particles, say electrons

41. 10 February 2010Modern Physics II Lecture 541 So, under interchange  2 possibilities  ±  If 2 identical particles interchange   = +  they are said to obey Bose-Einstein statistics and are called bosons. If 2 identical particles interchange   = - , they are said to obey Fermi-Dirac statistics and are called fermions. Bosons have integral spin, e.g. photons, mesons, some atoms and nuclei, ………

42. 10 February 2010Modern Physics II Lecture 542 Fermions have ½ integral spin, e.g. leptons, nucleons, some atoms and nuclei,……….. For fermions   Cannot have 2 identical particles with the same set of quantum numbers. Pauli Exclusion Principle You can stick as many bosons into a quantum state as you want.

43. 10 February 2010Modern Physics II Lecture 543 Electrons are fermions. Build some elements. As electrons are added the exclusion principle will have an effect. Hydrogen: 1 e in the 1s state1s 1 He: 2 e in the 1s state, m s =1 and -1, 1s 2 No more e can be added to the 1s state without violating the exclusion principle ! The K shell is filled

44. 10 February 2010Modern Physics II Lecture 544 Li: 3 rd e has to go in 2s state 2s 1 Be: 4 th e in the 2s state, m s =1 and -1, 2s 2 2s state (subshell) is now filled B: 5 th e has to go in the 2p state2p 1 p state has 2l+1 = 3 values of m l, each with 2 values of m s, accommodating C, N, O, F and Ne as 2p 2 – 2p 6. 2p subshell is now filled, as is the L shell Na: 11 th e has to go in 3s state3s 1 etc.

45. 10 February 2010Modern Physics II Lecture 545 3s and 3p each with 2l+1 m l values each having 2 values of m s. Weirdoes: 3d, 4d and 5d subshells fill up the transition metals followed by the lanthanides and the actinides Complicated inter electron interactions mess things up If electrons were bosons, they would all sit in the ground state, 1s, and chemistry would be very different.

46. 10 February 2010Modern Physics II Lecture 546