1. Last Time… Bohr model of Hydrogen atom Wave properties of matter Energy levels from wave properties

2. Thu. Nov. 29 2007Physics 208, Lecture 252 Hydrogen atom energies Zero energy n=1 n=2 n=3 n=4 Energy Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n

3. Thu. Nov. 29 2007Physics 208, Lecture 253 Quantum ‘Particle in a box’ Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Classically, ball bounces back and forth in tube. This is a ‘classical state’ of the ball. Identify each state by speed, momentum=(mass)x(speed), or kinetic energy. Classical: any momentum, energy is possible. Quantum: momenta, energy are quantized L

4. Thu. Nov. 29 2007Physics 208, Lecture 254 Classical vs Quantum Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right L Quantum mechanics: Particle represented by wave: p = mv = h / Different motions: waves traveling left and right Quantum wave function: superposition of both at same time

5. Thu. Nov. 29 2007Physics 208, Lecture 255 Quantum version Quantum state is both velocities at the same time Ground state is a standing wave, made equally of Wave traveling right ( p = +h/ ) Wave traveling left ( p = - h/ ) Determined by standing wave condition L=n( /2) : One half- wavelength momentum L Quantum wave function: superposition of both motions.

6. Thu. Nov. 29 2007Physics 208, Lecture 256 Different quantum states p = mv = h / Different speeds correspond to different subject to standing wave condition integer number of half-wavelengths fit in the tube. Two half- wavelengths momentum One half- wavelength momentum n=1 n=2 Wavefunction:

7. Thu. Nov. 29 2007Physics 208, Lecture 257 Particle in box question A particle in a box has a mass m. Its energy is all kinetic = p 2 /2m. Just saw that momentum in state n is np o. It’s energy levels A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy.

8. Thu. Nov. 29 2007Physics 208, Lecture 258 Particle in box energy levels Quantized momentum Energy = kinetic Or Quantized Energy Energy n=1 n=2 n=3 n=4 n=5 n=quantum number

9. Thu. Nov. 29 2007Physics 208, Lecture 259 Question A particle is in a particular quantum state in a box of length L. The box is now squeezed to a shorter length, L/2. The particle remains in the same quantum state. The energy of the particle is now A. 2 times bigger B. 2 times smaller C. 4 times bigger D. 4 times smaller E. unchanged

10. Thu. Nov. 29 2007Physics 208, Lecture 2510 Quantum dot: particle in 3D box Energy level spacing increases as particle size decreases. i.e CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Determined by energy- level spacing Decreasing particle size

11. Thu. Nov. 29 2007Physics 208, Lecture 2511 Interpreting the wavefunction Probabilistic interpretation The square magnitude of the wavefunction |  | 2 gives the probability of finding the particle at a particular spatial location WavefunctionProbability = (Wavefunction) 2

12. Thu. Nov. 29 2007Physics 208, Lecture 2512 Higher energy wave functions n=1 n=2 n=3 WavefunctionProbability L npE

13. Thu. Nov. 29 2007Physics 208, Lecture 2513 Probability of finding electron Classically, equally likely to find particle anywhere QM - true on average for high n Zeroes in the probability! Purely quantum, interference effect

14. Thu. Nov. 29 2007Physics 208, Lecture 2514 Quantum Corral 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). D. Eigler (IBM)

15. Thu. Nov. 29 2007Physics 208, Lecture 2515 Scanning Tunneling Microscopy Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Tip Sample

16. Thu. Nov. 29 2007Physics 208, Lecture 2516 Particle in a box, again L Wavefunction Probability = (Wavefunction) 2 Particle contained entirely within closed tube. Open top: particle can escape if we shake hard enough. But at low energies, particle stays entirely within box. Like an electron in metal (remember photoelectric effect)

17. Thu. Nov. 29 2007Physics 208, Lecture 2517 Quantum mechanics says something different! Quantum Mechanics: some probability of the particle penetrating walls of box! Low energy Classical state Low energy Quantum state Nonzero probability of being outside the box.

18. Thu. Nov. 29 2007Physics 208, Lecture 2518 Two neighboring boxes When another box is brought nearby, the electron may disappear from one well, and appear in the other! The reverse then happens, and the electron oscillates back an forth, without ‘traversing’ the intervening distance.

19. Thu. Nov. 29 2007Physics 208, Lecture 2519 Question ‘low’ probability ‘high’ probability Suppose separation between boxes increases by a factor of two. The tunneling probability A.Increases by 2 B.Decreases by 2 C.Decreases by <2 D.Decreases by >2 E.Stays same

20. Thu. Nov. 29 2007Physics 208, Lecture 2520 Example: Ammonia molecule Ammonia molecule: NH 3 Nitrogen (N) has two equivalent ‘stable’ positions. Quantum-mechanically tunnels 2.4x10 11 times per second (24 GHz) Known as ‘inversion line’ Basis of first ‘atomic’ clock (1949) N H H H

21. Thu. Nov. 29 2007Physics 208, Lecture 2521 Atomic clock question Suppose we changed the ammonia molecule so that the distance between the two stable positions of the nitrogen atom INCREASED. The clock would A. slow down. B. speed up. C. stay the same. N H H H

22. Thu. Nov. 29 2007Physics 208, Lecture 2522 Tunneling between conductors Make one well deeper: particle tunnels, then stays in other well. Well made deeper by applying electric field. This is the principle of scanning tunneling microscope.

23. Thu. Nov. 29 2007Physics 208, Lecture 2523 Scanning Tunneling Microscopy Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Tip Sample Tip, sample are quantum ‘boxes’ Potential difference induces tunneling Tunneling extremely sensitive to tip-sample spacing

24. Thu. Nov. 29 2007Physics 208, Lecture 2524 Surface steps on Si Images courtesy M. Lagally, Univ. Wisconsin

25. Thu. Nov. 29 2007Physics 208, Lecture 2525 Manipulation of atoms Take advantage of tip-atom interactions to physically move atoms around on the surface D. Eigler (IBM) This shows the assembly of a circular ‘corral’ by moving individual Iron atoms on the surface of Copper (111). The (111) orientation supports an electron surface state which can be ‘trapped’ in the corral

26. Thu. Nov. 29 2007Physics 208, Lecture 2526 Quantum Corral 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). D. Eigler (IBM)

27. Thu. Nov. 29 2007Physics 208, Lecture 2527 The Stadium Corral Again Iron on copper. This was assembled to investigate quantum chaos. The electron wavefunction leaked out beyond the stadium too much to to observe expected effects. D. Eigler (IBM)

28. Thu. Nov. 29 2007Physics 208, Lecture 2528 Some fun! Kanji for atom (lit. original child) Iron on copper (111) D. Eigler (IBM) Carbon Monoxide man Carbon Monoxide on Pt (111)

29. Thu. Nov. 29 2007Physics 208, Lecture 2529 Particle in box again: 2 dimensions Same velocity (energy), but details of motion are different. Motion in x direction Motion in y direction

30. Thu. Nov. 29 2007Physics 208, Lecture 2530 Quantum Wave Functions Ground state: same wavelength (longest) in both x and y Need two quantum #’s, one for x-motion one for y-motion Use a pair (n x, n y ) Ground state: (1,1) Probability (2D) Wavefunction Probability = (Wavefunction) 2 One-dimensional (1D) case

31. Thu. Nov. 29 2007Physics 208, Lecture 2531 2D excited states (n x, n y ) = (2,1) (n x, n y ) = (1,2) These have exactly the same energy, but the probabilities look different. The different states correspond to ball bouncing in x or in y direction.

32. Thu. Nov. 29 2007Physics 208, Lecture 2532 Particle in a box What quantum state could this be? A. n x =2, n y =2 B. n x =3, n y =2 C. n x =1, n y =2

33. Thu. Nov. 29 2007Physics 208, Lecture 2533 Next higher energy state The ball now has same bouncing motion in both x and in y. This is higher energy that having motion only in x or only in y. (n x, n y ) = (2,2)

34. Thu. Nov. 29 2007Physics 208, Lecture 2534 Three dimensions Object can have different velocity (hence wavelength) in x, y, or z directions. Need three quantum numbers to label state (n x, n y, n z ) labels each quantum state (a triplet of integers) Each point in three-dimensional space has a probability associated with it. Not enough dimensions to plot probability But can plot a surface of constant probability.

35. Thu. Nov. 29 2007Physics 208, Lecture 2535 Particle in 3D box Ground state surface of constant probability (n x, n y, n z )=(1,1,1) 2D case

36. Thu. Nov. 29 2007Physics 208, Lecture 2536 (211) (121) (112) All these states have the same energy, but different probabilities

37. Thu. Nov. 29 2007Physics 208, Lecture 2537 (221) (222)

38. Thu. Nov. 29 2007Physics 208, Lecture 2538 The ‘principal’ quantum number In Bohr model of atom, n is the principal quantum number. Arise from considering circular orbits. Total energy given by principal quantum number Orbital radius is

39. Thu. Nov. 29 2007Physics 208, Lecture 2539 Other quantum numbers? Hydrogen atom is three-dimensional structure Should have three quantum numbers Special consideration: Coulomb potential is spherically symmetric x, y, z not as useful as r, ,  Angular momentum warning!

40. Thu. Nov. 29 2007Physics 208, Lecture 2540 Sommerfeld: modified Bohr model Differently shaped orbits Big angular momentum Small angular momentum All these orbits have same energy… … but different angular momenta Energy is same as Bohr atom, but angular momentum quantization altered

41. Thu. Nov. 29 2007Physics 208, Lecture 2541 Angular momentum question Which angular momentum is largest?

42. Thu. Nov. 29 2007Physics 208, Lecture 2542 The orbital quantum number ℓ In quantum mechanics, the angular momentum can only have discrete values, 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 These states all have the same energy

43. Thu. Nov. 29 2007Physics 208, Lecture 2543 Orbital mag. moment Since Electron has an electric charge, And is moving in an orbit around nucleus… … it produces a loop of current, and hence a magnetic dipole field, very much like a bar magnet or a compass needle. Directly related to angular momentum Current electron Orbital magnetic moment

44. Thu. Nov. 29 2007Physics 208, Lecture 2544 Orbital magnetic dipole moment Dipole moment µ=IA Current = Area = Can calculate dipole moment for circular orbit magnitude of orb. mag. dipole moment In quantum mechanics,

45. Thu. Nov. 29 2007Physics 208, Lecture 2545 Orbital mag. quantum number m ℓ Possible directions of the ‘orbital bar magnet’ are quantized just like everything else! Orbital magnetic quantum number m ℓ ranges from - ℓ, to ℓ in integer steps Number of different directions = 2ℓ+1 Example: For ℓ=1, m ℓ = -1, 0, or -1, corresponding to three different directions of orbital bar magnet. N S m ℓ = +1 S N m ℓ = -1 ℓ=1 gives 3 states: m ℓ = 0 SN

46. Thu. Nov. 29 2007Physics 208, Lecture 2546 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

47. Thu. Nov. 29 2007Physics 208, Lecture 2547 Example: For ℓ=2, m ℓ = -2, -1, 0, +1, +2 corresponding to three different directions of orbital bar magnet.

48. Thu. Nov. 29 2007Physics 208, Lecture 2548 Interaction with applied B-field Like a compass needle, it interacts with an external magnetic field depending on its direction. Low energy when aligned with field, high energy when anti-aligned Total energy is then This means that spectral lines will split in a magnetic field

49. Thu. Nov. 29 2007Physics 208, Lecture 2549

50. Thu. Nov. 29 2007Physics 208, Lecture 2550 Summary of quantum numbers n describes the energy of the orbit ℓ describes the magnitude of angular momentum m ℓ describes the behavior in a magnetic field due to the magnetic dipole moment produced by orbital motion (Zeeman effect).

51. Thu. Nov. 29 2007Physics 208, Lecture 2551 Additional electron properties Free electron, by itself in space, not only has a charge, but also acts like a bar magnet with a N and S pole. Since electron has charge, could explain this if the electron is spinning. Then resulting current loops would produce magnetic field just like a bar magnet. But… Electron in NOT spinning. As far as we know, electron is a point particle. N S

52. Thu. Nov. 29 2007Physics 208, Lecture 2552 Electron magnetic moment Why does it have a magnetic moment? It is a property of the electron in the same way that charge is a property. But there are some differences Magnetic moment has a size and a direction It’s size is intrinsic to the electron, but the direction is variable. The ‘bar magnet’ can point in different directions.

53. Thu. Nov. 29 2007Physics 208, Lecture 2553 Electron spin orientations Spin up Spin down N S N S Only two possible orientations

54. Thu. Nov. 29 2007Physics 208, Lecture 2554

55. Thu. Nov. 29 2007Physics 208, Lecture 2555 Spin: another quantum number There is a quantum # associated with this property of the electron. Even though the electron is not spinning, the magnitude of this property is the spin. The quantum numbers for the two states are +1/2 for the up-spin state -1/2 for the down-spin state The proton is also a spin 1/2 particle. The photon is a spin 1 particle.

56. Thu. Nov. 29 2007Physics 208, Lecture 2556 Include spin We labeled the states by their quantum numbers. One quantum number for each spatial dimension. Now there is an extra quantum number: spin. A quantum state is specified four quantum numbers: An atom with several electrons filling quantum states starting with the lowest energy, filling quantum states until electrons are used.

57. Thu. Nov. 29 2007Physics 208, Lecture 2557 Quantum Number Question How many different quantum states exist with n=2? 1 2 4 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

58. Thu. Nov. 29 2007Physics 208, Lecture 2558 Pauli Exclusion Principle Where do the electrons go? In an atom with many electrons, only one electron is allowed in each quantum state (n,l,m l,m s ). Atoms with many electrons have many atomic orbitals filled. Chemical properties are determined by the configuration of the ‘outer’ electrons.

59. Thu. Nov. 29 2007Physics 208, Lecture 2559 Number of electrons Which of the following is a possible number of electrons in a 5g (n=5, l=4) sub-shell of an atom? 22 20 17 l=4, so 2(2l+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 17 will fit. And there is room left for 1 more !

60. Thu. Nov. 29 2007Physics 208, Lecture 2560 Putting electrons on atom Electrons are obey exclusion principle Only one electron per quantum state Hydrogen: 1 electron one quantum state occupied occupied unoccupied n=1 states Helium: 2 electrons two quantum states occupied n=1 states

61. Thu. Nov. 29 2007Physics 208, Lecture 2561 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

62. Thu. Nov. 29 2007Physics 208, Lecture 2562 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

63. Thu. Nov. 29 2007Physics 208, Lecture 2563 The periodic table Elements are arranged in the periodic table so that atoms in the same column have ‘similar’ chemical properties. Quantum mechanics explains this by similar ‘outer’ electron configurations. If not for Pauli exclusion principle, all electrons would be in the 1s state! H1s1H1s1 Be 2s 2 Li 2s 1 N2p3N2p3 Mg 3s 2 Na 3s 1 C2p2C2p2 B2p1B2p1 Ne 2p 6 F2p5F2p5 O2p4O2p4 P3p3P3p3 Si 3p 2 Al 3p 1 Ar 3p 6 Cl 3p 5 S3p4S3p4 H1s1H1s1 He 1s 2

64. Thu. Nov. 29 2007Physics 208, Lecture 2564 Wavefunctions and probability Probability of finding an electron is given by the square of the wavefunction. Probability large here Probability small here

65. Thu. Nov. 29 2007Physics 208, Lecture 2565 Hydrogen atom: Lowest energy (ground) state Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability 1s-state

66. Thu. Nov. 29 2007Physics 208, Lecture 2566 n=2: next highest energy 2s-state 2p-state Same energy, but different probabilities

67. Thu. Nov. 29 2007Physics 208, Lecture 2567 3s-state 3p-state n=3: two s-states, six p-states and…

68. Thu. Nov. 29 2007Physics 208, Lecture 2568 …ten d-states 3d-state

69. Thu. Nov. 29 2007Physics 208, Lecture 2569 Wave representing electron Electron wave around an atom Wave representing electron Electron wave extends around circumference of orbit. Only integer number of wavelengths around orbit allowed.

70. Thu. Nov. 29 2007Physics 208, Lecture 2570 Emitting and absorbing light Photon is emitted when electron drops from one quantum state to another Zero energy n=1 n=2 n=3 n=4 n=1 n=2 n=3 n=4 Absorbing a photon of correct energy makes electron jump to higher quantum state. Photon absorbed hf=E 2 -E 1 Photon emitted hf=E 2 -E 1

71. Thu. Nov. 29 2007Physics 208, Lecture 2571 The wavefunction Wavefunction =  = |moving to right> + |moving to left> The wavefunction is an equal ‘superposition’ of the two states of precise momentum. When we measure the momentum (speed), we find one of these two possibilities. Because they are equally weighted, we measure them with equal probability.

72. Thu. Nov. 29 2007Physics 208, Lecture 2572 Silicon 7x7 surface reconstruction These 10 nm scans show the individual atomic positions

73. Thu. Nov. 29 2007Physics 208, Lecture 2573 Particle in box wavefunction x=0x=Lx=L Prob. Of finding particle in region dx about x Particle is never here

74. Thu. Nov. 29 2007Physics 208, Lecture 2574 Making a measurement Suppose you measure the speed (hence, momentum) of the quantum particle in a tube. How likely are you to measure the particle moving to the left? A. 0% (never) B. 33% (1/3 of the time) C. 50% (1/2 of the time)