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Learn about the Bohr model of the hydrogen atom, wave properties of matter, and the quantized energy levels in a particle-in-a-box system. Understand the probabilistic interpretation of wave functions and the phenomenon of quantum tunneling.

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

  2. Hydrogen atom energies Zero energy n=4 n=3 n=2 Energy n=1 • Quantized energy levels: • Each corresponds to different • Orbit radius • Velocity • Particle wavefunction • Energy • Each described by a quantum number n Physics 208, Lecture 25

  3. Quantum ‘Particle in a box’ L Particle confined to a fixed region of spacee.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 Physics 208, Lecture 25

  4. Classical vs Quantum L • Classical: particle bounces back and forth. • Sometimes velocity is to left, sometimes to right • 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 Physics 208, Lecture 25

  5. Quantum version momentum L One half-wavelength • 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) : Quantum wave function: superposition of both motions. Physics 208, Lecture 25

  6. Different quantum states momentum Two half-wavelengths momentum One half-wavelength • p = mv = h /  • Different speeds correspond to different  • subject to standing wave condition integer number of half-wavelengths fit in the tube. Wavefunction: n=1 n=2 Physics 208, Lecture 25

  7. Particle in box question A particle in a box has a mass m. Its energy is all kinetic = p2/2m. Just saw that momentum in state n is npo. It’s energy levels A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. Physics 208, Lecture 25

  8. Particle in box energy levels n=5 Energy n=4 n=3 n=2 n=1 • Quantized momentum • Energy = kinetic • Or Quantized Energy n=quantum number Physics 208, Lecture 25

  9. 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 Physics 208, Lecture 25

  10. Quantum dot: particle in 3D box Decreasing particle size CdSe quantum dots dispersed in hexane(Bawendi group, MIT) Color from photon absorption Determined by energy-level spacing • Energy level spacing increases as particle size decreases. • i.e Physics 208, Lecture 25

  11. Interpreting the wavefunction • Probabilistic interpretation The square magnitude of the wavefunction ||2 gives the probability of finding the particle at a particular spatial location Wavefunction Probability = (Wavefunction)2 Physics 208, Lecture 25

  12. Higher energy wave functions L n=1 n p E Wavefunction Probability n=3 n=2 Physics 208, Lecture 25

  13. 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 Physics 208, Lecture 25

  14. 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) Physics 208, Lecture 25

  15. 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 Physics 208, Lecture 25

  16. Particle in a box, again Wavefunction Probability = (Wavefunction)2 L 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) Physics 208, Lecture 25

  17. 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. Physics 208, Lecture 25

  18. 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. Physics 208, Lecture 25

  19. Question ‘high’ probability ‘low’ probability Suppose separation between boxes increases by a factor of two. The tunneling probability Increases by 2 Decreases by 2 Decreases by <2 Decreases by >2 Stays same Physics 208, Lecture 25

  20. Example: Ammonia molecule N H H H • Ammonia molecule: NH3 • Nitrogen (N) has two equivalent ‘stable’ positions. • Quantum-mechanically tunnels 2.4x1011times per second (24 GHz) • Known as ‘inversion line’ • Basis of first ‘atomic’ clock (1949) Physics 208, Lecture 25

  21. Atomic clock question N H H H 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. Physics 208, Lecture 25

  22. 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. Physics 208, Lecture 25

  23. Scanning Tunneling Microscopy Tip, sample are quantum ‘boxes’ Potential difference induces tunneling Tunneling extremely sensitive to tip-sample spacing • Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. • Is a very useful microscope! Tip Sample Physics 208, Lecture 25

  24. Surface steps on Si Images courtesy M. Lagally, Univ. Wisconsin Physics 208, Lecture 25

  25. Manipulation of atoms • Take advantage of tip-atom interactions to physically move atoms around on the surface • 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 D. Eigler (IBM) Physics 208, Lecture 25

  26. 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) Physics 208, Lecture 25

  27. 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) Physics 208, Lecture 25

  28. Some fun! Kanji for atom (lit. original child) Iron on copper (111) Carbon Monoxide man Carbon Monoxide on Pt (111) Physics 208, Lecture 25 D. Eigler (IBM)

  29. Particle in box again: 2 dimensions Same velocity (energy), but details of motion are different. Motion in y direction Motion in x direction Physics 208, Lecture 25

  30. Quantum Wave Functions Probability(2D) Ground state: same wavelength (longest) in both x and y Need two quantum #’s,one for x-motionone for y-motion Use a pair (nx, ny) Ground state: (1,1) Wavefunction Probability = (Wavefunction)2 One-dimensional (1D) case Physics 208, Lecture 25

  31. 2D excited states (nx, ny) = (2,1) (nx, ny) = (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. Physics 208, Lecture 25

  32. Particle in a box What quantum state could this be? A. nx=2, ny=2 B. nx=3, ny=2 C. nx=1, ny=2 Physics 208, Lecture 25

  33. 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. (nx, ny) = (2,2) Physics 208, Lecture 25

  34. Three dimensions • Object can have different velocity (hence wavelength) in x, y, or z directions. • Need three quantum numbers to label state • (nx, ny , nz) 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. Physics 208, Lecture 25

  35. Particle in 3D box • Ground state surface of constant probability • (nx, ny, nz)=(1,1,1) 2D case Physics 208, Lecture 25

  36. (121) (112) (211) All these states have the same energy, but different probabilities Physics 208, Lecture 25

  37. (222) (221) Physics 208, Lecture 25

  38. 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 Physics 208, Lecture 25

  39. 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! Physics 208, Lecture 25

  40. Sommerfeld: modified Bohr model Big angular momentum Small angular momentum • Differently shaped orbits All these orbits have same energy…… but different angular momenta Energy is same as Bohr atom, but angular momentum quantization altered Physics 208, Lecture 25

  41. Angular momentum question • Which angular momentum is largest? Physics 208, Lecture 25

  42. 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 Physics 208, Lecture 25

  43. Orbital mag. moment Orbital magnetic moment electron Current • 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 Physics 208, Lecture 25

  44. Orbital magnetic dipole moment Current = Area = In quantum mechanics, magnitude of orb. mag. dipole moment Can calculate dipole moment for circular orbit Dipole moment µ=IA Physics 208, Lecture 25

  45. Orbital mag. quantum number mℓ N S S N S N mℓ = +1 mℓ = -1 mℓ = 0 • Possible directions of the ‘orbital bar magnet’ are quantized just like everything else! • Orbital magnetic quantum number • mℓ ranges from - ℓ, to ℓ in integer steps • Number of different directions = 2ℓ+1 Example: For ℓ=1, mℓ = -1, 0, or -1, corresponding to three different directionsof orbital bar magnet. ℓ=1 gives 3 states: Physics 208, Lecture 25

  46. Question • For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there?A. 1B. 2C. 3D. 4E. 5 Physics 208, Lecture 25

  47. Example: For ℓ=2, mℓ = -2, -1, 0, +1, +2 corresponding to three different directionsof orbital bar magnet. Physics 208, Lecture 25

  48. 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 splitin a magnetic field Physics 208, Lecture 25

  49. Physics 208, Lecture 25

  50. Summary of quantum numbers • n describes the energy of the orbit • ℓ describes themagnitude ofangular momentum • mℓ describes the behavior in a magnetic field due to the magnetic dipole moment produced by orbital motion (Zeeman effect). Physics 208, Lecture 25

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