Friday onors lecture Prof. Clint Sprott takes us on a tour of fractals. ydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n E n = " 13.6 n 2 ev n=4 Zero energy E 3 = " 13.6 3 2 ev E 2 = " 13.6 2 2 ev E 1 = " 13.6 1 2 ev Energy Thu. Nov. 29 2007 Physics 208, ecture 25 2 Quantum Particle in a box Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length Classical vs Quantum Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right Classically, ball bounces back and forth in tube. This is a classical state of the ball. Identify each state by speed, =(mass)x(speed), or kinetic energy. Classical: any, energy is possible. Quantum: momenta, energy are quantized Quantum mechanics: Particle represented by wave: p = mv = h / λ Different motions: waves traveling left and right Quantum wave function: superposition of both at same time Thu. Nov. 29 2007 Physics 208, ecture 25 3 Thu. Nov. 29 2007 Physics 208, ecture 25 4 Quantum version Quantum state is both velocities at the same time " = 2 One halfwavelength Ground state is a standing wave, made equally of Wave traveling right ( p = +h/λ ) Wave traveling left ( p = - h/λ ) Determined by standing wave condition =n(λ/2) : p = h " = h 2 Different quantum states p = mv = h / λ Different speeds correspond to different λ subject to standing wave condition integer number of half-wavelengths fit in the tube. " = 2 One halfwavelength "( x) = 2 sin % ' 2# & $ x ( Wavefunction: * ) p = h " = h 2 # p o "( x) = 2 sin % ' 2# & $ x ( * ) Quantum wave function: superposition of both motions. " = Two halfwavelengths p = h " = h = 2p o Thu. Nov. 29 2007 Physics 208, ecture 25 5 Thu. Nov. 29 2007 Physics 208, ecture 25 6 1
Particle in box question Particle in box energy levels A particle in a box has a mass m. Its energy is all kinetic = p 2 /2m. Just saw that 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. Quantized p = h " = n h 2 = np o Energy = kinetic ( ) 2 E = p2 2m = np o 2m Or Quantized Energy = n 2 E o Energy n=5 n=4 E n = n 2 E o n=quantum number Thu. Nov. 29 2007 Physics 208, ecture 25 7 Thu. Nov. 29 2007 Physics 208, ecture 25 8 Question A particle is in a particular quantum state in a box of length. The box is now squeezed to a shorter length, /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 Thu. Nov. 29 2007 Physics 208, ecture 25 9 Quantum dot: particle in 3D box Decreasing particle size Energy level spacing increases as particle size decreases. i.e E n +1 " E n = ( n +1)2 h 2 " n 2 h 2 8m 2 8m 2 CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Determined by energylevel spacing Thu. Nov. 29 2007 Physics 208, ecture 25 10 Interpreting the wavefunction igher energy wave functions Probabilistic interpretation The square magnitude of the wavefunction Ψ 2 gives the probability of finding the particle at a particular spatial location n p E 3 h 2 3 2 h 2 8m 2 Wavefunction Probability Wavefunction Probability = (Wavefunction) 2 2 h 2 2 2 h 2 8m 2 h 2 h 2 8m 2 Thu. Nov. 29 2007 Physics 208, ecture 25 11 Thu. Nov. 29 2007 Physics 208, ecture 25 12 2
Probability of finding electron Quantum Corral Classically, equally likely to find particle anywhere QM - true on average for high n Zeroes in the probability! Purely quantum, interference effect Thu. Nov. 29 2007 Physics 208, ecture 25 13 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). Thu. Nov. 29 2007 Physics 208, ecture 25 14 Scanning Tunneling Microscopy Tip Particle in a box, again Wavefunction Probability = (Wavefunction) 2 Particle contained entirely within closed tube. Sample Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Open top: particle can escape if we shake hard enough. But at low energies, particle stays entirely within box. ike an electron in metal (remember photoelectric effect) Thu. Nov. 29 2007 Physics 208, ecture 25 15 Thu. Nov. 29 2007 Physics 208, ecture 25 16 Quantum mechanics says something different! ow energy Classical state ow energy Quantum state Quantum Mechanics: some probability of the particle penetrating walls of box! 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. Nonzero probability of being outside the box. Thu. Nov. 29 2007 Physics 208, ecture 25 17 Thu. Nov. 29 2007 Physics 208, ecture 25 18 3
Question 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 high probability Example: Ammonia molecule Ammonia molecule: N 3 Nitrogen (N) has two equivalent stable positions. Quantum-mechanically tunnels 2.4x10 11 times per second (24 Gz) N Known as inversion line Basis of first atomic clock (1949) low probability Thu. Nov. 29 2007 Physics 208, ecture 25 19 Thu. Nov. 29 2007 Physics 208, ecture 25 20 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 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. A. slow down. B. speed up. C. stay the same. N Thu. Nov. 29 2007 Physics 208, ecture 25 21 Thu. Nov. 29 2007 Physics 208, ecture 25 22 Scanning Tunneling Microscopy Tip Tip, sample are quantum boxes Potential difference induces tunneling Tunneling extremely sensitive to tip-sample spacing Surface steps on Si Sample Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Images courtesy M. agally, Univ. Wisconsin Thu. Nov. 29 2007 Physics 208, ecture 25 23 Thu. Nov. 29 2007 Physics 208, ecture 25 24 4
Manipulation of atoms Take advantage of tip-atom interactions to physically move atoms around on the surface Quantum Corral 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 Thu. Nov. 29 2007 Physics 208, ecture 25 25 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). Thu. Nov. 29 2007 Physics 208, ecture 25 26 The Stadium Corral Some fun! 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. Thu. Nov. 29 2007 Physics 208, ecture 25 27 Kanji for atom (lit. original child) Iron on copper (111) Carbon Monoxide man Carbon Monoxide on Pt (111) Thu. Nov. 29 2007 Physics 208, ecture 25 28 5