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Hydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n Prof. Clint Sprott takes us on a tour of fractals. Zero energy n=4 n=3 n=2 E2 = " 13.6 eV 22 E1 = " 13.6 eV 12 ! 13.6 eV n2 Thu. Nov. 29 2007 13.6 eV 32 ! n=1 En = " E3 = " Energy Friday Honors lecture ! Physics 208, Lecture 25 2 ! Quantum ‘Particle in a box’ Classical vs Quantum Classical: particle bounces back and forth. Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Sometimes velocity is to left, sometimes to right L 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. Quantum mechanics: Classical: any momentum, energy is possible. Quantum: momenta, energy are quantized Particle represented by wave: p = mv = h / λ Different motions: waves traveling left and right Quantum wave function: Thu. Nov. 29 2007 Physics 208, Lecture 25 3 Quantum version Quantum state is both velocities at the same time " = 2L ! momentum h h p= = " 2L " = 2L Wave traveling right ( p = +h/λ ) ! Wave traveling left ( p = - h/λ ) Determined by standing wave condition L=n(λ/2) : "( x) = 2 % 2# ( sin' x* L & $ ) Thu. Nov. 29 2007 One halfwavelength ! Quantum wave function: superposition of both motions. Physics 208, Lecture 25 Different speeds correspond to different λ subject to standing wave condition integer number of half-wavelengths fit in the tube. Wavefunction: " ( x ) = Ground state is a standing wave, made equally of ! L 4 p = mv = h / λ One halfwavelength Physics 208, Lecture 25 Different quantum states superposition of both at same time Thu. Nov. 29 2007 "=L Two halfwavelengths 5 ! Thu. Nov. 29 2007 2 % 2# ( sin' x* L & $ ) momentum n=1 ! p= n=2 ! Physics 208, Lecture 25 h h = # po " 2L momentum h h p = = = 2 po " L 6 ! 1 Particle in box question Particle in box energy levels Quantized momentum p= A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. ! Energy = kinetic 2 p 2 ( npo ) E= = = n2Eo 2m 2m Or Quantized Energy ! Thu. Nov. 29 2007 Physics 208, Lecture 25 7 h h =n = npo " 2L n=5 Energy 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 npo. It’s energy levels n=4 En = n2Eo n=3 n=quantum number n=2 n=1 Thu. Nov. 29 2007 Physics 208, Lecture 25 8 ! Quantum dot: particle in 3D box Question CdSe quantum dots dispersed in hexane (Bawendi group, MIT) 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 Color from photon absorption Energy level spacing increases as particle size decreases. i.e E n +1 " E n = Thu. Nov. 29 2007 Determined by energylevel spacing Decreasing particle size A. 2 times bigger B. 2 times smaller C. 4 times bigger D. 4 times smaller E. unchanged Physics 208, Lecture 25 9 (n + 1) 2 8mL2 Thu. Nov. 29 2007 h2 " n 2h 2 8mL2 Physics 208, Lecture 25 10 ! Interpreting the wavefunction Higher 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 Wavefunction L n p n=3 h 3 2L E Probability h2 3 8mL2 2 Probability = (Wavefunction)2 ! n=2 ! 2 h 2L ! n=1 Thu. Nov. 29 2007 Wavefunction Physics 208, Lecture 25 11 ! h 2L Thu. Nov. 29 2007 ! 22 h2 8mL2 h2 8mL2 Physics 208, Lecture 25 12 ! 2 Probability of finding electron Quantum Corral Classically, equally likely to find particle anywhere QM - true on average for high n D. Eigler (IBM) Zeroes in the probability! Purely quantum, interference effect Thu. Nov. 29 2007 Physics 208, Lecture 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, Lecture 25 14 Particle in a box, again Scanning Tunneling Microscopy L Particle contained entirely within closed tube. Tip Wavefunction Open top: particle can escape if we shake hard enough. Sample But at low energies, particle stays entirely within box. Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Thu. Nov. 29 2007 Physics 208, Lecture 25 15 Like an electron in metal (remember photoelectric effect) Thu. Nov. 29 2007 Physics 208, Lecture 25 16 Two neighboring boxes Quantum mechanics says something different! Low energy Classical state Low energy Quantum state Probability = (Wavefunction) 2 Quantum Mechanics: some probability of the particle penetrating walls of box! 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, Lecture 25 17 Thu. Nov. 29 2007 Physics 208, Lecture 25 18 3 Question Suppose separation between boxes increases by a factor of two. The tunneling probability Example: Ammonia molecule N H A. Increases by 2 H H B. Decreases by 2 C. Decreases by <2 ‘high’ probability D. Decreases by >2 E. Stays same Ammonia molecule: NH3 Nitrogen (N) has two equivalent ‘stable’ positions. Quantum-mechanically tunnels 2.4x1011 times per second (24 GHz) Known as ‘inversion line’ Basis of first ‘atomic’ clock (1949) ‘low’ probability Thu. Nov. 29 2007 Physics 208, Lecture 25 19 Thu. Nov. 29 2007 20 Tunneling between conductors 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. Physics 208, Lecture 25 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. N H H H Thu. Nov. 29 2007 Physics 208, Lecture 25 21 Scanning Tunneling Microscopy Thu. Nov. 29 2007 Physics 208, Lecture 25 22 Surface steps on Si Tip, sample are quantum ‘boxes’ Tip Potential difference induces tunneling Tunneling extremely sensitive to tip-sample spacing Sample Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart. Is a very useful microscope! Thu. Nov. 29 2007 Physics 208, Lecture 25 23 Images courtesy M. Lagally, Univ. Wisconsin Thu. Nov. 29 2007 Physics 208, Lecture 25 24 4 Manipulation of atoms Quantum Corral 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 D. Eigler (IBM) D. Eigler (IBM) 25 The Stadium 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). Thu. Nov. 29 2007 Physics 208, Lecture 25 26 Some fun! D. Eigler (IBM) Again Iron on copper. This was assembled to investigate quantum chaos. Kanji for atom (lit. original child) Iron on copper (111) The electron wavefunction leaked out beyond the stadium too much to to observe expected effects. Thu. Nov. 29 2007 Physics 208, Lecture 25 27 Thu. Nov. 29 2007 Carbon Monoxide man Carbon Monoxide on Pt (111) Physics 208, Lecture 25 D. Eigler (IBM) 28 5