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Last Time… Bohr model of Hydrogen atom Wave properties of matter Energy levels from wave properties Hydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each described by a quantum number n Zero energy n=4 n=3 E3 13.6 eV 32 n=2 E2 13.6 eV 22 E1 13.6 eV 12 n=1 13.6 E n 2 eV n Thu. Nov. 29 2007 Energy Physics 208, Lecture 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 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. Classical: any momentum, energy is possible. Quantum: momenta, energy are quantized Thu. Nov. 29 2007 Physics 208, Lecture 25 3 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 4 Quantum version Quantum state is both velocities at the same time 2L One halfwavelength L momentum h h p 2L 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) : 2 2 x sin x L Thu. Nov. 29 2007 Quantum wave function: superposition of both motions. Physics 208, Lecture 25 5 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. 2 2 sin x Wavefunction: x L 2L One halfwavelength L Two halfwavelengths Thu. Nov. 29 2007 momentum h h p po 2L n=1 n=2 Physics 208, Lecture 25 momentum h h p 2 po L 6 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 7 Particle in box energy levels Quantized momentum h h p n npo 2L Energy = kinetic 2 2 npo p E n2Eo 2m 2m Or Quantized Energy Energy n=5 n=4 En n2Eo n=3 n=quantum number n=2 n=1 Thu. Nov. 29 2007 Physics 208, Lecture 25 8 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 9 Quantum dot: particle in 3D box CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Decreasing particle size Determined by energylevel spacing Energy level spacing increases as particle size decreases. i.e 2 E n 1 E n Thu. Nov. 29 2007 n 1 h 2 n 2 h 2 8mL2 8mL2 Physics 208, Lecture 25 10 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 Thu. Nov. 29 2007 Probability = (Wavefunction)2 Physics 208, Lecture 25 11 Higher energy wave functions L n p n=3 h 3 2L E Probability 2 h 32 8mL2 n=2 2 h 2L h2 2 8mL2 h 2L h2 8mL2 n=1 Wavefunction Thu. Nov. 29 2007 2 Physics 208, Lecture 25 12 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 13 Quantum Corral D. Eigler (IBM) 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 Scanning Tunneling Microscopy Tip 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 15 Particle in a box, again L Particle contained entirely within closed tube. Wavefunction Probability = (Wavefunction)2 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) Thu. Nov. 29 2007 Physics 208, Lecture 25 16 Quantum mechanics says something different! Low energy Classical state Low energy Quantum state Quantum Mechanics: some probability of the particle penetrating walls of box! Nonzero probability of being outside the box. Thu. Nov. 29 2007 Physics 208, Lecture 25 17 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 18 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 ‘high’ probability D. Decreases by >2 E. Stays same ‘low’ probability Thu. Nov. 29 2007 Physics 208, Lecture 25 19 Example: Ammonia molecule N H H H Thu. Nov. 29 2007 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) Physics 208, Lecture 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 A. slow down. B. speed up. C. stay the same. N H H H Thu. Nov. 29 2007 Physics 208, Lecture 25 21 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 22 Scanning Tunneling Microscopy 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 Surface steps on Si Images courtesy M. Lagally, Univ. Wisconsin Thu. Nov. 29 2007 Physics 208, Lecture 25 24 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 D. Eigler (IBM) 25 Quantum Corral D. Eigler (IBM) 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 The Stadium Corral D. Eigler (IBM) 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, Lecture 25 27 Some fun! Kanji for atom (lit. original child) Iron on copper (111) Thu. Nov. 29 2007 Carbon Monoxide man Carbon Monoxide on Pt (111) Physics 208, Lecture 25 D. Eigler (IBM) 28 Particle in box again: 2 dimensions Motion in x direction Motion in y direction Same velocity (energy), but details of motion are different. Thu. Nov. 29 2007 Physics 208, Lecture 25 29 Quantum Wave Functions Probability (2D) Wavefunction 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 (nx, ny) Ground state: (1,1) Probability = (Wavefunction)2 One-dimensional (1D) case Thu. Nov. 29 2007 Physics 208, Lecture 25 30 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 31 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 32 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) Thu. Nov. 29 2007 Physics 208, Lecture 25 33 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 34 Particle in 3D box Ground state surface of constant probability (nx, ny, nz)=(1,1,1) 2D case Thu. Nov. 29 2007 Physics 208, Lecture 25 35 (121) (112) (211) All these states have the same energy, but different probabilities Thu. Nov. 29 2007 Physics 208, Lecture 25 36 (222) (221) Thu. Nov. 29 2007 Physics 208, Lecture 25 37 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 13.6 E n 2 eV n • Orbital radius is Thu. Nov. 29 2007 rn n 2 ao Physics 208, Lecture 25 38 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! Thu. Nov. 29 2007 Physics 208, Lecture 25 39 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 40 Angular momentum question Which angular momentum is largest? Thu. Nov. 29 2007 Physics 208, Lecture 25 41 The orbital quantum number ℓ In quantum mechanics, the angular momentum can only have discrete values L ,1 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 42 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 43 Orbital magnetic dipole moment Can calculate dipole moment for circular orbit charge e ev Current = period 2r /v 2r Dipole moment µ=IA Area = r evr e mvr/ 2 2m B L / Thu. Nov. 29 2007 B In quantum mechanics, L B 2 e 0.927 1023 A m 2 2m 5.79 105 eV /Tesla 1 1 magnitude of orb. mag. dipole moment Physics 208, Lecture 25 44 Orbital mag. quantum number mℓ 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. ℓ=1 gives 3 states: Thu. Nov. 29 2007 m ℓ = +1 S N mℓ = 0 Physics 208, Lecture 25 m ℓ = -1 45 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 Thu. Nov. 29 2007 Physics 208, Lecture 25 46 Example: For ℓ=2, m ℓ = -2, -1, 0, +1, +2 corresponding to three different directions of orbital bar magnet. Thu. Nov. 29 2007 Physics 208, Lecture 25 47 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 13.6 E eV B Total energy is then 2 This means that spectral lines will split in a magnetic field Thu. Nov. 29 2007 n 13.6 2 eV z B n 13.6 2 eV m B B n Physics 208, Lecture 25 48 Thu. Nov. 29 2007 Physics 208, Lecture 25 49 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). Thu. Nov. 29 2007 Physics 208, Lecture 25 50 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 51 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 52 Electron spin orientations Spin up Spin down Only two possible orientations Thu. Nov. 29 2007 Physics 208, Lecture 25 53 Thu. Nov. 29 2007 Physics 208, Lecture 25 54 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 55 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: n, , m , ms An atom with several electrons filling quantum states starting with the lowest energy, filling quantum states until electrons are used. Thu. Nov. 29 2007 Physics 208, Lecture 25 56 Quantum Number Question How many different quantum states exist with n=2? 1 2 4 8 l = 0 : 2s2 ml = 0 : ms = 1/2 , -1/2 2 states l = 1 : 2p6 ml = +1: ms = 1/2 , -1/2 ml = 0: ms = 1/2 , -1/2 ml = -1: ms = 1/2 , -1/2 2 states 2 states 2 states There are a total of 8 states with n=2 Thu. Nov. 29 2007 Physics 208, Lecture 25 57 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,ml,ms). Atoms with many electrons have many atomic orbitals filled. Chemical properties are determined by the configuration of the ‘outer’ electrons. Thu. Nov. 29 2007 Physics 208, Lecture 25 58 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, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4 and ms=+1/2 or -1/2 for each. 18 available quantum states for electrons 17 will fit. And there is room left for 1 more ! Thu. Nov. 29 2007 Physics 208, Lecture 25 59 Putting electrons on atom Electrons are obey exclusion principle Only one electron per quantum state unoccupied occupied n=1 states Hydrogen: 1 electron one quantum state occupied Helium: 2 electrons n=1 states two quantum states occupied Thu. Nov. 29 2007 Physics 208, Lecture 25 60 Other elements: Li has 3 electrons n 2 0 m 0 1 ms 2 n 2 0 m 0 1 ms 2 n 2 1 m 0 1 ms 2 n 2 1 m 0 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n 2 1 m 1 1 ms 2 n=2 states, 8 total, 1 occupied n=1 states, 2 total, 2 occupied n 1 n 1 0 0 m 0 m 0 1/2 m s 2007 1/2 m sThu. 29 Nov. one spin up, one spin down Physics 208, Lecture 25 61 Electron Configurations Atom Configuration H 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 Ne Thu. Nov. 29 2007 etc 1s shell filled 1s22s22p6 (n=1 shell filled noble gas) 2s shell filled 2p shell filled Physics 208, Lecture 25 (n=2 shell filled noble gas) 62 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! H 1s1 Li 2s1 Na 3s1 Be 2s2 Mg 3s2 Thu. Nov. 29 2007 B 2p1 Al 3p1 C 2p2 Si 3p2 Physics 208, Lecture 25 N 2p3 P 3p3 O 2p4 S 3p4 H 1s1 F 2p5 Cl 3p5 He 1s2 Ne 2p6 Ar 3p6 63 Wavefunctions and probability Probability of finding an electron is given by the square of the wavefunction. Probability large here Probability small here Thu. Nov. 29 2007 Physics 208, Lecture 25 64 Hydrogen atom: Lowest energy (ground) state 1s-state n 1, Thu. Nov. 29 2007 Spherically symmetric. Probability decreases exponentially with radius. Shown here is a surface of constant probability 0, m 0 Physics 208, Lecture 25 65 n=2: next highest energy 2s-state 2p-state n 2, 0, m 0 n 2, 1, m 0 2p-state n 2, 1, m 1 Same energy, but different probabilities Thu. Nov. 29 2007 25 Physics 208, Lecture 66 n=3: two s-states, six p-states and… 3p-state 3s-state 3p-state n 3, 0, m 0 Thu. Nov. 29 2007 n 3, 1, m 0 Physics 208, Lecture 25 n 3, 1, m 1 67 …ten d-states 3d-state 3d-state n 3, 3d-state 2, m 0 Thu. Nov. 29 2007 n 3, 2, m 1 Physics 208, Lecture 25 n 3, 2, m 2 68 Electron wave around an atom Wave representing electron Electron wave extends around circumference of orbit. Only integer number of wavelengths around orbit allowed. Thu. Nov. 29 2007 Wave representing electron Physics 208, Lecture 25 69 Emitting and absorbing light Zero energy n=4 n=3 13.6 E 3 2 eV 3 n=2 13.6 E 2 2 eV 2 Photon emitted hf=E2-E1 n=1 E3 13.6 eV 32 n=2 E2 13.6 eV 22 E1 13.6 eV 12 Photon absorbed hf=E2-E1 E1 13.6 eV 12 Photon is emitted when electron drops fromone quantum state to another Thu. Nov. 29 2007 n=4 n=3 n=1 Absorbing a photon of correct energy makeselectron jump to higher quantum state. Physics 208, Lecture 25 70 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. Thu. Nov. 29 2007 Physics 208, Lecture 25 71 Silicon Thu. Nov. 29 2007 Physics 208, Lecture 25 7x7 surface reconstruction These 10 nm scans show the individual atomic positions 72 Particle in box wavefunction x dx Prob. Of finding particle in region dx about x 2 x L ? Particle is never here x=0 x=L Particle is never here x 0 ? Thu. Nov. 29 2007 Physics 208, Lecture 25 73 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) Thu. Nov. 29 2007 Physics 208, Lecture 25 74