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by: Miss Alberto Prohaska


Miss Alberto Prohaska
GPA 3.94


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This 41 page Class Notes was uploaded by Miss Alberto Prohaska on Thursday October 22, 2015. The Class Notes belongs to MATRL 162A at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/226928/matrl-162a-university-of-california-santa-barbara in Material Science and Engineering at University of California Santa Barbara.

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Date Created: 10/22/15
ECE 162A Mat 162A Lecture 6 Stationary Solutions Read Chapter 56 of EisbergResnick John Bowers Bowerseceucsbedu Eigenvalue Equation Using 0 erators Schroedinver s e luation can be expressed as an eigenvalue equation where hz d2 019 de The solution of the equation involves finding the particular solutions w called eigenfunctions and En called eigenvalues Solutions to SE Free particle Step potential Infinite box Finite box Harmonic oscillator Free particle VO i W Em 2m dxz Solution tube 2 eXpikx 2sz where E 2m The complete solution is LIJCXquot wxe 1Elh eikZ iEhl There are no constraints on E any value is allowed at this point This corresponds to a wave moving to the right k solutions are also valid Step Potential f O Sketch solutions for EgtVO and EltVO Step potential im Vxlx Eelx 2m dxz Solution For E gt V0 For x lt 0 lC A eXpik1x B eXp ik1x For x gt 0 lC C eXpik2x DeXp ik2x fisz 2 E hzkj m 2m where E V0 Boundary conditions M96 gt 0 M96 gt 0 dlx gt 0 dyx gt 0 dx dx The wave is entering from the left Step Potential EgtVO F or x lt 0 Ax A eXpik1x B eXp ik1x F or x gt O wx C eXpik2x D eXp ik2x Wave om left D O szBC dwdx ik1A B 2 1 sz 1 k A E 1 fC 1 3211k2c 2 k1 Normalization C can be normalized if the density of electrons is known or the problem is limited by either A large box Periodic boundary conditions In general though what matters is the reflection coefficient R and transmission coefficient T 33 AA gtllt T C C AA Step Potential EltVO F or x lt 0 wx A expikx B eXp ikx F or x gt 0 wx C eXp Ioc D eXpIoc F iniie wave function D 0 szBC dwdx ikA B KC 1 iK A 1 C 2 k 1 iK B 1 C 2 k Sketc e golution Explain the difference between EgtVO and EltVO Problem Particle in an infinite bOX VOf0r0ltxlta V 00 otherwise Solutions wx Boundary Conditions w0 wa 0 Eigenvalues kL n 2122232 22 nh 2 8ma E n Square Well 2 2 2 2 h k E h K V0 E 2m 2m V0 For x lt 612 0 W00 A 111006 B 003kx a2 32 F or x lt a 2 Ax C eXpIoc D eXp Ioc Boundary condition D 0 F or x gt a 2 Ax F eXpIoc G eXp Ioc Boundary condition F 0 ECE 162A Mat 162A Lecture 15Spin ER Chapter 8 John Bowers Bowerseceucsbedu ECEMat 162A Accomplished Chapter 1 Thermal radiation Planck s postulate Chapter 2 Light Wavelike and particlelike photon Chapter 3 Matter Wavelike and particlelike Chapter 4 Thompson Rutherford Bohr Model of Atom Chapter 5 Schroedinger Theory Time dependent Time independent Schroedinger Equations Chapter 6 Solutions of Time Independent SE Free particle Step potential Barrier potential Infinite square well finite square well harmonic oscillator 3D Solutions Chapter 7 Hydrogen atom Quantum numbers and degeneracy Angular momentum Commutator Simultaneous eigenvalues 2D harmonic oscillator ECEMat 162A Left to do Chapter 8 Spin Numerical solutions Chapter 9 Exclusion principle periodic table Chapter 11 111 to 113 Quantum statistics Chapter 13 Free electrons in metals Bonds Periodic potentials Energy bands Semiconductors Band offsets Ternary Quaternary Quantum Wells ECEMat 162A Under what conditions ca two V Ve observable properties of a quantum systs an amuse s5swws fv a given quantum state ECEMat 162A If two operators commute then the eigenvalues associated with those operators are simultaneous eigenvalues If two operators do not commute then the eigenvalues associated with those two operators typically exhibit an uncertainty relation In general for every system one may identify at least one complete set of commuting observables ECEMat 162A Specific Case 2D Harmonic Oscillator VxyCx2y2EAIa2x2y2 hz 621 62W 1 2 2 2 Ma x E 2Max2 ayz 2 Mk V WOW f xgy h2 62 62 1 g 6 f ay gt Mw2ltx2y2gtfgEfg 2 2 2 2 h a1Mafx2 h af lezy2E 2Mf 6x 2 2Mf 6y 2 Constant Constant E h2 62f 1 2 2 Max E 2M 6x2 2 f xf h2 62g 1 Ma2 2 E 2M6y2 2 yg yg ExEyE ECEMat 162A 2D Harmonic Oscilla 0 x x 2 2 2 2 anny anane x W a E nx ny 1ha nx O12 my 2 O12 ECEMat 162A Are these solutions of ALZ Yes if I W L W A h 5 L2 8 We need to find linear combinations of degenerate solutions that satisfy the above equation Note Degenerate solutions solutions with the same energy do not change in time and are called stationary solutions ECEMat 162A Lowest energy solution n20 nxznyzO x2yz2at2 r22at2 6 quot h a 2 2 r 2a sz e 1 69 This is a solution of energy and LZ ECEMat 162A Solutions These are not solutions that satisfy izw LZ 91 2 ii i M ECEMat 162A N1 Solutions 2x 2 2 r a 11 1 nx l ny O pm 76 Note requot rcos irsin xiy re i rcos z rsin x z y So 2Xly 2 2 2 2 2 r a z r a W Wm HWm e e e a a 2X ly 2 2 2 2 2 r a 1 r a W Wm le e e e a a These are both solutions with LZ 1 and 1 respectively ECEMat162A Dirac Notation anny s represented by the Dirac ket vector This notation is a useful shorthand n1m1gt10gtz 01gt lnxny gt The projection of onto a pos ptos the wave function lt xy nxny gt wnxny ECEMat 162A Magnetic moments CLASSICALLY an electron moving in a loop produces a current 0 e l period dis tan 66 2727 perlod o veloczly v 8V 1 2727 ECEMat 162A Magnetic moments CLASSICALLY an electron moving in a loop produces a current 0 e l period dis tan ce 27zr perlod veloczly v 8V 1 27zr A current in a loop produces a magnetic dipole moment ev 2 evr u 2 current gtlt area 21A 2 7zr ECEMat162A 7Z7 2 Bohr Magneton Classically L 2 mm i 2 2m If Bohr Magneton lab E i 927 X1023Am2 2m ubL u h ECEMat 162A Bohr Magneton Classically L mrv evr eL 7 E If eh 23 2 Bohr Magneton U 2m 927gtlt10 Am L Ll i The correct quantum mechanical result is g is the orbital 7 factor and lb b 1 ggzz ECEMat 162A Dipole in a Magnetic N The effect of a magnetic field on a magnetic dipole is to exert a torque fz x The potential energy is lowest when the dipole is aligned with the magnetic moment AE2 f3 ECEMat 162A Uniform magnetic field precession but no translation Converging magnetic field translational force 53 F Z 52 2 ECEMat 162A Stern Gerlach Experiment Stern Gerlach Exp Pass a beam of silver atoms through a nonuniform magnetic field and record deflections Classical prediction Quantum mechanical prediction ECEMat 162A Stern Gerlach Experiment Stern Gerlach Exp Pass a beam of silver atoms through a nonuniform magnetic field and record deflections Classical prediction a range of deflections corresponding to uz ranging from u to p Quantum mechanical prediction discrete deflections corresponding to mI l0l Result 2 discrete components one positive one negative ECEMat162A PhippsTaylor 1927 Experimen Repeat Stern Gerlach Exp with hydrogen atoms in the ground state l0 m0 Quantum mechanical prediction No deflection corres ondin39 to m0 ECEMat 162A PhippsTaylor 1927 Experimen Repeat Stern Gerlach Exp with hydrogen atoms in the ground state l0 m0 Quantum mechanical prediction No deflection corres ondin39 to m0 Result 2 beams one deflected positive one negative Something is missing in the theory Size of deflection 2000x bigger than Bohr magneton for a proton The atom is not responsible for the deflection The electron is ECEMat 162A Electron Spin Goudsmit and Uhlenbeck 1925 grad students Explain fine splitting of Hydrogen lines by assuming the electron is a small spinning sphere with surface cnarge Le a magnetic moment Quantum numbers S 1 2 Intrinsic angular momentum S SS 1h The 2 component of spin is quantized S2 msh ms 1212 ECEMat 162A gs 2 gs is the spin g factor Spin Orbit Interaction The spin orbit interaction is the result of the electron spin magnetic moment and the internal magnetic field of the atom clue to the electrons angular momentum 1 dV a AE L 2m 0 r dr This is typically about 104 eV This splitting is called fine structure ECEMat 162A Total Angular Momentum 32 J Due to the spin orbit interaction L and S are not independent The spin orbit interaction causes a coupling between L and S and a precessmn abom tne axes The total angular momentum is fixed and quan zed The 2 component of total angular momentum is quan zed ECEMat 162A Total Angular Momentum J E 3 The total angular momentum in terms of quantum numberj J jj1h The 2 component of angular momentum is JZ mjh Where the quantum number m is m j0j J ECEMat 162A How doesj relate to l and s li l2 2lZ l Jzz1Jss12Jjj12Jzz1 Jss1 The result of this inequality is that can have two values jl12l 12 When l0 then there is only one value of j 12 S F S J L L I J ECEMat162A Example for mi 2 N 52 N a 4 i m I Q 32 a 324 m N N II N Q II N N m 12 12 f 12 12 32 32 52 ECEMat 162A Hydrogen spectra The spectra of hydrogen snow a c structure which is well explained by Schroedinger s equation with spin There is also a hyperfine structure which is due to the spin of the nucleus ECEMat 162A


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