Download ECE 162A: Lecture 9 - 3D Solutions in Quantum Mechanics and more Study notes Criminal Justice in PDF only on Docsity! ECE 162A Mat 162A Lecture #9: 3D Solutions. E pectation Val esx u Read Chapter 3,8 of French/Taylor John Bowers Bowers@ece.ucsb.edu Engineering I, Room 4163 ECE/Mat 162A Midterm Material • Lecture • Eisberg/Resnick Chapters 1-6 • French/Taylor 1-5,8,9 (except 5-5) • Background of quantum theory – Wave/particle basis of light: • Planck’s postulate • Planck’s constant – Wave/particle basis of matter • Photoelectric effect, Compton effect, pair production,… – Wave/particle duality – Uncertainty principle – Atom models: • Thompson model • Rutherford model ECE/Mat 162A • Bohr model Square Well V022 0 2222 −== EV m E m k κhh 0 /2 /2)()i ()( 2/< kBkA axFor -a a 2/ coss n −< += axFor xxxψ 0: )exp()exp()( = −+= DconditionBoundary xDxCx κκψ )exp()exp()( 2/ −+= > xGxFx axFor κκψ 0: =FconditionBoundary Solution in Appendix H • 4 Equations (ψ and dψ/dx at two interfaces) • 4 Unknowns (A,B,D,G) • Solution for : 22tan R εεε where −= 2 2 02 2 22 2 2 h h amVR ma E == ε ε 3 Dimensional Time Independent S h di E tic roe nger qua on ψψ zyxzyxVzyx zyxm + ∂ ∂ + ∂ ∂ + ∂ ∂ − 2 2 2 2 2 22 ),,(),,(),,()( 2 h ψ zyxE= ),,( ψψψ EV m =+∇− 2 2 2 h ECE/Mat 162A 3D Particle in a Box ZdXYYdXZXdYZ 222222 hhh XYZbDi id EXYZ dzmdymdxm =−−− 222 222 EZdYdXd yv e =−−− 2 22 2 22 2 22 hhh ConstZFunctionofYFunctionofXFunctionof dzmZdymYdxmX = . 222 zyx XEXd EEEE =− =++ 22h ikx x eX dxm = 22 ECE/Mat 162A ikzikyikxikzikyikx eeee ++==ψ Simplest to use Sines and Cosines h b d di ito matc oun ary con t ons ,...3,2,1sin)( nxnxX x == π 321sin)( yn Y a y x π ,...,, zn n b y y == π ,...3,2,1sin)( 2 nnn n c zZ zz == πππh ))()()(( 2 2222 222 h cbam E zyx ++= ECE/Mat 162A )( 8 222 c n b n a n m E zyx ++= Time Dependence of Solution • Solve for stationary solutions ψn(x) with energies En )()( tiAtΨ −∑ ω h/ , n nn E exx n = = ω ψ nn • Find the values for An that satisfy the intial conditions. ECE/Mat 162A Time Evolution xx ππ 2 ff LL xψ 21)sin()sin()( +=+= titi efeftx ωω 421),( +=Ψ )/)cos((2),( 1221 2 2 2 1 2 htEEfffftx −++=Ψ ECE/Mat 162A Expectation Values dd ∫∫ )()()( * 2 d xxxxxxxx ∫ == )()( 2*2 ψψψ d xxxxx ∫ = * ψψ dxx dx ixp ∫ −= )())(( ψψ h dxxxVxV = )()()(* ψψ ECE/Mat 162A Expectation Values dd ∫∫ )()()( * 2 d xxxxxxxx ∫ == )()( 2*2 ψψψ d xxxxx ∫ = * ψψ dxx dx ixp ∫ −= )())(( ψψ h dxxxVxV = )()()(* ψψ Calculate these for the lowest order solution for an infinite square well ECE/Mat 162A Spherical Coordinates φθ 222 cossin= ++= rx zyxr θ φθ cos sinsin = = rz ry φθθ θ θθ 2 2 222 2 2 2 sin 1)(sin sin 1)(1 ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =∇ rrr r rr ECE/Mat 162A Transform 222 ++= zyxr φθ φθ sinsin cossin = = ry rx φθ θcos ∂∂ + ∂∂ + ∂∂∂ = r rz θ φθ 2 22 1)(sin1)(1 ∂+∂∂+∂∂=∇ ∂∂∂∂∂∂ = ∂ r xxrxx φθθθθ 22222 sinsin ∂∂∂∂∂ rrrrr ECE/Mat 162A Solution to SE in Spherical C dioor nates EV+∇2 2 ψψψh ZerVrVIf m −== =− 4 1)(),,( 2 2 φθ R iablesofseparationtryThen r ΦΘ )()()()( var 0 φθφθ πε ddddRd RbydivideandSubstitute rr ΦΘ ΘΦ = 111 ,, 22 ψ h ErV drddrdr r drrmR =+ Φ + Θ +− )( sin )(sin sin ))(( 2 2222 2 2 φθθ θ θθ ECE/Mat 162A
