3.23 Electrical, Optical, and Magnetic Properties of Materials

Transcription

1 MIT OpenCouseWae Electical, Optical, and Magnetic Popeties of Mateials Fall 7 Fo infomation about citing these mateials o ou Tems of Use, visit:

2 3.3 Fall 7 Lectue 4 CLOSE TO COLLAPSE The collapse of the wavefunction Tavel Office hou (this time only): This Fiday, Sep 1, 4pm (instead of Mon, Sep 4, 4pm) 1

3 Last time: Wave mechanics 1. The ket Ψ descibe the system. The evolution is deteministic, but it applies to stochastic events 3. Classical quantities ae eplaced by opeatos 4. The esults of measuements ae eigenvalues, andth the ket collapses in an eigenvecto Commuting Hemitian opeatos have a set of common eigenfunctions

4 Fifth postulate If the measuement of the physical quantity A gives the esult a n, the wavefunction of the system immediately afte the measuement is the eigenvecto ϕ n Position and pobability Gaphs of the pobability density fo positions of a paticle in a one-dimensional had box accoding to classical mechanics emoved fo copyight easons. See Motime, R. G. Physical Chemisty. nd ed. San Diego, CA: Elsevie,, page 555, Figue Diagam showing the pobability densities of the fist 3 enegy states in a 1D quantum well of width L. 3

5 Collapse of the wavefunction x (a) a x (b) a x (c) The wave function of a paticle in a box. (a) Befoe a position measuement (schematic). The pobability density is nonzeo ove the entie box (except fo the endpoints). (b) Immediately afte the position measuement (schematic). In a vey shot time, the paticle cannot have moved fa fom the position given by the measuement, and the pobability density must be a shaply peaked function. (c) Shotly afte a position measuement (schematic). Afte a shot time, the pobability density can be nonzeo ove a lage egion. a Figue by MIT OpenCouseWae. Quantum double-slit Image emoved due to copyight estictions. Please see any expeimental veification of the double-slit expeiment, such as Image of a double-slit expeiment simulation emoved due to copyight estictions. Please see "Double Slit Expeiment." in Visual Quantum Mechanics. 4

6 Deteministic vs. stochastic Classical, macoscopic objects: we have well- defined values fo all dynamical vaiables at evey instant (position, momentum, kinetic enegy ) Quantum objects: we have well-defined pobabilities of measuing a cetain value fo a dynamical vaiable, when a lage numbe of identical, independent, identically pepaed physical systems ae subject to a measuement. When scientists tun bad Image fom Wikimedia Commons, 5

7 Cat wavefunction 1 1 exp t exp t cat + Ψ dead 1 τ τ Ψ () t = Ψ alive Thee is not a value of the obsevable until it s measued (a conceptually diffeent statistics fom themodynamics) Uncetainties, and Heisenbeg s Indetemination Pinciple ( ) ( ) A = A A = A A 1 A B [A, B] d i x, h ih dx = 6

8 Linewidth Boadening E t h Image emoved due to copyight estictions. Please see: Fig. in Uhlenbeg, G., et al. "Magneto-optical Tapping of Silve Ions." Physical Review A 6 (Novembe ): Top Thee List Albet Einstein: Gott wufelt nicht! nicht! [God does not play dice!] Wene Heisenbeg I myself... only came to believe in the uncetainty elations afte many pangs of conscience... Ewin Schödinge: Had I known that we wee not going to get id of this damned quantum jumping, I neve would have involved myself in this business! 7

9 Spheical Coodinates z x φ θ = P y x= sinθ cos ϕ y = sinθ sin ϕ z = cos θ Figue by MIT OpenCouseWae. Angula Momentum Classical Quantum L p Lˆ = x = yˆˆpz ẑˆ p y = i h y z z y Lˆ = zp ˆˆ xˆˆ y x p z = ih z x x z L ˆ xˆˆp ŷˆ p ih = = x y z y x y x 8

10 Commutation Relation Lˆ = L ˆ + L ˆ + L ˆ x y z ˆ ˆ ˆ L L L L ˆ ˆ ˆ, x =, y = L, L z = Lˆ ˆ = ˆ L, L x y ihlh z Angula Momentum in Spheical Coodinates Lˆ = ih z ϕ h 1 Lˆ = h 1 sinθ + sinθ θ θ sin θ ϕ 9

11 Eigenfunctions of L z, L m m m LY ˆ z l ( θ, ϕ )= i h Y θ, ϕ mh Y θ, ϕ ϕ l ( )= l ( ) 1 1 m m h θ + Y θ, h l, l l sinθ θ θ sin θ ϕ sin (,ϕ ) = (l +1)Y (θ,ϕ ) Simultaneous eigenfunctions of L, L z ˆ m m LY z l ( θϕ, )= mhy l ( θϕ, ) LY ˆ m (, ) h l( 1) Y m l θ ϕ = l + l ( θ,ϕ ) m m Y l (θ,ϕ ) =Θ l (θ )Φ m(ϕ ) 1

12 Spheical Hamonics in Real Fom Figue by MIT OpenCouseWae. Same as a beating dum Copyight 1999, David M. Haison. This mateial may be distibuted only subject to the tems and conditions set foth in the Open Publication License, vx.y o late (the latest vesion is pesently available at 11

13 fo the caee helioseismologist Image coutesy of NSO/AURA/NSF. Used with pemission. Nomal modes (i.e. sound, o seismic waves) fo the Sun (basically jello in a 3d spheical box) Angula Momentum, then m = m = x m = - m = 1 y m = -1 L = l( l+1) h =, h, 6h... L z =, ± h, ± h, ± 3h... Cones of possible angula momentum diections fo l =. These cones ae simila to the cones of pecession of a gyoscope, and epesent possible diections fo the angula momentum vecto. The z component is abitaily chosen as the one component that can have a definite value. Figue by MIT OpenCouseWae. 1

14 An electon in a cental potential (I) ˆ h V( * H = + ) µ needs to be in spheical coodinates h Hˆ = + sinϑ + + V( ) µ sin ϑ ϑ ϑ sin ϑ ϕ ˆ h 1 L = H + V () µ h An electon in a cental potential (II) h 1 d d L Ĥ = V() µ d + + d µ ψ nlm ( ) = R ( ) Y (ϑ, ϕ) nlm lm h 1 d d h l(l +1) + + V() R () = E R () nl nl nl µ d d µ 13

15 An electon in a cental potential (III) u () = R + Ze () V () = h ll ( 1) µ 4πε nl nl eff h d + V () eff u () = E u nl nl nl () µ d What is the V eff () potential? 1 V centipetal () 1 18 v() (J) 1 1 (m) V eff () V Coulomb () - Figue by MIT OpenCouseWae. 14

16 Z The Radial Wavefunctions fo Coulomb V() R 1 R R 1 R R 1 R Thickness d Y R R R 31 R 31 X R 3 R 3 Figues by MIT OpenCouseWae The Gand Table Radial functions R nl () and adial distibution functions R nl () fo atomic hydogen. The unit of length is a µ = (m/ µ ) a, whee a is the fist Boh adius. Shell Quantum numbes n l m Spectoscopic notation Wave function Ψ nlm (, θ, φ) K 1 1s 1 (Z/a ) 3/ exp (-Z/a ) π L _ s p p +1 _ 1 (Z/a ) 3/ (1-Z/a ) exp (-Z/a ) π 1 (Z/a ) 3/ (Z/a ) exp (-Z/a ) cos θ 4 π + _ 1 (Z/a 8 ) 3/ (Z/a ) exp (-Z/a ) sin θ exp (+iφ) _ π M 3 3s 1 (Z/a 3 ) 3/ (1-Z/3a + Z /7a ) exp (-Z/3a ) 3π _ _ _ 3p 3p +1 _ 3d 3d +1 _ 3d _ + (Z/a ) 7 3/ (1-Z/6a ) (Z/a ) exp (-Z/3a ) cos θ π + _ (Z/a 7 ) 3/ (1-Z/6a ) (Z/a ) exp (-Z/3a ) sin θ exp (+iφ) _ π 1 (Z/a ) 81 3/ (Z /a ) exp (-Z/3a ) (3 cos θ 1) 6π + _ 81 1 (Z/a ) 3/ (Z /a ) exp (-Z/3a ) sin θ cos θ exp (+iφ) _ π 1 16 π (Z/a ) 3/ (Z /a ) exp (-Z/3a ) sin θ exp (+iφ) _ The complete nomalised hydogenic wave functions coesponding to the fist thee shells, fo an 'infinitely heavy' nucleus. The quantity a = 4πε h /me is the fist Boh adius. In ode to take into account the educed mass effect one should eplace a by a µ = a (m/µ) 15 Figue by MIT OpenCouseWae.

17 Solutions in the cental Coulomb Potential: the Alphabet Soup Coutesy of David Manthey. Used with pemission. Souce: 16