Physics 2D Lecture Slides Lecture 26: March 2nd 2005 Vivek Sharma UCSD Physics Measurement Expectation: Statistics Lesson • Ensemble & probable outcome of a single measurement or the average outcome of a large # of measurements � n � � xP x dx ( ) n x + + + i i n x n x n x .... n x < >= = = = �� x 1 1 2 2 3 3 i i i 1 � + + + n n n ... n N � 1 2 3 i P x dx ( ) � � For a general Fn f(x) Sharpness of a distribution : � n � � = scatter around the average � � * ( ) ( ) ( ) x f x x dx n f x ( ) i i < >= = � f x ( ) = �� i 1 � 2 (x x ) � N � i = � P x dx ( ) N �� � � 2 2 = ( x ) ( ) x � � = small Sharp distr. � � Uncertainty X = 1
Particle in the Box, n=1, find <x> & Δ x ? � 2 � � � (x)= sin x � � L � L � � � � 2 sin � � 2 � � � <x>= x x sin x dx � � � � L � L � L � L � � - L � � � � � � 2 � � 2 = x sin x dx , change variable = x � � � � L � L � � L � 0 � 2 1 � � � � � = � � <x>= sin 2 , use sin 2 (1 cos2 ) � 2 L 2 0 � � � � 2L � � � � � � � � � � <x>= d - cos2 d use ud v=uv- vdu � � � 2 2 � � 0 0 � � � = L 2 L � � 2 <x>= (same result as from graphing ( )) x � � � 2 2 2 � � L � 2 2 L L � 2 2 2 = � Similarly <x >= x s in ( x dx ) � L 3 2 2 0 2 2 2 L L L � 2 > � < > = 2 � � = and X= <x x 0.18 L � 3 2 2 4 � X= 20% of L, Particle not sharply confi ned in Box 2
The Case of a Rusty “Twisted Pair” of Naked Wires & How Quantum Mechanics Saved ECE Majors ! Oxide layer Wire #1 Wire #2 • Twisted pair of Cu Wire (metal) in virgin form • Does not stay that way for long in the atmosphere •Gets oxidized in dry air quickly Cu Cu 2 O •In wet air Cu Cu(OH) 2 (the green stuff on wires) • Oxides or Hydride are non-conducting ..so no current can flow across the junction between two metal wires • No current means no circuits no EE, no ECE !! • All ECE majors must now switch to Chemistry instead & play with benzene !!! Bad news ! Potential Barrier U Transmitted ? E<U x Description of Potential U = 0 x < 0 (Region I ) U = U 0 < x < L (Region II) U = 0 x > L (Region III) Consider George as a “free Particle/Wave” with Energy E incident from Left Free particle are under no Force; have wavefunctions like Ψ = A e i(kx-wt) or B e i(-kx-wt) 3
Tunneling Through A Potential Barrier Region I Region III U Prob ? II E<U •Classical & Quantum Pictures compared: When E>U & when E<U •Classically , an particle or a beam of particles incident from left encounters barrier: •when E > U Particle just goes over the barrier (gets transmitted ) •When E<U particle is stuck in region I, gets entirely reflected, no transmission (T) •What happens in a Quantum Mechanical barrier ? No region is inaccessible for particle since the potential is (sometimes small) but finite Beam Of Particles With E < U Incident On Barrier From Left Region I II Region III U A Incident Beam F B Reflected Beam Transmitted Beam x 0 L Description Of WaveFunctions in Various regions: Simple Ones first � � � � � � = i kx ( t ) + i ( kx t ) = In Region I : ( x t , ) Ae Be incident + reflected Waves I 2 2 = � k � with E = � 2 m 2 |B| = def ine Reflection Coefficient : R = frac of incident wave intensity reflected back 2 |A| � � � � � � = i kx ( t ) + i ( kx t ) = In Region III: ( , ) x t F e G e transmitted III i ( � kx � � t ) Note : Ge corresponds to wave incident from righ t ! This piece does not exist in the scattering picture we are thinking of now (G=0) |F| 2 � = � � So ( , ) x t Fe i kx ( t ) represents transmitted beam. Define T = |A| III 2 � Unitarity Condition R + T= 1 (particle i s either reflected or transmitted) 4
Wave Function Across The Potential Barrier In Region II of Potential U 2 2 � � d ( ) x + � = � TISE: - U ( ) x E ( ) x 2 2m dx 2 � d ( ) x 2 m U � = � � ( E ) ( ) x 2 2 dx � � � 2 = ( ) x 2m(U-E) � � � > with 2 = ; U> E 2 0 � � � ± � Solutions are of for m ( ) x e x + � � � � � � � � = x i t + x i t ( , ) x t Ce De 0< x<L I I � To determine C & D apply matching cond . � = ( , ) x t continuous acro ss barrie r (x=0,L) II � d ( , ) x t I I = continuous across barrier (x=0,L) dx Continuity Conditions Across Barrier � � At x = 0 , continuity of (x) A+B=C+D (1) � d (x) � At x = 0 , continuity of dx � = � � � ikA i kB C D (2 ) � � Similarly at x=L continuity of (x) � � + + � = Ce L De L F e ikL (3) � d (x) � at x=L, continuity of dx � � � � � = L L ikL -( C) e + ( D) e ikFe (4) Four equations & four unknow ns Cant determine A,B,C,D but if you Divide thruout by A in all 4 eq uations : � � ratio of amplitudes rel ations f or R & T That' s wh at we ne ed a ny way 5
Potential Barrier when E < U Expression for Transmissi on Coeff T=T(E) : Depends on barrier Height U, barrier Width L and particle Energy E � 1 � � � � 2 � 1 U 2 ( m U E ) 2 � � = T(E) = 1+ sinh ( L ) ; � � � � � 4 E U ( E ) � � � � � and R(E)=1- T(E).. .......what's not transmitted is reflected Above equation holds only for E < U For E>U, α =imaginary# Sinh( α L) becomes oscillatory This leads to an Oscillatory T(E) and Transmission resonances occur where General Solutions for R & T: For some specific energy ONLY, T(E) =1 At other values of E, some particles are reflected back ..even though E>U !! That’s the Wave nature of the Quantum particle Ceparated in Coppertino Oxide layer Wire #1 Wire #2 Solved Example 6.1 (...that I made such a big deal about yesterday) Q: 2 Cu wires are seperated by insulating Oxide layer. Modeling the Oxide layer as a square barrier of height U=10.0eV, estimate the transmission coeff for an incident beam of electrons of E=7.0 eV when the layer thickness is (a) 5.0 nm (b) 1.0nm Q: If a 1.0 mA current in one of the intwined wires is incident on Oxide layer, how much of this current passes thru the Oxide layer on to the adjacent wire if the layer thickness is 1.0nm? What becomes of the remaining current? � 1 � � � � 2 1 U 2m(U-E) 2mE � � = 2 = , k T(E) = 1+ sinh ( L ) � � � � � � � 4 E U ( E ) � � � � 6
� 1 � � � � 2 1 U 2 � T(E) = 1+ � sinh ( L ) � � � � 4 E U ( E ) � � � � � = 2 Use =1.973 keV.A/c , m � 511 keV/c e Oxide layer � � 2 � � 3 2 m U ( E ) 2 511 kev c / (3.0 10 keV ) � � � = e = = 0.8875A -1 � � 1.973 keV.A/c Substitute in expression for T=T(E) Wire #1 Wire #2 � 1 � � � � 2 1 10 � � = � � 2 -1 38 T = � 1+ si nh (0.8875 A )(50A ) � 0.963 10 ( small )!! � � � 4 7 (10 7) � � � � � � -7 However, for L=10A; T=0.657 1 0 � Reducing barrier width by 5 leads to Trans. Coeff enhancement by 31 orders of ma gnitude !! ! Q=Nq � � 15 1 mA current =I= e N =6.25 10 electrons t N =# of electrons that escape to the adjacent wire (past oxide layer) T = = � 15 � Oxide thickness makes all the difference ! N N . T (6.25 10 electrons ) T ; T That’s why from time-to-time one needs to � � -7 � = � � = For L=10 A, T=0.657 1 0 N 4.11 10 I 65.7 pA !! T T Scrape off the green stuff off the naked wires Cur en r t Measured on the first wire is sum of incident+reflected currents and current measured on "adjacent" wire is the I T QM in 3 Dimensions • Learn to extend S. Eq and its solutions from “toy” examples in 1-Dimension (x) → three orthogonal dimensions (r ≡ x,y,z) � ˆ ˆ ˆ = + + r ix jy kz • Then transform the systems – Particle in 1D rigid box 3D rigid box – 1D Harmonic Oscillator 3D Harmonic Oscillator z • Keep an eye on the number of different integers needed to specify system 1 3 (corresponding to 3 available degrees of freedom y x,y,z) x 7
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