Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lecture 12 Matter waves (Quantum mechanics). Aims: Massive particles as dispersive waves. Phase and group velocity Evanescent waves (tunneling). Schrödinger equation. Time-dependent equation; Time-independent equation. Interpretation of the wave function. Heisenberg’s Uncertainty Principle. Applications: Reflection/transmission at potential step. 1-D potential well Energy quantisation 1 Waves 11 Matter waves de Broglie (1924) Proposed wavelike properties for particles (since light showed particle properties) de Broglie wavelength l = h/p (or p k ) Davisson and Germer (1927) Electron diffraction. Now the basis of the use of LEED (Low Energy Electron Diffraction) as a probe for surface structure. At 100eV, K.E.=p2/2m is such that p=5.4x10-24 Kg m s-1. l=1.2x10-10m, i.e. of order of atomic dimensions. d sin l GP Thompson (1927) directed 15keV electrons through a thin metal foil. Again discrete scattering angles were observed. 2 Waves 11 LEED and diffraction Modern electron diffraction Results from the (111) surface of a Nickel crystal: Left panel: native Ni(111) Right panel: with adsorbed hydrogen Ni(111)-(2x2)H Shadow of sample Additional spots 3 Waves 11 Dispersion relation Dispersion relation By analogy with light: p k E w ( N .B. h / 2 ) For a particle of mass m, E KE PE p 2 2m V 2k 2 w V 2m 9.1 Dispersion relation for massive paerticles “Free electron parabola” Arbitrary constant V implies freqency and phase velocity (w/k) of a massive particle are not uniquely defined. 4 Waves 11 Group velocity Group velocity follows from the dispersion relation: 2 dw 2 k dk 2m k p vg v m m Classical velocity of the particle The same is true relativistically. Classiclly forbidden region. Classically, particles cannot enter regions where their total energy, E, is less than V. Under such circumstances, eq [9.1] gives 2m k 2 2 E V a 2 i.e. k ia where a is real The wave becomes evanescent. The wave penetrates into the classically forbidden region but the amplitude decays exponentially. 5 Waves 11 Schrödinger equation “Derivation” Start from expression for the energy of a nonrelativistic particle E p 2 2m V 9.2 Take a plane wave (travelling in +ve x-direction) Y x, t Yoei kx wt (note we use the Q.M. “convention” of kx-wt) 2 Y x, t p2 2 k Y x, t 2 Y x, t 2 x Y x, t iE iwY x, t Y x, t t Inserting into [9.2] and multiplying by Y gives: Y x, t 2 2 Y x, t i VY x, t 2 t 2m x 9.3 Time dependent Schrödinger equation It is a linear equation so we can superpose solutions 6 Waves 11 Time independent equation V=V(x) V(x,t). Potential is a function of position only. Separate variables Y x, t y x T t dT 2 d 2y iy T 2 VyT EyT dt 2m dx divide by yT 1 1 dT 2 d 2y E i V y y T dt 2m dx 2 Function of x Function of t E must be constant (cannot simultaneously be a function of x and t). Integrate equation for t. iEt / iwt T Ae Ae Equation for x is: 2 2 d y E V y 0 2 2m dx 9.4 Time independent Schrödinger equation. 7 Waves 11 Wave function Interpretation Born (1927): The probability that a particle with wave function Y(x,t) will be found at a position between x and x+dx at time t is given by |Y(x,t)|2dx. Normalisation 2 Yx, t dx 1 Particle must be somewhere Y itself is unobservable, only |Y|2 has physical significance. Plane wave: Y x, t Yoei (kx wt ) YY Yo2 complex form of Y is necessary (not a mathematical convenience) to ensure uniform probability. 8 Waves 11 Heisenberg’s Uncertainty Principle Localised wavepacket Probability of finding particle localised in space. MUST be a corresponding spread in the Fourier Transform. Wavepacket of width Dx corresponds to Dk~2/Dx. DkDx 2 DpDx h It is impossible to measure the position and momentum of a particle with arbitrary precision simultaneously. Since particles with mass are dispersive waves, the packet will spread with time and the particle becomes less well localised. Uncertainty in k means we loose knowledge of the particle’s position as time goes on. Computer animation. 9 Waves 11 Applications: simple systems Potential steps Scattering Potential barriers Tunelling radioactive decay Potential wells Stationary states atoms (crude model) electrons in metals (surprisingly good model) Boundary conditions at a potential discontinuity (1) The wavefunction, Y, is continuous (2) The gradient , dY/dx, is continuous (See notes and QM course for justification) 10 Waves 11 Reflection/transmission Finite potential step. x0 V 0 v Vo 2 2m k1 2 E 2 2m k1 2 E Vo Boundary conditions at x=0. Y continuous A+r=t dy/dx continuous ik1A-ik1r=ik2t k1 k2 r k1 k2 2k1 A t k1 k2 Vo<0. Classical: accelerate, (no reflection) Q.M.: Some reflection; phase change of . 0<Vo<E. Classical: decelerate, (no reflection) Q.M.: Some reflection. Vo>E. Classically forbidden, all reflect. Q.M.: y a e-ax penetrates barrier. i.e. finite probability for being in classically forbidden region, though all reflect. 11 Waves 11 Potential barrier Potential barrier height, Vo; width a. Full solution given in Q.M. course. Here we consider two distinct situations: Vo<E Classically no reflection; Q.M. some reflection. Except when k2a=n, when all particles transmitted (c.f. l/4 coupler, but with phase change of at one of the boundaries). Vo>E Classically no transmission; Q.M. gives some transmission 12 Waves 11 1-D, infinite potential-well Particle in a box: V(x) = 0 if 0< x <l V(x) = if x< 0 or x>l. Probability of particle being outside box is zero. Total reflection at the barrier, gives a standing wave. y x A sin kx k n / l A 2/l Normalisation Quantisation condition Corresponding energies are: 2 2 2k 2 2 E n 2m 2ml 2 Quantum number, n The end 13 Waves 11