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Matter Waves and Quantum Mechanics

This lecture discusses the properties of matter waves, including the wave-particle duality, Schrödinger equation, uncertainty principle, and applications such as reflection, transmission, and tunneling.

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Matter Waves and Quantum Mechanics

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  1. 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

  2. Matter waves • de Broglie (1924) • Proposed wavelike properties for particles (since light showed particle properties) • de Broglie wavelengthl = h/p (or ) • 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 thatp=5.4x10-24Kg m s-1. l=1.2x10-10m, i.e. of order of atomic dimensions. • GP Thompson (1927) directed 15keV electrons through a thin metal foil. Again discrete scattering angles were observed.

  3. 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 Shadow of sample Additional spots

  4. Dispersion relation • Dispersion relation • By analogy with light: • For a particle of mass m, • Arbitrary constant V implies freqency and phase velocity (w/k) of a massive particle are not uniquely defined. Dispersion relation for massive paerticles “Free electron parabola”

  5. Group velocity • Group velocity • follows from the dispersion relation: • 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] giveswhere a is real • The wave becomes evanescent. • The wave penetrates into the classically forbidden region but the amplitude decays exponentially. Classical velocityof the particle

  6. Schrödinger equation • “Derivation” • Start from expression for the energy of a non-relativistic particle • Take a plane wave (travelling in +ve x-direction)(note we use the Q.M. “convention” of kx-wt)Inserting into [9.2] and multiplying by Y gives: • Time dependent Schrödinger equation • It is a linear equation so we can superpose solutions

  7. Time independent equation • V=V(x)¹V(x,t). • Potential is a function of position only. • Separate variablesdivide by yT • E must be constant (cannot simultaneously be a function of x and t). • Integrate equation for t. • Equation for x is: • Time independent Schrödinger equation. Function of x Function of t

  8. 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 • Y itself is unobservable, only |Y|2 has physical significance. • Plane wave:complex form of Y is necessary (not a mathematical convenience) to ensure uniform probability. Particle must be somewhere

  9. 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~2p/Dx. • 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.

  10. 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)

  11. Reflection/transmission • Finite potential step. • Boundary conditions at x=0.Y continuous A + r = tdy/dx continuous ik1A-ik1r=ik2t • Vo<0. Classical: accelerate, (no reflection) • Q.M.: Some reflection; phase change of p. • 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.

  12. 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=np, when all particles transmitted (c.f. l/4 coupler, but with phase change of p at one of the boundaries). • Vo>E Classically no transmission; • Q.M. gives some transmission

  13. 1-D, infinite potential-well • Particle in a box: • V(x) = 0 if 0< x <lV(x) = ¥ if x< 0 or x>l. • Probability of particlebeing outside box iszero. Total reflectionat the barrier, gives astanding wave. • Corresponding energies are: Normalisation Quantisation condition Quantum number, n The end

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