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Lecture_08: Outline Matter Waves de Broglie hypothesis Experimental verifications Wave functions de Broglie hypothesis Photons: E ph h p k h Wave → Particle What about the opposite? Particle → Wave ? h p de Broglie hypothesis Matter waves: h h h B p mv 2mK Wavelength for a walking man? Wavelength for a moving electron? What is the wavelengths difference for 5 eV electron and 5 eV photon? How to reveal the wave properties of electrons? Experimental verifications X-rays diffraction: X-rays are electromagnetic waves with λ = 10-8 to 10-12 m = 10 – 0.001 nm Experimental verifications Bragg’s law: 2d sin n If θ and λ are known, d can be determined Experimental verifications Davisson-Germer experiment: • Electrons were directed onto nickel crystals • Accelerating voltage is used to control electron energy: E = |e|V • The scattering angle and intensity (electron current) are detected – φ is the scattering angle Experimental verifications Electron scattering: From X-ray experiments: d = 0.091 nm 2d sin n For φ = 50° (θ = 65°): λ = 0.165 nm Experimental verifications Experimental results: For φ ~ 50°, the B maximum is at ~54V E = 54 eV = 8.64·10-18J h 0.167nm 2mE Experimental verifications X-rays and electrons: Experimental verifications Electrons: Experimental verifications Other particles: He atoms Neutrons C60 molecules Experimental verifications Double-slit experiments: Light: Experimental verifications Double-slit experiments: Light: Electrons: Experimental verifications Individual electrons: • In previous experiments many electrons were diffracted (or show interference) • Will one get the same result for a single electron? • Such experiments were performed – Intensity of the electron beam was so low that only one electron at a time proceeds – Still diffraction (and interference) patterns, and not diffused scattering, were observed, confirming that Thus individual electrons possess wave properties!!! Wave functions Complimentarity Principle: The particle and the wave models are COMPLIMENTARY No measurements can simultaneously reveal the particle and the wave properties of matter Wave functions Photons: E(r , t ) Emax cos(k r t ) E ph h p k h Free Particles: B h p p E (r , t ) A cos( r t ) Wave functions Superposition principle: Electromagnetic waves: E(r , t ) E1 (r , t ) E2 (r , t ) Matter waves: (r , t ) 1 (r , t ) 2 (r , t ) Wave functions Probability: Matter waves: The value of |Ψ(x,t)|2 gives the probability to find the particle in point x at time t. dx ( x, t ) 2 1