Lecture_08: Outline Matter Waves  de Broglie hypothesis  Experimental verifications  Wave functions

de Broglie hypothesis Photons: Wave → Particle What about the opposite? Particle → Wave ?

de Broglie hypothesis Matter waves: Wavelength for a walking man? Wavelength for a moving electron? How to reveal the wave properties of electrons? What is the wavelengths difference for 5 eV electron and 5 eV photon?

Experimental verifications X-rays diffraction: X-rays are electromagnetic waves with λ = to m = 10 – nm

Experimental verifications Bragg’s law: 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 = nm For φ = 50° (θ = 65°): λ = nm

Experimental verifications Experimental results: For φ ~ 50°, the maximum is at ~54V E = 54 eV = 8.64· J

Experimental verifications X-rays and electrons:

Experimental verifications Electrons:

Experimental verifications Other particles: Neutrons He atoms C 60 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: Free Particles:

Wave functions Superposition principle: Electromagnetic waves: Matter waves:

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.