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Modern Physics (PHY 251) Lecture 18 Joanna Kiryluk Fall Semester Lectures Department of Physics and Astronomy, Stony Brook University • Matter waves • Rutherford Scattering • Models of an Atom Duality of light and matter … 1 Matter Waves De Broglie: the dual (wave-particle) behavior of radiation applies to matter (symmetry of nature) • A photon has a light wave associated with it that governs its motion • A particle with non-zero rest mass has an associated matter wave that governs its motion. https://www.youtube.com/watch?v=Vsm2ubwuA_g https://www.youtube.com/watch?v=JIGI-eXK0tg (documentary/biography < 5.33min) 2 Matter Waves De Broglie: the total energy of an entity E (momentum p) is related to the frequency n (wavelength l) of the wave associated with its motion: Note: E = hν , p = h λ for massive (rest mass >0) particles: ln=c predicts the de Broglie wavelength l of a matter wave λ = h p associated with the motion of a particle with momentum p. h ≈ 6.6 ×10 −34 J ⋅ s h ≈ 4.1×10 −15 eV ⋅ s 3 Example: What’s the de Broglie’s wavelength (in meters) of a) a baseball of m=1kg moving at a speed of v=10m/s? b) an electron whose kinetic energy is 100 eV (i.e. non-relativistic)? m0= 0.511 MeV/c 2 E = hν , p = h λ 1eV ≈ 1.6 ×10 −19 J h ≈ 6.6 ×10 −34 J ⋅ s h ≈ 4.1×10 −15 eV ⋅ s The wave nature of light/matter propagation is not revealed by experiments when the important dimensions of the apparatus used are very large compared to the wavelength of light/matter. 4 Double-Slit Experiment: cannot predict where electron would land (Quantum particle) or Light source The same result! http://freevideos.astronomycentral.net/video/Xmq_FJd1oUQ/Quantum-Physics-made-simple-WaveParticle-Duality-Animation.html https://www.youtube.com/watch?v=M4_0obIwQ_U (Episode: Quantum Leap) Interactive simulations (play@ home): http://phet.colorado.edu/en/simulation/quantum-wave-interference 7 5 Group work: 1eV ≈ 1.6 ×10 −19 J h ≈ 6.6 ×10 −34 J ⋅ s h ≈ 4.1×10 −15 eV ⋅ s 6 Matter Waves – Do they exist? Yes! 1927 Experiments by Thompson (UK), Davisson and Germer (USA): diffraction of electron beams confirmed de Broglie relation λ=h p Constructive Destructive interference 7 Matter Waves – Do they exist? Yes! 1927 Experiments by Thompson (UK), Davisson and Germer (USA): diffraction of electron beams confirmed de Broglie relation λ=h p http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html 8 @home (practice) In an experiment an electron beam of 54 eV kinetic energy falls on a single crystal of nickel. The Bragg angle is q = 65o . We know the lattice spacing d= 0.91x10-10 m from X-rays experiment. The electron’s rest mass is 0.511 MeV/c2. Calculate electron’s wavelength using: a) Bragg’s law (assume n=1, i.e. the most intense diffraction maximum) b) de Broglie’s wavelength Do they agree? Useful formulae: E = hν , p = h λ 9 Source: Ch. Baily 10 Duality of light and matter: Summary https://www.youtube.com/watch?v=FlIrgE5T_g0 21:49min – 26.07min Niels Bohr, principle of complementarity: § The wave and particle models are complementary. § If a measurement proves the wave character of radiation or matter, then it is impossible to prove the particle character in the measurement and conversely. § Our understanding of radiation or matter is incomplete unless we take into account measurements which reveal the wave aspects and those which reveal particle aspects. The radiation and matter are neither waves nor particles. Probability interpretation of the wave-particle duality: the link between wave and particle models. Born, Schrodinger (will be discussed in the upcoming lectures) 12 Quantum Mechanics (Future Lecture) Schroedinger’s theory of quantum mechanics The theory specifies the laws of wave motion that particles of any microscopic system obey. § Maxwell theory of electromagnetism: e-m waves § Schroedinger theory: wave functions but what is really waving and how??? Need a quantitative relation between the properties of particle and properties of the wave function that describes the wave. How does the wave governs the particle? 13 Quantum Mechanics (Future Lectures) Born’s interpretation of the wave function Probability density function connects the properties of the wave function and the behavior of the associated particle: ∗ P ( x, t ) = Ψ ( x, t ) Ψ ( x, t ) = Ψ ( x, t ) 2 Ψ ∗ ( x, t ) - complex conjugate of Ψ ( x, t ) Born’s postulate: if at the instant t, a measurement is made to locate the particle associated with the wave function Ψ ( x, t ), then the probability P ( x, t ) dx that the particle will be found at a coordinate 2 between x and x+dx is equal to Ψ ( x, t ) dx ∞ ∞ ∫ P ( x, t )dx = ∫ Ψ ( x, t ) −∞ 2 dx = 1 −∞ Normalized, i.e. probability to find the particle somewhere is 100% 14 Bohr Model of the Atom 22 Models of the Atom Ernest Rutherford – “the father of nuclear physics” § 1911: Rutherford proposed the existence of a massive nucleus as a small central part of an atom J.J. Thompson’s Plum Pudding Model (1904) Rutherford’s Planetary Model (1911) 15 Rutherford experiment § Rutherford designed an experiment to use a particles as atomic bullets (then new technology). § The “gold-foil experiment” was performed in 1909 by Hans Geiger and Ernest Marsden (under Rutherford’s direction) 16 Rutherford experiment § Rutherford designed an experiment to use a particles as atomic bullets (then new technology). § The “gold-foil experiment” was performed in 1909 by Hans Geiger and Ernest Marsden (under Rutherford’s direction) Radium 17 Rutherford experiment “It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive center, carrying a charge.” — Ernest Rutherford 18 Interactive Simulations: Rutherford Scattering http://phet.colorado.edu/en/simulation/rutherford-scattering Download it and run it! 19 Rutherford scattering Coulomb force Nα α Ekin ≈ 8MeV Au n (target density) dΩ = 2π sin θ dθ dN For a given number of incoming a particles Na what is dN, where dN is the number of scattered a particles into an angle dW at angle q. dN dσ = Nα ⋅ n (last lecture) dΩ dΩ dσ D2 20 = 4 Rutherford scattering - Differential Cross Section Here a nucleus = point like particle (i.e. no structure) dN dσ = Nα ⋅ n, dΩ dΩ 2 2 &2 # & # dσ 1 zZe 1 =% × ( ×% ( dΩ $ 4πε 0 ' $ 2Mv 2 ' sin 4 (θ 2 ) Coulomb interaction dA ( r sin θ ) × dΩ = 2 = r dΩ = 2π sin θ dθ In the spherical coordinate system 21 Result: Rutherford model of Atom Summary: How protons, electrons and neutrons were discovered: (10 min) https://www.youtube.com/watch?v=kBgIMRV895w Rutherford type of experiment has been applied with higher energy beams to study structure of nuclei and nucleons 22 2nd midterm exam: Tuesday, November 8 Material – Lectures 12,13,14,15,16,17 PLACE (NOT P118): • Physics P130 (R01) • Light Engineering 152 (R02 and R03)