Outlines of Quantum Physics

Transcription

1 Duality Outlines of 1 Wave-Particle Duality Bohr s Theory Wave-Particle Duality Probability interpretation of the Wave Function

2 Spectrum of Atomic Hydrogen Q: How do we know the energy levels of the hydrogen atom? Spectrum Fingerprints of atoms & molecules... Atomic Hydrogen J. Balmer, = R( 1 1 ), λ 2 2 n 2 n =3,4,5,6 J. Rydberg, = R( 1 1 ) λ n 2 n 2 ν = 1 (in λ cm 1 )

3 Bohr s Theory Electron in an Atom >fe: Faraday, Millikan, Ò4 : J J Thomson, (.Planetary model, E Rutherford, 1911 f1ì Balmerúª, J Balmer, 1885 fþfzu?quantization energy to H atom, Niels Bohr, 1913 Bohr s Assumption There are certain allowed orbits for which the electron has a fixed energy. The electron loses energy only when it jumps between the allowed orbits and the atoms emits this energy as light of a given wavelength.

4 Bohr s Theory m ev 2 r = e2 4πɛ 0r 2 E = e2 /4πɛ 0 2r r = a 0 n 2, E = e2 /4πɛ 0 1 2a 0 n, 2 ν = R( 1 n 1 2 n ), 2 a 0 = 2 (e 2 /4πɛ 0)m e, hcr = (e2 /4πɛ 0) 2 m e 2 2 M R M = R m R e+m (1 me M ) Bohr s success f1ì k X a 1Ì He + Pickering X Ÿþ1: Ù y fsþf3 µfrank-hertz 1914 fx Ì balance between centripetal acceleration and Column attraction E = 1 2 m ev 2 e2 /4πɛ 0 r Assumption: m e vr = n m = mem m e+m

5 Moseley and the Atomic Number Henry G.J. Moseley, Phil. Mag., 27,703(1914) X-ray of elements: f Z

6 Moseley and the Atomic Number X-ray, K- to L-shell transitions, 1 λ = R { (Z σ K ) 2 1 (Z σ L) } 2 1 (@Z = 1 92(U) "7 2 RutherfordµÚ ƒ±ïl Œ

7 Problems in Bohr s theory ØU)ºÙ õ>f f ØU)º f1ì [( Sommerfeld, relativistic Effects /The integral of the momentum associated with a coordinate around one period of the motion associated with that coordinate is an integral multiple of Planck s constant. For any physical system where the classical motion is periodic.0 Circular orbit: m e v 2πr = nh taking r = n 2 a 0 and a 0 = 2 (e 2 /4πɛ 0)m e, v c = α n fine-structure constant α = e2 /4πɛ 0 c Electrons in elliptical orbits with relativistic corrections, E = hcr[ 1 n 2 ÔnÆ ŠŸå } + α2 n ( n 4 k 3 4 )], Sommerfeld, 1916

8 Problems in Bohr s theory ØU)ºÙ õ>f f ØU)º f1ì [( Bohr ½ Maxwell>^nØmÀâ How can a hydrogen atom be stable? Quantization? Why? Electron is a wave? Einsteinµ g f[ uñ 1f XÛÀJÙ º Rutherfordµ >fxûû½ ±ŸoªÇ ĺ 3Ÿožÿl ½[$Ä, ½º q 7Lb½>f k ò Û?º Bohr)ºŸ Ï ²;Ôn Maxwell>^nØ gñ ½ [ Ø gñ [ Ø L Ûž XÛ û½?1[ (½5Ô

9 Radiative Decay An electric dipole moment ed oscillating at angular frequency ω radiates a power P, Total energy E of an electron in harmonic motion, This energy decreases at a rate equal to the power radiated, Classical radiative lifetime τ, ²;>^nØØ#N ½ ±Ï; 3œ P = e2 D 2 ω 4 12πɛ 0c 3 E = m e ω 2 D 2 /2 de dt = e2 ω 2 6πɛ 0m ec E = E 3 τ 1 τ = e2 ω 2 6πɛ 0m ec 3 Na D line (3s-3p): λ = 589nm, τ = 16ns. Sÿ-uÆ µ16.25 ns ܺœ

10 Wave-particle duality Åâ 5 Classical Physics Object Particles Fields and Waves Govern Laws Newton s Law Maxwell s Eq. Phenomena Mechanics, Heat Optics, Electromagnetism é ] º Ÿ/ïÓ3²;Vgƒþ" N. Bohr, þfåæ Œ?ØŒ±ÿþÔnþ"?ÛÔnnØ A?ØÔnþŒ±*ÿÔnþ éuïá *y (nø cùa5 ù:... Werner Heisenberg

11 Wave-particle duality Åâ 5 Photon >^ÅElectromagnetic wave, James Clerk Maxwell: Maxwell s Equations, 1860; Heinrich Hertz, 1888 çnëblackbody radiation, Max Planck: Planck s constant, > APhotoelectric effect, Albert Einstein: photons, 1905 y3...kü«1æ` Ñ ØŒ" < 7L«@ +nø[ s cãœãå vk?ûéx " A. Einsein, Berliner Tageblatt, 20 April 1924.

12 Wave-particle duality Åâ 5 Breakthrough Bohr s Hydrogen Atom, Niels Bohr, 1913 xêîñphoton-electron scattering, Arthur Compton, 1923 W Bothe and H Geiger, ÔŸÅb` Louis de Broglie, 1923: λ = h p >fûdiffraction of electrons, Clinton Davisson and Lester Germer, 1927 Compton Scattering Crystalline Diffraction pattern

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14 First Solvay Conference, 1911 Walther Nernst, Marcel Brillouin, Ernest Solvay, Hendrik Lorentz, Emil Warburg, Jean Baptiste Perrin, Wilhelm Wien, Marie Curie, and Henri Poincaré Robert Goldschmidt, Max Planck, Heinrich Rubens, Arnold Sommerfeld, Frederick Lindemann, Maurice de Broglie, Martin Knudsen, Friedrich Hasenörl, Georges Hostelet, Edouard Herzen, James Hopwood Jeans, Ernest Rutherford, Heike Kamerlingh Onnes, Albert Einstein, Paul Langevin Photograph by Benjamin Couprie, 1911

15 Wave-particle duality Åâ 5 ²;åÆn) ÅÄ5µŒU\5 Ônþ3 m Ù" âf5µøœ 5 ; ²;n) öƒmkøœnúgñœ Üä F"SU "ÁU Bohr:... Ù` ±1þfÆ` ØX`q éuþ ÚÄþÅðéËL ^5JÑ~ " Einsteinµ Š n KéÏJ5 ï k3 4à;½œ¹eâ AT#N"

16 Duality Bohr Duality Probability

17 1927 Solvay Conference on Quantum Mechanics A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schröinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin, P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson Photograph by Benjamin Couprie, Institut International de Physique Solvay, Brussels, Belgium

18 Wave-particle duality Åâ 5 þfåæn) ÅÄ5µƒZU\5 Coherent superposition ØI kônþ3 m Ù" âf5µˆâ5!øœ 5 Corpuscularity ïøœÿþ ; Well, an electron is also a... wave. pö5 n Complementarity Principle Niels Bohr

19 ^żê ã *âf żê âfg ã Ψ( r ) AÇÅ)º Probability Interpretation of the Wave Function τ r Max Born, 1926 Ψ( r ), Ψ( r ) 2 τ: τ éâfaç" ýï Ø kƒ " Let there be math! Here is the math for an electron...

20 ^żê ã *âf żêÚO)ºéżê 1 ψ² ŒÈ 2 ψk 3 ψüš ùø î ~ µ 1 ² Å Ψ e i( p r )/ ψ 2 dτuñ 2 ψœ±káû: 3 ψœ±kø½ƒïfe iφ

21 U\ nprinciple of Superposition Ψ = c 1 Ψ 1 + c 2 Ψ 2 K ÿþψ 1!Ψ 2 O ÿþša 1 Úa 2 µψ 1 a 1, Ψ 2 a 2, ÿþψ = c 1 Ψ 1 + c 2 Ψ 2 Ÿo(JºΨ? ü1fu 1 0??

22 ësnjj SK SN ëö SK f1ì => 1-5 =>1-8 BohrnØ => 7-9 =>2-9, 2-11, 2-12 => 10 OÏd"Xê => 3A =>2-5 Åâ 5 =Lv> 1.2 =Lv>1.1, 1.2 => 12 =Q> 2.1 żêAÇ)º =Lv> 1.6 => 14 =Q> 2.2 =>3-9, 3-10

23 Questions How do we know the energy levels of the hydrogen atom? How can a hydrogen atom be stable? What does a transition mean? The electron is also a... wave? What is the math for an electron?