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12 量子物理 Sections 1. 2. 3. 4. 5. 6. 7. 8. 9. Photon and Matter Waves Compton Effect Light as a Probability Wave Electrons and Matter Waves Schrodinger’s Equation Waves on Strings and Matter Waves Trapping an Electron Three Electron Traps The Hydrogen Atom 12-1 Photon and Matter Waves (光子和物質波) • Light Waves and Photons c f E hf (photon energy ) h 6.63 10 34 J s The Photoelectric Effect 光電效應 The experiment • First Experiment (adjusting V)– the stopping potential Vstop K max eVstop 光電子的最大動能與 光強度無關 •Second Experiment (adjusting f)– the cutoff frequency f0 低於截止頻率時即使光再 強也不會有光電效應 The plot of Vstop against f The Photoelectric Equation hf K max h Vstop ( ) f e e 34 h 6.6 10 J s Work function 12-2 Compton Effect hf h p c (photon momentum) 康 普 吞 效 應 實 驗 圖 表 康普吞效應圖示 Energy and momentum conservation hf hf K K mc ( 1) 2 2 hf hf mc ( 1) h h mc( 1) p X h / pe mv Frequency shift h h cos mv cos h 0 sin mv sin h (1 cos ) mc Compton wavelength 12-3 Light as a Probability Wave The standard version The Single-Photon Version First by Taylor in 1909 The single-photon, double-slit experiment is a phenomenon which is impossible, absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics - Richard Feynman The Single-Photon, WideAngle Version (1992) 50μm The postulate Light is generated in the source as photons Light is absorbed in the detector as photons Light travels between source and detector as a probability wave 12-4 Electrons and Matter Waves •The de Broglie wave length •Experimental verification in 1927 •Iodine molecule beam in 1994 h p 1989 double-slit experiment 7,100,3000, 20,000 and 70,000 electrons Experimental Verifications X-ray Electron beam 苯 環 的 中 子 繞 射 12-5 Schrodinger’s Equation • Matter waves and the wave function ( x, y, z, t ) ( x, y, z )e i t •The probability (per unit time) is 2 ie. * Complex conjugate 共軛複數 The Schrodinger Equation from A Simple Wave Function i t ( x, y , z , t ) ( x, y , z ) e A sin( kx) B cos( kx) (1D) p h / k E p / 2m k / 2m 2 2 2 1D Time-independent SE A sin( kx) B cos( kx) 1d d / dx k k 2 dx 2 2 2 2 1 d E 2 2m dx 2 2 d E 2 2m dx 2 2 2 3D Time-dependent SE d 2 E 2 2m dx 2 2 2 2 ( 2 2 2 ) E 2m x y z 2 i 2m t 2 2 V i 2m t 2 2 12-6 Waves on Strings and Matter Waves 駐波與量子化 Quantization 駐波: 2L v v = f n n = 0,1,2, n 2L Confinement of a Wave leads to Quantization – discrete states and discrete energies 12-7 Trapping an Electron For a string: n L 2 n 1,2,3, n yn A sin( ) x, n 1,2,3, L n : quantum number Finding the Quantized Energies of an infinitely deep potential energy well h / p h / 2mE , L n / 2 En n h / 8mL , 2 2 2 n 1,2,3, The Energy Levels 能階 The ground state and excited states The Zero-Point Energy n can’t be 0 The Wave Function and Probability Density For a string n n 1,2,3, ) x, yn A sin( L n n 1,2,3, n A sin( ) x, L 2 n 2 2 n A sin ( ) x, n 1,2,3, L The Probability Density Correspondence principle (對應原理) At large enough quantum numbers, the predictions of quantum mechanics merge smoothly with those of classical physics •Normalization (歸一化) ( x)dx 1 A 2 / L 2 n A Finite Well 有限位能井 d 8 m 2 [ E E pot ( x)] 0 2 dx h 2 2 The probability densities and energy levels Barrier Tunneling 穿隧效應 •Transmission coefficient T e 2 kL k 8 2m( E pot ( x) E ) h2 STM 掃描式穿隧顯微鏡 Piezoelectricity of quartz 12-8 Three Electron Traps • Nanocrystallites 硒化鎘奈米晶粒 那種顏色的顆粒比較小 2 2 nh En 8mL2 c ch t f t Et A Quantum Dot An Artificial Atom The number of electrons can be controlled Quantum Corral 量子圍欄 12-1.9 The Hydrogen Atom •The Energies 2 1 q1q2 1 e U 40 r 40 r 4 me 1 13.6ev En 2 2 2 , 2 8 0 h n n n 1,2,3, 氫 原 子 能 階 與 光 譜 線 Bohr’s Theory of the Hydrogen Atom The Ground State Wave Function 1 r / a e (r ) 3/ 2 a 2 h 0 5.29pm a 2 me (Bohr radius) Quantum Numbers for the Hydrogen Atom The Ground State Dot Plot 氫原子的量子數 N=2, l=0, ml=0 N=2, l=1