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lecture_10_07b

lecture_10_07b - EM wave-particle duality E c x Maxwell...

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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 1 EM wave-particle duality Maxwell: E nergy/vol ~ E 2 (or E 2 + B 2 ) But quanta have E = h ν . How to reconcile? Note that 100 W light bulb 30 W visible light with peak λ near 500 nm, so ν =c/ λ =6x10 14 s -1 and each γ has E=6.6x10 -34 J•s • 6x10 14 s -1 ~4x10 -19 J. Hence no. γ ’s ~ 30J/s÷ 4x10 -19 J ~ 8x10 19 γ ’s/sec E x c
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 2 EM wave interference E = E 1 + E 2 and | E | 2 | E 1 | 2 + | E 2 | 2 As | E | 2 (~ Probability for γ ’s) decreases to few γ ’s get same pattern on the average monochromatic -no γ ’s, E 2 =0 - γ ’s, E 2 >0 red=dark, white=bright
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 3 Light, bb’s, electrons Compare EM interference with bb’s shot through 2 slits ( robability = 1 + 2 ) Electrons are like light, not bb’s. 2200 Ψ = Ψ 1 + Ψ 2 and ℘∝ | Ψ | 2 =| Ψ 1 + Ψ 2 | 2 | Ψ 1 | 2 + | Ψ 2 | 2 2200 Ψ (x ,t) is the amplitude for an electron at x ,t. | Ψ (x ,t)| 2 is Probability Distribution Function or Probability Density, with | Ψ (x ,t)| 2 V the Probability for finding e - in Volume V
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 4 Normalizing probability Necessary mathematical requirement for physical sense 2200 (x ,t)=| Ψ (x ,t)| 2 in 3-dim space In 1-dim (x,t)=| Ψ (x,t)| 2 and (x,t)dx is probability in length dx So need All x dx (x,t) =1 since single electron is somewhere This is normalization of probability and amplitude Ψ
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 5 Travelling waves Time development of Ψ is the dynamics in QM Travelling plane wave (1-dim) is a solution for a free particle : x V Real( Ψ ) λ
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 6 Plane wave variables Ψ ( x , t ) = A sin( kx - wt ) where k = 2 p l and w = 2 pn for a travelling plane wave. But for Quantum Physics l = h p so h k = p . Also p 2 2 m = E h w = h 2 k 2 2 m dispersion velocity V = l n = w k depends on k (= 1 2 p m ) - h 2 2m 2 x 2 Y( x , t ) + U ( x )Y( x , t ) = ih t Y( x , t ) U(x)=0 for free particles
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05/12/09 Physics 13 - Fall 07 - G.R. Goldstein 7 Interpretation of plane wave Infinite in extent completely unlocalized particle with definite wavelength & frequency (for precise spatial shape & time dependence) or momentum & energy To localize need Fourier Superposition Consider 2 waves - both solutions (linearity allows superposition)
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