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Unformatted text preview: 05/12/09 Physics 13  Fall 07  G.R. Goldstein 1 EM waveparticle 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 s1 • and each γ has E=6.6x1034 J•s • 6x10 14 s1 • ~4x1019 J. • Hence no. γ ’s ~ 30J/s÷ 4x1019 J • ~ 8x10 19 γ ’s/sec E x c 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 monochromaticno γ ’s, E 2 =0 γ ’s, E 2 >0 red=dark, white=bright 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  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 3dim space • In 1dim ℘ (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 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 (1dim) is a solution for a free particle : x V Real( Ψ ) λ 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 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...
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This note was uploaded on 03/27/2008 for the course PHY 13 taught by Professor Garyr.goldstein during the Fall '04 term at Tufts.
 Fall '04
 GaryR.Goldstein
 Physics, Energy, Light, WaveParticle Duality

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