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class36 - What brought us to the Schrd equ Today Review...

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Today: Review Exam: Tomorrow 7:30 - 9:15pm, MEUN E-050 Topics: All we have done so far (TZD: 1-7) Allowed materials: Two letter paper-sized, hand- written cheat-sheets (all four sides) and a Calculator. But: Textbooks, cell phones, laptops, radios, TVs, pot-plants, aquarium, smokes, beer, etc. are not allowed. What brought us to the Schröd. equ? EM-waves Wave equation: Probability to find photon: E(x,t) EM wave function: |E| 2 dx De Broglie: ‘Nature is symmetric’ ‘Matter is a wave’ λ =h/p 2 2 2 2 2 1 t E c x E = (1D In free space) E = hf Energy x V(x) n=1 n=2 n=3 ψ (x) t t x i t x t x V x t x m Ψ = Ψ + Ψ ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h Wave equation: Probability to find particle: Wave function: t t x const x t x Ψ = Ψ ) , ( . ) , ( 2 2 1D In free space: Matter waves: Ψ(x,t) |Ψ| 2 dx Vibrations on a string: Electromagnetic waves: Classical wave Equations 2 2 1 y y = 2 2 1 E E = x y E x 2 2 2 t v x v = speed of wave 2 2 2 t c x c = speed of light Magnitude is non-spatial: = Strength of Electric field Magnitude is spatial: = Vertical displacement of String Solutions: E(x,t) Solutions: y(x,t) EM waves vs. matter waves EM Waves (light/photons): Amplitude E = electric field E 2 tells us probability of finding photon. Maxwell’s Equations: Matter Waves (electrons, etc.): Amplitude ψ = “wave function” | Ψ | 2 tells us probability of finding particle. Schrödinger Equation: Ψ Ψ (for V=0) Solutions are sin/cosine waves: Solutions are complex sine/cosine waves: 2 2 2 2 2 1 t E c x E = t i x m = h h 2 2 2 2 ) cos( ) , ( ) sin( ) , ( t kx A t x E t kx A t x E ω ω = = )) sin( ) (cos( ) , ( ) ( t kx i t kx A Ae t x t kx i ω ω ω + = = Ψ
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