Physics Lecture 16

Physics Lecture 16 - a. works iii. If x > L: ψ(s) =...

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Lecture 16: What doe it mean for a particle to have wave-like properties? I. Uncertainty a. Born – wave Probability b. Heisenburg: i. ΔxΔp ≥ ħ/2 ħ = h/2π ii. If we know the momentum definitively, we cannot know the position. iii. In general, a particle does not even have a well-definded position and momentum c. Energy-time Uncertainty i. ΔEΔt ~ ħ ii. Energy is not constant in quantum world, but it fluctuates II. Schrödinger’s Equation a. d 2 ψ/dx 2 + (2m/ħ)(E – U)ψ = 0 b. U = potential Energy E – U = K c. If E<U => K is negative NOT POSSIBLE CLASSICALLY d. Semi-Infinite Well e. d 2 ψ/dx 2 = infinity*ψ i. ψ must be 0 for x < 0 ii. If 0<x<L: d 2 ψ/dx 2 = -(2m/ħ 2 )(E – 0)ψ 1. Ψ = Asin(kx) dψ/dx =Akcos(kx) d 2 ψ/dx 2 = -Ak 2 sin(kx) 2. -Ak 2 sin(kx) = -(2mE/ħ 2 )(Asin(kx)) k = sqrt(2mE/ħ 2 ) real number (positive)
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Unformatted text preview: a. works iii. If x > L: ψ(s) = Ae-κx III. Tunneling a. If the classically-forbidden region is narrow or high goes through b. Radioactivity i. Transmission Coefficient T = e-2αa α = sqrt(2m(U – E)/ħ 2 ) ii. Semi-classical ΔEΔt ~ ħ 1. Particle “borrows” energy to get over the barrier has to pay it back in Δt ~ ħ/E a. Can particle do it in time? IV. II. Again a. If E>U Schrodinger Equation d 2 ψ/dx 2 = -constant*ψ, curvature opposite to ψ (towards axis) b. If E < U constant*ψ curves away from axis Uncertainty Principles ΔxΔp ≥ ħ/2 ħ = h/2π Schrödinger Equation d 2 ψ/dx 2 + (2m/ħ 2 )(E – 0)ψ = 0 Tunneling Probability E U E Allowed Forbidden a U...
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This note was uploaded on 04/07/2008 for the course PHYS 212 taught by Professor Ladd during the Spring '08 term at Bucknell.

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