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5-Wave Particle Duality of Radiation and Matter

# 5-Wave Particle Duality of Radiation and Matter - 5.111...

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5.111 Lecture Summary #5 September 18, 2009 5.1 Reading for today: Section 1.3, Section 1.7 up to equation 9b Reading for Lecture 6: Section 1.9 Wave-Particle Duality of Radiation and Matter III. Matter as a wave The Schrödinger Equation & H atom states Let’s go back to classical waves, like those for light. We saw that they were described by expressions such as E(x,t) = a cos(2 π x/ λ - 2 πν t) This equation is actually a solution to a more general description of a light wave as a differential equation ( ) ( ) ∂∂ = 22 2 Ex ,t 1 xv t where v is the velocity of the wave (v=c) This differential equation is the wave equation, and its solutions are classical waves with velocity v. Shrödinger suggested that matter waves could be described using a similar looking wave equation, which he wrote down. ( ) () () () ∂ψ −+ ψ = ψ = 2 2 2 x Ux x E x 2m x Also written as ψ ˆ HE where () =− + = 2 ˆ HU x 2m x is the Hamiltonian operator where = π = h 2 , U(x) is the potential energy, E is the total energy, and ψ (x) is the “wavefunction” for the matter wave (similar to E(x,t) for light)

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5.111 Lecture Summary #5 September 18, 2009 5.2 This is the time independent Schrödinger equation in one dimension, which governs the “motion” of matter at microscopic scales, like an electron in an atom! We will not prove Schrödinger’s equation (it can’t be proven), but we can motivate why it might be reasonable. Let’s start by describing an e - with a wavefunction similar to a wave of light π ⎛⎞ ψ= ⎜⎟ λ ⎝⎠ 2x (x) acos Is there a differential equation, like the wave equation for light, that governs this wavefunction?
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5-Wave Particle Duality of Radiation and Matter - 5.111...

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