ps5Xtra - x t in quantum mechanics i ∂u ∂t = ∂ 2 u...

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MATH 405: Problem Set 5 Additional Problems due Monday 10/23/2006 1. Consider Laplace’s equation: 2 φ = 0 in 2D Cartesian coordinates, i.e. φ = φ ( x, y ). Assume a separation of variables form φ ( x, y ) = f ( x ) g ( y ) and find the ODEs for f and g . Solve these explicitly for every case of the separation constant - positive, negative, and zero! How do things change if you consider this problem on a square domain, with boundary conditions φ = 0 when x = 0 or x = L , and also φ = 0 when y = 0 or y = L (all edges of the square)? 2. Consider the 1D Schr¨ odinger equation for a wavefunction u (
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Unformatted text preview: x, t ) in quantum mechanics: i ∂u ∂t = ∂ 2 u ∂x 2 Use separation of variables to show that this can have travelling waves solutions, i.e. that u ( x, t ) can look like e i ( kx-wt ) , where k and ω are constants (which is equivalent to sin( kx-ωt ) and cosines). 3. Show that Laplace’s equation ∇ 2 φ = 0 in 2D cylindrical coordinates for φ = φ ( r, θ ) leads to Bessel’s equation (Kreyszig Section 5.5 Eq 1, with ν = 0) in one of the variables. What is the other ODE which results?...
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This note was uploaded on 07/23/2008 for the course MATH 405 taught by Professor Belmonte during the Fall '06 term at Penn State.

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