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HMWR2 - 2 Particle in a square box(40 pts 2.1 Consider a...

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110A Physical Chemistry: Quantum Mechanics Homework #2: assigned Oct. 11 th , 2010; due Oct. 18 th 1. Eigenvalues and eigenfunctions (40 pts) 1.1 Show that ψ (x,y,z) = cos(ax)*cos(by)*cos(cz) is an eigenfunction of the Laplacian operator in three dimensions and find the eigenvalue; a, b, c are constants. 1.2 Find a linear, differential operator of which the function cos ( ω x) is an eigenfunction; find a linear, differential operator of which the function exp (i ω t ) is an eigenfunction. Find the corresponding eigenvalues. 1.3 If ψ (x) is an eigenfunction of the Hamiltonian H (particle in a box, 1D), is any function [a ψ (x)], with a = generic constant, also an eigenfunction? Explain why.
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Unformatted text preview: 2. Particle in a square box (40 pts) 2.1 Consider a planar molecule with 18 electrons and approximate its energy levels with those of a free particle in a two-dimensional, square box of side a . What are the energy levels of such a molecule and their degeneracies? Consider a = 1000 pm: what is the predicted lowest energy absorption of the molecule? Would the lowest absorption increase or decrease for a bigger planar molecule? Explain why. 3. Uncertainty Principle (20 pts) 3.1 Show that when the quantum number n is large, the mean square deviation of the position of a quantum particle in a box agrees with its classical value....
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