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hw10 - Chemistry 550 475 Homework Assignment#10 due Friday...

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Chemistry 550 / 475 – Homework Assignment #10 due: Friday, December 9th at 5:00 pm Readings : Chapters 10 and 11 in Levine Problem 1 - Perturbative treatment of Helium. Compute the electronic ground-state energy of the Helium atom up to first-order in perturbation theory where the electron- electron repulsion, e 2 / | x 1 x 2 | , is taken as the perturbation Hamiltonian. You may find the following expansion 1 | x 1 x 2 | = summationdisplay lm 4 π 2 l + 1 r l < r l +1 > Y lm ( θ 1 1 ) Y lm ( θ 2 2 ) useful in your calculations. Here, r < ( r > ) are the lesser (greater) of | x 1 | = r 1 and | x 2 | = r 2 and the sum runs over the usual bounds for l and m . How does your energy compare to the observed value of 79 . 01 eV? Problem 2 - More on Helium. [550 only] Problem 9.12 in Levine. In working this problem note that the (arbitrary) choice of unperturbed Hamiltonian H 0 (and, consequently, perturbation Hamiltonian H = H H 0 ) affects the numerical values obtained for the perturbative approximations to the energy and wave function at every order in perturbation theory. Problem 3 - Anharmonic oscillator. Problem 9.10 in Levine. Problem 4 - Energy variance. [550 only] Given a complete set of states { φ n } labeled by n we may expand an arbitrary function χ sharing the same boundary conditions as | χ ) = 1 | χ ) = parenleftBig summationdisplay n | φ n )( φ n | parenrightBig | χ ) = summationdisplay n | φ n )( φ n | χ ) , where the expansion coefficients ( φ n | χ ) are the projections of χ onto the basis { φ n } .
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