pset2 - perturbation Use second order perturbation theory...

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PH 216, Problem set 2, Due 4/13 1. A mass m is attached by a massless rod of length l to a pivot p and swings in a vertical plane under the inFuence of gravity, with deFection angle θ from the vertical. (a) In the small angle approximation ±nd the energy levels of the system. (b) ²ind the lowest order correction to the ground state energy resulting from inac- curacy of the small angle approximation. 2. A diatomic molecule behaves like a rigid rotator with a moment of inertia I = Mr 2 0 , where M is the reduced mass and r 0 is the distance between the atoms. The Hamil- tonian can be approximated by: H = ° L 2 2 I . (1) Assume that the molecule consists of two atoms of charge ± e , separated by a distance r 0 . Suppose that a uniform electric ±eld is present which points in a ±xed direc- tion. Compute the energy eigenvalues of the system, treating the electric ±eld as a
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Unformatted text preview: perturbation. Use second order perturbation theory. HINT : There is one hard integral. This can be done trivially, if you know the following Clebsch-Gordon coefficient: ° ° m , 1 0 | ° − 1 m , ° 1 ² = − ° ° 2 − m 2 ° (2 ° + 1) ± 1 / 2 . 3. Two non-identical particles, each of mass m , are con±ned in one dimension to an impenetrable box of length L . What are the wave functions and energies of the three lowest-energy states (i.e. in which at most one particle is excited out of its ground state)? If an interaction potential of the form V 12 = λδ ( x 1 − x 2 ) is added, calculate to ±rst order in λ the energies of these three lowest states, along with their wave functions to zeroth order in λ . 2...
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This document was uploaded on 10/31/2011.

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