852HW4 - − 5 3 3 − 5 3 . ±ind the eigenvalues of H = H...

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PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 4: 1. [20 points] Consider the shifted harmonic oscillator: H = P 2 2 m + 1 2 2 X 2 + aX. Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to Frst-order. 2. [20 points] ±ind the Frst non-vanishing corrections to the ground-state energy and wavefunction of the harmonic oscillator when an anharmonic term is added to the potential. ±irst consider the asymmetric anharmonic oscillator H = P 2 2 m + 1 2 2 X 2 + λX 3 , then do the same for the symmetric anharmonic oscillator H = P 2 2 m + 1 2 2 X 2 + λX 4 . 3. [30 points] Consider a 3-dimensional Hilbert space spanned by the states | 1 a , | 2 a , and | 3 a . Let the bare Hamiltonian be H 0 = 16 4 28 4 1 7 28 7 49 , and consider the situation where this system is perturbed by the operator V = 0 3
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Unformatted text preview: − 5 3 3 − 5 3 . ±ind the eigenvalues of H = H + λV to second-order in λ , and Fnd the corresponding eigenstates to Frst-order. Purely numerical solutions are su²cient. 4. [30 points] Consider a symmetric two-dimensional Harmonic Oscillator, described by H = 1 2 M ( P 2 x + P 2 y ) + 1 2 Mω 2 ( X 2 + Y 2 ) . a.) Determine the Frst three energy levels, and give the degeneracy of each level. b.) Consider the symmetry breaking perturbation V = p ω 2 XY, so that H = H + λV . Use perturbation theory to compute the e³ects of the perturbation of the second energy level. Compute the level splittings to second order in λ , and the corresponding eigenstates to Frst order. 1...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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