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Fall2009_final_exam_QM_I_5250

# Fall2009_final_exam_QM_I_5250 - Final Examination QM I...

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Final Examination, QM I, Physics 5250, Fall 2009 Problem 1. Consider a particle of mass m that has the following two-dimensional Hamiltonian: H = p 2 x 2 m + p 2 y 2 m + 1 2 2 ( x 2 + y 2 ) . Suppose you are given the one-dimensional energy eigenfunctions for an oscillator of the same mass and frequency, denoted u n ( x ) for the eigenstate at energy E n = ( n + 1 2 ) ~ ω. Using this information, answer the following questions: (a) What are the energy eigenvalues and eigenfunctions of this 2-dimensional Hamiltonian, and what is the degeneracy of each energy level? (b) Suppose there are 4 identical noninteracting fermions, each experiencing the above Hamiltonian and each in the same intrinsic spin substate. Find the ground state energy and wavefunction of the system, and state its degeneracy. Problem 2. . Prove that for a 1-dimensional system, e ipa/ ~ ψ ( x ) = ψ ( x + a ) , where p = - i ~ ∂x is the momentum operator in the position representation. Problem 3 . A coherent state | α i of a one-dimensional harmonic oscillator was written

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