486_HW8 - Physics 486 Homework Set 8 Due: Nov 8, 2007 1....

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Physics 486 Homework Set 8 Due: Nov 8, 2007 1. The Anisotropic 2D Simple Harmonic Oscillator – Consider a particle of mass m moving in an anisotropic two-dimensional simple harmonic oscillator potential: ( ) 2 2 2 2 1 1 , 2 2 x y V x y m x m y ω = + , where ω x ω y . (a) Write down the eigenstates for a particle confined to the anisotropic 2D harmonic oscillator (No calculations required!). (b) Write an expression for the energy eigenvalues of this system (No calculations required!). Are there degenerate states associated with this potential? (c) Rewrite the potential V(x,y) above in terms of polar coordinates, r and ϕ , i.e., V(r, ϕ ). Calculate the commutator between the anisotropic 2D harmonic oscillator Hamiltonian and the angular momentum operator , i.e., calculate [H,L z ] (Hint: Write the Hamiltonian in polar coordinates). Do these operators commute? If not, describe any conditions under which the operators would commute. 2) Another problem in higher dimensions.
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This note was uploaded on 09/20/2010 for the course PHYS 486 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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486_HW8 - Physics 486 Homework Set 8 Due: Nov 8, 2007 1....

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