7. A harmonic oscillator in the two-dimensional (2D) space has a mass m and is in a potential V 2 also:2 + yz) a) What are the energy and degeneracy of the ground state and rst excited state? (4 pt) b) The ground state of a one-dimensional (1D) harmonic oscillator is a Gaussian Jnction, that is 1/139 (x) = cue\"r"2 for x space. Ca and a are constants dependent on m and k, which you do not need to solve for this problem. Please use these constants to write down the wave1nction( s) of the ground state(s) of the 2D harmonic oscillator 1,030 (x, y). (1 pt) c) The rst excited state ofa 1D harmonic oscillator along the x axis can be written as a Gaussian function times x, that is 111330;?) = Clare\"x2, where a is the same constant as that in c). Use the constants in b) and c) to write down the wavefunction(s) of the first excited state( s) of the 2D harmonic oscillator. ( 1 pt) (1) The gradient operator can be expressed in the 2D polar coordinates as V: f' :7 + 9 % . What is the angular momentum operator along the z axis, L2, in the polar coordinates? (2 pts) (5) Please construct an eigenfunction osz using the wavefunctions you found in c) and identify its corresponding eigenvalue. (2 pts) (Hint: Is V a central potential? What does that imply?) Q7 7a) Eground = hw, degeneracy = 1. The first excited state has energy Eexc = 2hw, degeneracy = 2. 7b) The wavefunction is 100) = 4," (x, y) = che-a(x7+>?) 7c) The wavefunctions are (10) = Cocixe "( +y), 101) = cocive (12+2) 7d) Lz = -in- 7e) 15 ( 110) + 1 101)) = 091 ide with eigenvalue h, and [10) - 7101)) = COC1-it p-arz V2 with eigenvalue - h