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1) A particle in two dimensions is described by a wave function v(r, y). We can make a variable substitution to cylindrical variables (x, y) - (r, ) by defining an alternative wave function in terms of the new variables which describes the exact same state: D(r, 6) = w(x,y) = (rcos(), rsino). Using the definition of L, in position (c, y) space, show that its representation in terms of these new variables is L. = -ihe . Hint: Simply apply this form of L, to O(r, $) and show that you get the same answer as applying L, in Cartesian coordinates to v(r, y). 2) A particle of mass m is constrained to move in a circle of fixed radius R around the origin in the x - y plane in the absence of any potential energy. a) Using the generalized coordinates q = o and po = Ly, write down the classical Hamiltonian for this system. b) Now assume that the same system in quantum mechanics is governed by a Hamil- tonian operator of the same functional form, i.e., by replacing L, with the opera- tor L,. Find the eigenfunctions o($) and the eigenvalues E; of this Hamiltonian. Make sure your solutions are "single-valued" (meaning for the same physical point in space, the eigenfunctions have the same value). What are allowed energies E;? c) What can you say about the difference in energy of two eigenfunctions? What regular pattern would you observe in the frequency spectrum of photons emitted by this system when it undergoes a transition from some excited state to the ground state? d) Which kinematic quantities (operator expectation values) are conserved (inde- pendent of time) for this system (even when it is not in an eigenstate of the Hamiltonian)? List two examples