Question

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Problem 1: Cartesian vs. spherical coordinates: the amazing isotropic oscillator. A particle of mass m moves in a three-dimensional isotropic harmonic oscillator potential V(n)= Kr2 (1) with K = mwa positive constant. a) Solve the time-independent Schrödinger equation by doing separation of variables in Cartesian coordinates and determine the energy eigenvalues and their degeneracy. b) If the particle has energy E = 5hw/2 write all the linearly independent wavefunctions that can describe it in separated Cartesian coordinate form. c) Solve the time-independent Schrödinger equation by doing separation of variables in spherical coordinates. For each value of the angular momentum l solve the radial equation by expanding the radial wavefunction u(r) as a power series starting with power pl+1 and find the corresponding energy eigenvalues and degeneracies. d) If the particle has energy E = 5hw/2 write all the linearly independent wavefunctions that can describe it in separated spherical coordinate form. e) Express each wavefunction of part (b) as a linear combination of wavefunctions of part (d), and each wavefunction of part (d) as a linear combination of wavefunctions of part (b).

Answer 1