(7, 38) The potential energy for the isotropic three-dimensional harmonicoscillator in Cartesian coordinates is U(x,y,where the force constant k is the same in all directions. (a) Guidedby the discussions of the one-dimensional harmonic oscillator and thetwo-dimensional infinite well in Chapter 5, show that the energies of thethree-dimensional oscillator are given by Ennynz(b) What are the quantum numbers and corresponding degeneracies ofthe ground state and the first two excited states? (c) We can also solvethis problem in spherical polar coordinates, with U(r)x² +y? + z?. The solution is of the form of Eq. 7.11 with radial functionsdifferent from those of the hydrogen atom but the same angular functions.z) = }kx² + ky? + kz2,= (nx + Ny +nz +)hw.+ nz2= }kr² and p2is En = (n + hw independent of l and m. The relationshipbetween n and l is different from what it is for hydrogen: 1 < n, and nand l are either both odd or both even. Explain how the degeneraciesof the ground state and the first two excited states arise in the sphericalpolar solutions.

Solution: The energy of the three-dimensional oscillator is given as follows:

Answered: (7, 38) The potential energy for the…

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