Three-dimensional (3D) isotropic harmonic oscillator. Consider the Hamiltonian of a 3D harmonic oscillator: 4 = p 1 2m+mwx, (1) with p and the 3D momentum and position operators respectively, m the mass, and the w the angular frequency of oscillation. (a) (b) By separating variables, show that the 3D harmonic oscillator can be decomposed into three independent 1D harmonic oscillators. Write down the cor- responding time-independent Schrdinger equations for motions along the x-, y- and z-axes, respectively. The eigenfunctions of the 3D harmonic oscillator can be written as the product of normalized eigenfunctions of three 1-D harmonic oscillators, i.e., Vnz,ny,nz (r) = 4nz (x)4ny (y)4n. (2), (2) where the quantum numbers n; = 0, 1, 2, ... with (i = x, y or 2). Write down the eigenenergies of the 3D harmonic oscillator and determine the degeneracy of each energy level for a given N(= n + ny + nz). (c) Show that the z-component of the angular momentum of the 3D har- monic oscillator can be expressed in terms of the creation and annihilation oper- ators of the 1D harmonic oscillators as follows - (3) (d) (e) Show that the squared total angular momentum operator of the 3D harmonic oscillator can be expressed as =-h [( ) + (! ) + (z x)]. (4) Show that 0,0,1) is a simultaneous eigenket of the angular momentum operators L2 and L (= + + ). Find the respective eigenvalues. Comment: In spherical coordinates, the eigenkets of the Hamiltonian (1) can be expressed in a different basis set: { ,,l,m)}, where n, (= 0, 1, 2, ...) is the quantum number characterizing the radial motion of the electron, and I and my are the conventional azimuthal and magnetic quantum numbers corresponding to and L respec- tively. In this basis, the energy eigenvalues are given by E = (2n, +1+3/2)hw.