Consider a 3D harmonic oscillator for which the potential is 1 U(x, y, z) = -mw(x...

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Consider a 3D harmonic oscillator for which the potential is 1 U(x, y, z) = -mw²(x² + y² + z²), (1) where m is the mass of the oscillator, and wo is the angular frequency. The term "isotropic” means that the same wo applies to all the three directions. (a) Write down the time-independent Schrödinger equation in Cartesian coordinates for this problem. (b) Show that by separation of variables, the 3D problem can be turned into three 1D harmonic oscillator problems. (c) Recall the eigenvalues En = ħwo(n+1/2) and eigenfunctions un(x), where n = 0, 1, 2, ..., of a 1D harmonic oscillator. Write down the energy eigenvalues and eigenfunctions for the 3D harmonic oscillator. (d) Write down the lowest four allowed energies (different values) in ascending order and identify the degeneracy of each value. Consider a 3D harmonic oscillator for which the potential is 1 U(x, y, z) = -mw²(x² + y² + z²), (1) where m is the mass of the oscillator, and wo is the angular frequency. The term "isotropic” means that the same wo applies to all the three directions. (a) Write down the time-independent Schrödinger equation in Cartesian coordinates for this problem. (b) Show that by separation of variables, the 3D problem can be turned into three 1D harmonic oscillator problems. (c) Recall the eigenvalues En = ħwo(n+1/2) and eigenfunctions un(x), where n = 0, 1, 2, ..., of a 1D harmonic oscillator. Write down the energy eigenvalues and eigenfunctions for the 3D harmonic oscillator. (d) Write down the lowest four allowed energies (different values) in ascending order and identify the degeneracy of each value.