# Isotropic harmonic oscillator in polar versus cartesian

I read another Phys.SE post here: 3D Quantum harmonic oscillator that I believe says the wave function in Cartesian coordinates for a 3D harmonic oscillator is the product of the 3 one dimensional wave functions. If so, are the angular and radial equations combined within it? Is it possible to separate the radial part from the angular part?

The wavefunction for 1D in the ground state is $$\psi_{0}(x)=(\frac{m\omega}{\pi\hbar})^{\frac{1}{4}}e^{-\frac{m\omega x^{2}}{2\hbar}}.$$

I think this product would then be: $$\psi_{0}(r)=(\frac{m\omega}{\pi\hbar})^{\frac{3}{4}}e^{-\frac{m\omega r^{2}}{2\hbar}}.$$

Is this the radial equation for the ground state? Would I then need to multiply by the angular equation for the ground state to get the full wave equation?