(Solved):   1. A three-dimensional isotropic harmonic oscillator with Hamiltonian \[ \widehat{H}=\frac ...

 

1. A three-dimensional isotropic harmonic oscillator with Hamiltonian \[ \widehat{H}=\frac{\widehat{\mathrm{p}}^{2}}{2 m}+\frac{K r^{2}}{2}, \quad K>0, r^{2}=x^{2}+y^{2}+z^{2}, \] is an instance of quantum-mechanical systems, where a particle of mass \( m \) is exposed to a spherically-symmetric potential. The specific nature of the harmonic potential however also results in extra degeneracies, beyond mandated by the spherical symmetry. In this problem you are given stationary state \( |\psi\rangle \) of this system: \[ \widehat{H}|\psi\rangle=E|\psi\rangle, \] which is known to be the lowest-energy state with the \( z \)-projection of the orbital angular momentum equal to \( 4 \hbar \) : \[ \widehat{L}_{z}|\psi\rangle=4 \hbar|\psi\rangle \] (a) What is energy \( E \) of this state in Eq. (2) in terms of the Hamiltonian parameters \( m \) and \( K \) ? (b) How many linearly independent stationary states \( |\varphi\rangle \) of the same energy \( E \) are there with the \( z \)-projection of the orbital angular momentum equal to \( \hbar \) : \[ \widehat{L}_{z}|\varphi\rangle=\hbar|\varphi\rangle ? \]