Question 3. Problem 3Consider a particle of mass , whose Hamiltonian is:1Ho =2+2▬▬2(an isotropic three-dimensional harmonic oscillator), where wo is a given positive constant.a) Find the energy levels of the particle and their degree of degeneracy. Is it possible toconstruct a basis of eigenstates common to Ho, L2, L₂?Q SearchIb) Now, assume that the particle with a charge q is placed in a uniform magnetic fieldB parallel to Oz. We set w = -qB/2u. Choosing the gauge A = -x B,the Hamiltonian H of the particle is then, (Note: you will need to use minimalsubstitution to include the effect of the gauge field in the Hamiltonian).H = Ho+ H1(L)where H₁ is the sum of an operator that is linearly dependent on wL (a paramagneticterm) and an operator that is quadratically dependent on wL (a diamagnetic term).Write explicitly the form of the Hamiltonian H and show that the new stationarystates of the system and their degrees of degeneracy can be determined exactly.

Consider a particle of mass u, whose Hamiltonian is: p2 1 Ho = 27+R² Но (an isotropic three-dimensional harmonic oscillator), where wo is a given positive constant. a) Find the energy levels of the particle and their degree of degeneracy. Is it possible to construct a basis of eigenstates common to Ho, L², L₂? b) Now, assume that the particle with a charge q is placed in a uniform magnetic field B parallel to Oz. We set w = -qB/2u. Choosing the gauge A = - x B,

Consider a particle of mass u, whose Hamiltonian is: p2 1 Ho = 27+R² Но (an isotropic three-dimensional harmonic oscillator), where wo is a given positive constant. a) Find the energy levels of the particle and their degree of degeneracy. Is it possible to construct a basis of eigenstates common to Ho, L², L₂? b) Now, assume that the particle with a charge q is placed in a uniform magnetic field B parallel to Oz. We set w = -qB/2u. Choosing the gauge A = - x B,