I'm studying a hole quantum dot in Germanium. The problem is described in the Luttinger Spin-3/2 basis and takes Fock-Darwin states as solutions in the Quantum Dot plane(xy plane). Furthermore, I assume an external artificial harmonic oscillator laboratory confinement on the Quantum dot in the out-of-plane z-direction. And so the z-wavefuctions are the ground state and the first excited states of a harmonic oscillator. These represent ground state and first excited state of this artificial confinement. There is also an external static magnetic field in the z direction.
My question: What is the relationship between the Zeeman Energy and the excited states of the artificial laboratory imposed confinement ? How would the Zeeman Energy change from its usual g $\mu_B B m_j$ value when the Zeeman Hamiltonian is being evaluated between excited states in the confinement ? For e.g. $<\psi_{z_{0}}|H_Z|\psi_{z_{1}}>$ or $<\psi_{z_{1}}|H_Z|\psi_{z_{0}}>$ or $<\psi_{z_{1}}|H_Z|\psi_{z_{1}}>$ ?
The Zeeman effect is calculated by $<n\,l\,j\,m_j|H_Z|n\,l\,j\,m_j>$ and none of those quantum numbers change with higher order in confinement, so I don't know how to come up some theory that couples the two.
I can't find any literature on this nor anything that is covered in any textbook, but my advisor is convinced there is some interaction here which needs to be found. Any help/guidance will be greatly appreciated
Thanks for reading
