In previous post, the number of combination between the core and the single particle state becomes massive as the number of m-states are huge. To simplify the coupling, assuming the interaction between the state and are the same for all . The Hamiltonian matrix can be reduced.

To demonstrate the idea, suppose the core only has a ground state. And the single-particle state has degenerated state with different m-state, say,

Lets the coupled states be

The Hamiltonian matrix is

To give a concrete example, say

The eigenstates and eigenvalues are

We can see, there are 2 states the energies do not change and the coefficient just re-configured.

Let’s look at the eigen system of this matrix:

I wish to prove it more general as a theorem, but I can just worked on few cases.