Started from independent particle model, the Hamiltonian of a nucleus with mass number is

we can rewrite the Hamiltonian by isolating a nucleon

than, we can use the basis of and to construct the wavefunction of the nucleus A as

where the square bracket is anti-symmetric angular coupling between single particle wavefunction and wavefunction . The is the spectroscopic amplitude.

The square of the spectroscopic amplitude times number of particle at state is the spectroscopic factor of the nucleon at state and nucleus B at state

The occupation number is the sum of the spectroscopic factors of the nucleus B

After the definition, we can see, when the nucleon-core interaction is neglected, the spectroscopic factor is 1 and the occupation number is also 1.

Since the is coefficient for changing basis from into basis of Thus, the matrix is unitary

thus, each column of row vector of is normalized.

The properties of can be found by solving the eigen system of the from the core . The core hamiltonian is diagonal. The nucleon-core interaction introduce diagonal terms and off-diagonal terms. When only diagonal terms or monopole term exist, only the eigen energy changes but the eignestate unchanges. Therefore, the configuration mixing is due to the off-diagonal terms.

However, when there are off-diagonal terms, the change of diagonal terms will changes the mixing.

In degenerate 2 states system, the Hamiltonian be

The eigen energy are , where and , The eigen vector are

When the states are degenerated, , the eigen energy is , and the eigen state is

As we can see, the eigen state only depends on the difference of the energy level, thus, we can always subtract the core energy and only focus on a single shell. For example, when we consider 18O, we can subtract the 16O binding energy.

In above figure, we fixed the and . We can see the spectroscopic factor decreases for the lower energy state (red line), and the state mixing increase.

We now fixed the and . When , the particle stays more on the state, which is lower energy. The below plot is the eigen energy. Even though the mixing is mixed more on the excited state, but the eigen energy did not cross.

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