Exercise 4: First simple shell-model calculation

We will now consider a simple three-level problem, depicted in the figure below. This is our first and very simple model of a possible many-nucleon (or just fermion) problem and the shell-model. The single-particle states are labelled by the quantum number $p$ and can accomodate up to two single particles, viz., every single-particle state is doubly degenerate (you could think of this as one state having spin up and the other spin down). We let the spacing between the doubly degenerate single-particle states be constant, with value $d$. The first state has energy $d$. There are only three available single-particle states, $p=1$, $p=2$ and $p=3$, as illustrated in the figure.

a) How many two-particle Slater determinants can we construct in this space? We limit ourselves to a system with only the two lowest single-particle orbits and two particles, $p=1$ and $p=2$. We assume that we can write the Hamiltonian as $$\hat{H}=\hat{H}_0+\hat{H}_I,$$ and that the onebody part of the Hamiltonian with single-particle operator $\hat{h}_0$ has the property $$\hat{h}_0\psi_{p\sigma} = p\times d \psi_{p\sigma},$$ where we have added a spin quantum number $\sigma$. We assume also that the only two-particle states that can exist are those where two particles are in the same state $p$, as shown by the two possibilities to the left in the figure. The two-particle matrix elements of $\hat{H}_I$ have all a constant value, $-g$.

b) Show then that the Hamiltonian matrix can be written as $$\left(\begin{array}{cc}2d-g &-g \\ -g &4d-g \end{array}\right),$$

c) Find the eigenvalues and eigenvectors. What is mixing of the state with two particles in $p=2$ to the wave function with two-particles in $p=1$? Discuss your results in terms of a linear combination of Slater determinants.

d) Add the possibility that the two particles can be in the state with $p=3$ as well and find the Hamiltonian matrix, the eigenvalues and the eigenvectors. We still insist that we only have two-particle states composed of two particles being in the same level $p$. You can diagonalize numerically your $3\times 3$ matrix.

This simple model catches several birds with a stone. It demonstrates how we can build linear combinations of Slater determinants and interpret these as different admixtures to a given state. It represents also the way we are going to interpret these contributions. The two-particle states above $p=1$ will be interpreted as excitations from the ground state configuration, $p=1$ here. The reliability of this ansatz for the ground state, with two particles in $p=1$, depends on the strength of the interaction $g$ and the single-particle spacing $d$. Finally, this model is a simple schematic ansatz for studies of pairing correlations and thereby superfluidity/superconductivity in fermionic systems.

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