# Finding the eigenstates and eigenvalues of a system of three not interacting electrons

I am trying to solve a problem on identical particles where there are three not interacting electrons. It's known that the Hamiltonian of a particle $h_{0}$ acts only on the orbital variables with non degenerated eigenvalues $0$,$\hbar\omega$ and $2\hbar\omega$ (where $\omega$ is a positive constant). The Hamiltonian of the system composed by three independent electrons can be written as

$$H=h_{0}(1)+h_{0}(2)+h_{0}(3)$$

Since they are fermions I started defining a state $|u \rangle$ given by

$$|u \rangle =|1,\phi ;2,\alpha; 3,\beta\, \rangle$$

I wrote the antisymmetric normalized state $|\phi ;\alpha;\beta\, \rangle$ obtained from $|u \rangle$ using the Slater determinant.

So I applied the operator $\hat H$ to $|\phi ;\alpha;\beta\, \rangle$

$$\hat H |\phi ;\alpha;\beta\, \rangle = [h_{0}(1)+h_{0}(2)+h_{0}(3)]|\phi ;\alpha;\beta\, \rangle$$

From this point on I am struggling to find the eigenvalues and eigenstates of the system. I am facing difficulties to understand where are the eigenvectors in this case and once they are found, how I find the eigenvalues.

• Since the particles are non-interacting, the wave function is a product state (a Slater determinant). This is a linear combination of single-particle eigenstates $\phi(1) \alpha(2) \beta(3)$. Each part of the Hamiltonian only acts on one of the three single-particle wave functions. – Praan Nov 29 '15 at 10:32