# Correct way to write the eigenvector in second quantization

1. Mar 29, 2014

### sam12

I am studying diagonalization of a quadratic bosonic Hamiltonian of the type:

$$H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\dagger a_j^\dagger + B_{ij}^* a_j a_i ]$$

in second quantization.

I can write this in matrix form as
$$H = \frac{1}{2} \alpha^\dagger M \alpha - \frac{1}{2} tr(A)$$
where $$\alpha = \begin{pmatrix} a \\ a^\dagger \\ \end{pmatrix}$$
, $$\alpha^\dagger = \begin{pmatrix} a & a^\dagger \end{pmatrix}$$
and M is $$M = \begin{pmatrix} A & B\\ B^* & A^* \\ \end{pmatrix}$$

The Hamiltonian is called diagonal when it is expressed as:

$$H = \beta^\dagger N \beta - \frac{1}{2} tr(A)$$
where $$\beta = \begin{pmatrix} b \\ b^\dagger \\ \end{pmatrix}$$
, $$\beta^\dagger = \begin{pmatrix} b & b^\dagger \end{pmatrix}$$
and N is a 2-by-2 matrix .

Question: Can I numerically diagonalize the matrix M to get eigenvalues and eigenvectors of the Hamiltonian?
If yes, then what would be the right way to write the eigenvector in second quantization?

e.g. If for 2$\times$2 matrix M, one of the numerically calculated eigenvectors is $$\begin{pmatrix} p\\ q \\ \end{pmatrix}$$ , then, should it be written as $$p \,a|0> + q \, a^\dagger|0>$$
(where the column $\alpha$ has been used as the basis)
or
$$p \,a^\dagger|0> + q \, a|0>$$
(where $\alpha^\dagger$ has been used as the basis)?

Note: where |0> is the vacuum state for 'a' type (bosonic) particles.
End of Question.

Note : Consider a simpler Hamiltonian
$$H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j$$
and note that its eigenvectors are of the form
$$(a_1^\dagger a_2^\dagger ... ) |0>$$