Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Correct way to write the eigenvector in second quantization

  1. Mar 29, 2014 #1
    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
    $$(a_1^\dagger a_2^\dagger ... ) |0> $$
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Correct way to write the eigenvector in second quantization
  1. Second Quantization (Replies: 4)

  2. Second quantization (Replies: 6)

  3. Second Quantization (Replies: 1)

  4. Second Quantization (Replies: 8)

  5. Second quantization (Replies: 9)

Loading...