(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have been given the Hamiltonian

[tex]H = \sum_{k}\left(\epsilon_k - \mu\right) c_k^{\dag} c_k + \gamma \sum_{kp}c_k^{\dag} c_p[/tex]

and also that

[tex]c_p = \sum_{q} U_{pq} b_q[/tex]

I have to prove that this matrix [tex]U_{pq}[/tex] is unitary, and find an equation for [tex]U_{pq}[/tex].

2. Relevant equations

This is equivalent to proving that

[tex]\{b_q, b_p\} = 0[/tex]

and

[tex]\{b_q , b_p^{\dag}\} = \delta_{pq}[/tex]

where [tex]b[/tex] and [tex]c[/tex] are creation and annihiliation operators.

3. The attempt at a solution

Knowing that

[tex]c_p = \sum_{q} U_{pq} b_q[/tex]

then

[tex]c_q = \sum_{p} U_{pq} b_p[/tex]

and

[tex]\{b_q , b_p\} = b_q b_p + b_p b_q[/tex]

[tex]c_p b_p = \left(\sum_{q} U_{pq} b_q\right) b_p[/tex]

[tex]b_q c_q = b_q \left(\sum_{p} U_{pq} b_p\right)[/tex]

So that

[tex]c_p b_p + b_q c_q = \left(\sum_{q} U_{pq} b_q\right) b_p + b_q \left(\sum_{p} U_{pq} b_p\right)[/tex]

Hmm, now what?

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# Homework Help: Proving a matrix is unitary

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