# Understanding completeness relation and writing Hamiltonian in matrix form

A three level system hamiltonian I found where it is written as:

$$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |g_2\rangle \langle e|$$

Where the $$\Omega_1$$, $$\Omega_2$$, are two complex coupling parameters for the normalized states $$|g_1\rangle$$,$$|g_2\rangle$$ and $$|e\rangle$$.

I do not understand how they write the Hamiltonian in the matrix form like this?? $$\begin{vmatrix} 0&0&\Omega\\ 0&0&\Omega\\ \Omega&\Omega&0\\ \end{vmatrix}$$

For real $$Ω_1 = Ω_2 = Ω$$.

It would be helpful if you disintegrate the the Hamiltonian in much simpler form so I can understand it better.

• Can you give me example of just one component for example $H_{11}$ matrix ? Jul 31 '19 at 6:24
• @user193422 : Just multiply the Hamiltonian by $<g_1|$ from the left and by $|g_1>$ from the right. Jul 31 '19 at 6:45