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.


You can just check how the Hamiltonian acts on the three normalized states.

  • $\begingroup$ Can you give me example of just one component for example $H_{11}$ matrix ? $\endgroup$
    – user193422
    Jul 31 '19 at 6:24
  • $\begingroup$ @user193422 : Just multiply the Hamiltonian by $<g_1|$ from the left and by $|g_1>$ from the right. $\endgroup$
    – akhmeteli
    Jul 31 '19 at 6:45
  • $\begingroup$ Be careful, that is the form of the Hamiltonian in the base {|e>, |g_1>, |g_2>}. When you write a matrix explicitely you have always to specify in which base this is done. $\endgroup$ Jul 31 '19 at 9:53

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