# Jaynes-Cummings Model Lagrangian

1. Nov 10, 2012

### lxhrk9

1. The problem statement, all variables and given/known data

I am trying to write down the path integral for the Jaynes-Cummings Model which involves obtaining the Lagrangian.

2. Relevant equations

$$\hat{H}_{\text{JC}} = \hbar \nu \hat{a}^{\dagger}\hat{a} +\hbar \omega \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right)$$

3. The attempt at a solution

To get the Lagrangian from the Hamiltonian is it reasonable to write the creation and annihilation operators in terms of x and p, solve
$$\dot{x}=\frac{\partial H(x,p)}{\partial p}$$
for p and plug this into
$$L(x,\dot{x};t)=\dot{x}p−H(x,p;t)$$?

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