Is the S-matrix Consistent with Time Dependent Hamiltonian?

[Petar Mali]

When I have time dependent Hamiltonian

$$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$

Is then this relation correct?


$$\psi_S(t)=e^{\frac{1}{i\hbar}\hat{H}_0t}\hat{S}(t)\psi_S(0)$$

where $$\hat{S}(t)$$ is S-matrix.


$$\psi_S(t)$$ - wave function in Schrödinger picture

 

[Petar Mali]

If I want time dependent density matrix is it

$$\hat{\rho}_t=e^{\frac{1}{i\hbar}\hat{H}_0t}\hat{S}(t)\hat{\rho}_H\hat{S}^{-1}(t)e^{-\frac{1}{i\hbar}\hat{H}_0t}$$

Is this expression OK?

So I have


$$\left\langle \hat{A} \right\rangle_t=Tr(\hat{A}_S\hat{\rho}_t)=\left\langle \hat{S}^{-1}(t)\hat{A}_H(t)\hat{S}(t) \right\rangle_{\hat{\rho}_H}$$

Does it make sense?