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 Schrodinger 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?

 

[Entropia]

, describes the state of a system at time t

\hat{H}(t) - time dependent Hamiltonian, describes the energy of the system at time t

\psi_S(0) - initial wave function at t=0

\hat{S}(t) - S-matrix, describes the scattering of particles in a system

The relation \psi_S(t)=e^{\frac{1}{i\hbar}\hat{H}_0t}\hat{S}(t)\psi_S(0) is correct in the context of the S-matrix formalism. This formalism is used to describe the scattering of particles in a system, and the time dependent Hamiltonian allows for the inclusion of time varying potentials or interactions. The S-matrix itself is independent of time, and is related to the time independent Hamiltonian \hat{H}_0 through the scattering operator \hat{S}(t). The wave function in the Schrodinger picture, \psi_S(t), is related to the wave function in the Heisenberg picture, \psi_H(t), through the unitary transformation \psi_H(t)=e^{\frac{i}{\hbar}\hat{H}t}\psi_S(t). Therefore, the relation \psi_S(t)=e^{\frac{1}{i\hbar}\hat{H}_0t}\hat{S}(t)\psi_S(0) is consistent with the time dependent Hamiltonian and the S-matrix formalism.