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S - matrix

  1. Nov 4, 2009 #1
    When I have time dependent Hamiltonian

    [tex]\hat{H}(t)=\hat{H}_0+\hat{V}(t)[/tex]

    Is then this relation correct?


    [tex]\psi_S(t)=e^{\frac{1}{i\hbar}\hat{H}_0t}\hat{S}(t)\psi_S(0)[/tex]

    where [tex]\hat{S}(t)[/tex] is S-matrix.


    [tex]\psi_S(t)[/tex] - wave function in Schrodinger picture
     
  2. jcsd
  3. Nov 4, 2009 #2
    If I want time dependent density matrix is it

    [tex]\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}[/tex]

    Is this expression OK?

    So I have


    [tex]\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}[/tex]

    Does it make sense?
     
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