Those Pictures (Representations) in QM and the density equations

[Robert_G]

Hi there:

I am reading a book (Atom-Photon interaction by Claude Cohen-Tannoudji, Page 448) and the following things gave a big headache.

(1) Is there a density equation in Schrödinger Picture. because I encounter one, like:
$i \hbar \frac{d \sigma}{dt}=[\hat{H}, \sigma]$
and $\hat{H}$ contains the Hamiltonian of the atom, photon, and there interaction. So this is in Schrödinger Representations. right?

(2) The correlation $\langle \mathscr{L}_+(\tau)\mathscr{L}_-(0)\rangle$ is calculated step by step, from the equation in (1). So this is also in Schrödinger Representation. But the "double" correlation $\langle \mathscr{L}_+(t)\mathscr{L}_+(t+\tau)\mathscr{L}_-(t+\tau)\mathscr{L}_-(t)\rangle$ is in Heisenberg Representation, and this is clearly stated in the book, because, as the book said, the operators in that "double" correlation are in Heisenberg Representation. So those two correlations are from different Represetations?

Ps: $\mathscr{L}_+$ is the atomic upper operator, and $\mathscr{L}_-$ is the atomic lower operator.

HELP ME!

 

[Robert_G]

It's a rough journey to learn this things, oh, my brain.

 

[kith]

(1) Supposing that sigma is the density matrix, yes, you are using the Schrödinger picture

(2) Expectation values don't really specify which picture is being used. You can convert between pictures by rearranging the time evolution operator: <C(t)> = tr{σ C(t)} = tr{σ U(t)⁺CU(t)} = tr {U(t)σU⁺(t) C} = tr {σ(t) C} = <C>_t