# Those Pictures (Representations) in QM and the density equations

1. Jul 10, 2014

### 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 Schrodinger 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 Schrodinger 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 Schrodinger 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!

2. Jul 10, 2014

### Robert_G

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

3. Jul 10, 2014

### 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