# Interpretation of density matrix

I think what he means is with reference to his equation (5.1) $$\psi = \sum_n c_n \psi_n.$$ These $$\psi_n$$ states are in a superposition and this superposition is not the same as saying the system has some probability of being in one state or the other, even though $$|c_n|^2$$ is really the probability of the system being in state $$\psi_n$$ (regular quantum mechanics interference effects). This is manifested in the off diagonal terms of the density matrix. This lead us to the other part of the question.

As you mentioned, density matrix is always interpreted as being a probability of the system of being in some quantum mechanical state. The definition Landau gave for the density matrix is $$w_{mn} = c^*_n c_m,$$ which is clearly a Hermitian matrix and so can be diagonalised, $$w_{\alpha \beta} = c_{\alpha} \delta_{\alpha \beta}.$$ Now in this new basis you can really think of the diagonal terms $$c_{\alpha}$$ as being a probability of the system being in state $$\alpha$$.

It's important to notice that in this diagonal basis you cannot think of the system as being in a superposition of the $$\alpha$$'s, $$\psi \neq \sum_{\alpha} c_{\alpha} \psi_{\alpha}.$$

If this bothers you, I can give you another way to think about this. Instead of equation (5.1) think about the bigger state containing the system and the environment, $$\Psi =\sum_{ij} C_{ij} \theta_i \psi_j,$$ where with Landau, $$\psi_j$$'s are the wavefunctions for the system in consideration, and $$\theta_i$$'s are for the environment. With this, following exactly what Landau has in the book, but for an operator $$f$$ that only act on the system and leave the environment unchanged, we can write the density matrix as, $$w_{j^{\prime} j } = \sum_{i} C^*_{ij^{\prime}} C_{ij}.$$ Notice that this is also Hermitian. Now this sum can be diagonal in $$j$$ and $$j^{\prime}$$ with no problem.