Density matrix

1. Aug 4, 2009

KFC

Hi there,
In all text of QM I have, they tells that the density operator is hermitian. But without considering the math, from the physics base, why density operator must be hermitian? What's the physical significane of the eigenvalue of density matrix?

Thanks

2. Aug 4, 2009

Demystifier

Eigenvalues of the density matrix are probabilities. They clearly must be real, which is why the matrix must be hermitian. (In addition, they also must be non-negative and not larger than 1.)

3. Aug 4, 2009

KFC

Thanks. So before diagonalization, the diagonal elements of density matrix don't tell anything , right?

4. Aug 4, 2009

KFC

Well, I understand now how the diagonalized density matrix works. But for the following case how do I use it? For example, I have a two-level system without interaction, and hamiltonian gives

$$H|\Psi_n\rangle = \hbar\omega_n |\Psi_n\rangle$$

Now I consider the heisenberg equation of the density operator

$$\dot{\rho} = -\frac{i}{\hbar}[H,\rho]$$

to findout the element of $$\dot{\rho}_{nm}$$, I apply an eigenstate of H on both side

$$\dot{\rho}|\Psi_i\rangle = -\frac{i}{\hbar}(H\rho - \rho H)|\Psi_i\rangle = -\frac{i}{\hbar}(H-\omega_i)\rho|\Psi_i\rangle$$

So, what is $$\rho|\Psi_i\rangle$$? In the text, it gives

$$\dot{\rho}_{nm} = -i\omega_{nm}\rho_{nm}$$ ???

5. Aug 5, 2009

Jano L.

Real eigenvalues does not imply hermiticity - for example, look at "[URL [Broken] Hermiticity of the density matrix follows from its definition, which is

$$\rho := \sum_k p_k |\psi_k\rangle\langle\psi_k|$$

where $$p_k$$ is the probability that system will be found in the state $$\psi_k$$ (states $$\psi_k$$ can be arbitrary possible states of the system).

Last edited by a moderator: May 4, 2017