Density matrix

  • Thread starter KFC
  • Start date
  • #1
KFC
488
4

Main Question or Discussion Point

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
 

Answers and Replies

  • #2
Demystifier
Science Advisor
Insights Author
10,558
3,319
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
KFC
488
4
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.)
Thanks. So before diagonalization, the diagonal elements of density matrix don't tell anything , right?
 
  • #4
KFC
488
4
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

[tex]H|\Psi_n\rangle = \hbar\omega_n |\Psi_n\rangle[/tex]

Now I consider the heisenberg equation of the density operator

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

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

[tex]\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
[/tex]

So, what is [tex]\rho|\Psi_i\rangle[/tex]? In the text, it gives

[tex]\dot{\rho}_{nm} = -i\omega_{nm}\rho_{nm}[/tex] ???
 
  • #5
Jano L.
Gold Member
1,333
74
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.)
Real eigenvalues does not imply hermiticity - for example, look at "[URL [Broken] Hermiticity of the density matrix follows from its definition, which is

[tex]
\rho := \sum_k p_k |\psi_k\rangle\langle\psi_k|
[/tex]

where [tex]p_k[/tex] is the probability that system will be found in the state [tex]\psi_k[/tex] (states [tex]\psi_k[/tex] can be arbitrary possible states of the system).
 
Last edited by a moderator:

Related Threads for: Density matrix

  • Last Post
Replies
5
Views
932
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
16
Views
651
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
1
Views
991
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
1K
Top