Expressing the density matrix in matrix form

[Morbidly_Green]

Homework Statement

Given the above lambda system, is it wrong to say that the density matrix is of the form $\rho = c_1|1> + c_2|2> + c_3|3>$ ? Hence when written in matrix form (basis of $|i>$), $\rho$ is a diagonal matrix who's elements are the $c_i$s?

 

[DrClaude]

Given the above lambda system, is it wrong to say that the density matrix is of the form $\rho = c_1|1> + c_2|2> + c_3|3>$ ? Hence when written in matrix form (basis of $|i>$), $\rho$ is a diagonal matrix who's elements are the $c_i$s?

Yes, it is wrong because that $\rho$ is not a density operator, just a quantum state (ket). For pure states, the density operator will be a projection operator,
$$
| \psi \rangle \rightarrow \rho = | \psi \rangle \langle \psi |
$$
In the general case, one will have (for a basis $| i \rangle$)
$$
\rho = \sum_i \sum_j c_{ij} | i \rangle \langle j|
$$
with the complex coefficients $c_{ij}$ to be determined for a particular state.

 

[Morbidly_Green]

Yes, it is wrong because that $\rho$ is not a density operator, just a quantum state (ket). For pure states, the density operator will be a projection operator,
$$
| \psi \rangle \rightarrow \rho = | \psi \rangle \langle \psi |
$$
In the general case, one will have (for a basis $| i \rangle$)
$$
\rho = \sum_i \sum_j c_{ij} | i \rangle \langle j|
$$
with the complex coefficients $c_{ij}$ to be determined for a particular state.

I see okay thank you