Expressing the density matrix in matrix form

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The discussion clarifies that the expression of the density matrix as ## \rho = c_1|1> + c_2|2> + c_3|3> ## is incorrect, as this represents a quantum state (ket) rather than a density operator. For pure states, the correct form of the density operator is a projection operator given by ## \rho = | \psi \rangle \langle \psi | ##. In general, the density matrix should be expressed as ## \rho = \sum_i \sum_j c_{ij} | i \rangle \langle j| ##, where the coefficients ##c_{ij}## are complex numbers specific to the state. The conversation emphasizes the distinction between quantum states and density operators in quantum mechanics. Understanding this difference is crucial for accurately representing quantum systems.
Morbidly_Green
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page1-800px-Lambda-type_system.pdf.jpg


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?
 

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Morbidly_Green said:
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.
 
DrClaude said:
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
 
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