Density matrix of spin 1 system

In summary, the conversation discusses the number of independent parameters needed to characterize a density matrix of a spin 1 system. It is mentioned that 8 parameters are required, with the explanation being that there are 9 elements in the matrix but the trace is known to be one. However, there is uncertainty about whether the counting was done correctly and it is suggested to consider other operators that can be constructed from Sx, Sy, and Sz. The conversation also mentions a nonsensical solution found through Google search and the books where the treatment of the density matrix was found.
  • #1
Dishsoap
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Homework Statement


Consider an ensemble of spin 1 systems (a mixed state made of the spin 1 system). The density matrix is now a 3x3 matrix. How many independent parameters are needed to characterize the density matrix? What must we know in addition to Sx, Sy and Sz to characterize the mixed state completely?

2. The attempt at a solution

I know that 8 parameters are needed to characterize the density matrix (9 elements, minus one because we know the trace of the density matrix is one.) However, I'm not sure what else is needed to characterize it... Sx, Sy and Sz are all that is needed for the spin 1/2 state, which is all that is treated in all of the books in which I've found a treatment of the density matrix (Sakurai, Gottfried/Yan, Shankar).

When I Google the problem, I get a nonsensical solution that suggests things to do with polarization and quadripole moments, but no explanation of the answer.
 
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  • #2
Dishsoap said:
I know that 8 parameters are needed to characterize the density matrix (9 elements, minus one because we know the trace of the density matrix is one.)
This is actually a problem from Sakurai. While it is correct that 8 parameters are required to parameterise the density matrix, could I check whether you did the counting correctly? (because your 8 = 9 - 1 explanation seems suspicious and you have not mentioned whether you've made use of the fact that the density matrix must be Hermitian)

The hint is to consider what other operators you can construct from ##S_x, S_y, S_z## (and then, if you want to, you can prove that they are sufficient by considering an explicit form of the density matrix in terms of the 8 parameters)
 

1. What is a density matrix of a spin 1 system?

A density matrix is a mathematical representation of the state of a quantum mechanical system. In the case of a spin 1 system, it describes the probabilities of different spin orientations of the particles in the system.

2. How is the density matrix of a spin 1 system calculated?

The density matrix of a spin 1 system is calculated by taking the outer product of the state vector, which represents the state of the system, with its conjugate transpose. This results in a square matrix with dimensions equal to the number of possible spin orientations in the system.

3. What information can be obtained from the density matrix of a spin 1 system?

The density matrix of a spin 1 system contains information about the probabilities of different spin orientations, as well as the coherence between different spin states. It can also be used to calculate average values of spin measurements.

4. How does the density matrix of a spin 1 system change over time?

The density matrix of a spin 1 system can change over time due to interactions with the environment or external forces. This results in changes to the probabilities of different spin orientations and the coherence between different spin states.

5. What are some applications of the density matrix of a spin 1 system?

The density matrix of a spin 1 system is used in a variety of applications, such as quantum computing, quantum information processing, and quantum state engineering. It is also important in understanding the behavior of systems at the nanoscale and in studying quantum entanglement.

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