How to Calculate Probability using Density Operator?

In summary, the conversation discusses the calculation of the probability of finding a system in a specific eigenstate using the density operator. The density operator is defined as the expectation value of the projector in the orientation of the eigenstate. The issue arises when trying to calculate the trace of the product between the density matrix and the projector, as the projector must be a matrix of the same rank as the density matrix. The correct way to construct the projection is to transform the state vector into a matrix with only one element in the main diagonal for each of the eigenstate directions. The conversation also touches on the dual use of the projector as an observable for the response of a system to a measurement of a pure state.
  • #1
Guilherme Vieira
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Hello, I'm trying to understand how to calculate de probability of finding a system in a specific eigenstate using the density operator. In the book of Balian, Haar, Gregg I've found a good definition of it being the expectation value of the projector Pr in the orientation of the eingenstate.
P(a) = tr(D.Pra)
The problem is, since I have then a product between de density matrix tr(D.Pra), Pra would have to be a matrix of the same rank of D, write ? To calculate then the tr. What is the right way to construct the projection ?
Transform the state, a vector, into a matrix ? Beeing a matrix with just one element in the main diagonal for each of the a directions relative to each eingenstate ?

I'm a new member, so I'm not used to ask questions online. Thank you.
 
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  • #2
Hi. I'll start from the beginning. If u have a density operator, call it $$\hat{D},$$ and some eigenstate, call it $$|\psi>.$$ You can simply make a projector to that eigenstate: $$ \hat{P}= | \psi> <\psi|.$$Then the probability of finding your system in that eigenstate is defined as $$ Tr( \hat{D} \hat{P}). $$ If u have a problem of writing that projector operator, here's help. U must write that bra state in some basis, and just multiply bra and ket vectors, u get a matrix with the same dimensions as your density operator and that's it. If you have problems with that multiplication of bra and ket vectors, search an article on wikipedia and that will help you. You can then multiply density operator and that projector and take trace of that final matrix.
 
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  • #3
If your density operator is ##\rho## and your pure state is ##\psi##, the projector is ##P=\psi\psi^*## and the probability is
##p## = Trace ##\rho P## = Trace ##\rho\psi\psi^*=\psi^*\rho\psi##
since the trace of ##uv^*## is ##v^*u##.
 
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  • #4
Thank you.
But, the definition of the density operator is not
ρ=|ψ><ψ| or ρ=∑|ψi><ψi| ?

Then it'is like the projector and the density operator are the same thing, or almost the same thing.
 
  • #5
Guilherme Vieira said:
But, the definition of the density operator is not
ρ=|ψ><ψ| or ρ=∑|ψi><ψi| ?

Then it'is like the projector and the density operator are the same thing, or almost the same thing.
The density operator of a pure state is indeed its projector. But the projector has a second use as an observable for the response of a system in an arbitrary state to a measurement of the pure state.
 
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  • #6
A. Neumaier said:
The density operator of a pure state is indeed its projector. But the projector has a second use as an observable for the response of a system in an arbitrary state to a measurement of the pure state.
Thanks, that is enlightening
 

1. What is the density operator in quantum mechanics?

The density operator is a mathematical representation used in quantum mechanics to describe the state of a quantum system. It is a Hermitian operator, meaning it is equal to its own conjugate transpose, and it is used to calculate probabilities of a system being in a particular state.

2. How is probability calculated using the density operator?

To calculate the probability of a quantum system being in a particular state, the density operator is multiplied by the projection operator for that state. The trace of this product gives the probability of the system being in that state.

3. What is the role of the density operator in quantum mechanics?

The density operator is an important tool in quantum mechanics as it allows for the calculation of probabilities and expectation values of a quantum system. It also provides a way to describe the state of a system that is in a mixed state, meaning it is not in a pure state.

4. Can the density operator be used for all types of quantum systems?

Yes, the density operator can be used for all types of quantum systems, including single particles, multiple particles, and entangled particles. It is a versatile tool that is used in many areas of quantum mechanics, including quantum information and quantum computing.

5. What is the difference between the density operator and the wave function in quantum mechanics?

The density operator and wave function both describe the state of a quantum system, but they are different mathematical representations. The density operator is a Hermitian operator, while the wave function is a complex-valued function. The density operator is used for calculating probabilities and expectation values, while the wave function is used for calculating amplitudes and probabilities.

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