Complex coefficents in density operator expansion?

In summary, the conversation discusses a discrepancy between the proposed solution for a quantum state exam question and the correct solution. The proposed solution does not include complex coefficients in the density operator expansion, while the correct solution does. The speaker confirms that the correct solution, with the complex coefficients, is indeed correct and explains that the density matrix must be Hermitian.
  • #1
center o bass
560
2
Hey, I recently had an exam where the quantum state were on the form

[tex] |\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle )[/tex]

Here I formed the density operator for the pure state

[tex] \rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle -| ) = \frac{1}{2} ( |+\rangle \langle + | + |- \rangle \langle - | + i(|-\rangle \langle + | - |+\rangle \langle -|)). [/tex]

However in the solutions for the exam the complex i's were not there, i.e the solutions states that

[tex] \rho = \frac{1}{2} ( |+\rangle \langle + | + |- \rangle \langle - | + |-\rangle \langle +| - |+\rangle \langle - |).[/tex]

Have I missed something here or is the suggested solution erroneous? Is there a reason why a density operator expansion should not have complex coefficients?
 
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  • #2
Your solution seems correct - I've got i's on anti-diagonal as well. The second expression cannot be right, because the density matrix has to be hermitian.
 

1. What are complex coefficients in density operator expansion?

Complex coefficients in density operator expansion refer to the numerical values assigned to each basis state in the density matrix. These coefficients are complex numbers and determine the probability of the system being in a particular state.

2. How are complex coefficients calculated in density operator expansion?

The complex coefficients in density operator expansion are calculated by taking the inner product of the basis states with the density matrix. This involves multiplying the basis state vector with the density matrix and taking the complex conjugate of the result.

3. What is the significance of complex coefficients in density operator expansion?

The complex coefficients in density operator expansion are significant as they represent the probability amplitudes of the system being in a particular state. They also play a crucial role in determining the expectation values and measurements of observables in quantum mechanics.

4. Can complex coefficients in density operator expansion be negative?

Yes, complex coefficients in density operator expansion can be negative. This is because they are complex numbers and can have both real and imaginary parts. The sign of the coefficient does not affect its physical interpretation as a probability amplitude.

5. How do complex coefficients in density operator expansion differ from classical probabilities?

Complex coefficients in density operator expansion differ from classical probabilities in that they are not limited to values between 0 and 1. In quantum mechanics, probabilities can be represented by complex numbers, allowing for the possibility of negative values and interference effects between states.

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