Calculating Time Evolution of Density Matrix

[dg88]

Hi,

I am trying to calculate the time evolution of a density matrix. Like if there is a mixed state with 50% of |x, 0> and 50% of |y, 0>. After time t due to time evolution, the kets become:

|x,t>= e^(-i/h Ht) |x,0> and so on.

Is it ok to use these kets instead of the original ket to calculate the density matrix after time t? Or is there another method to do it?

 

[Count Iblis]

Hi,

I am trying to calculate the time evolution of a density matrix. Like if there is a mixed state with 50% of |x, 0> and 50% of |y, 0>. After time t due to time evolution, the kets become:

|x,t>= e^(-i/h Ht) |x,0> and so on.

Is it ok to use these kets instead of the original ket to calculate the density matrix after time t? Or is there another method to do it?

Yes, but don't forget to take the Hermitian conjugate:

<x,t| = <x,0| e^(i/h Ht)

So, you see that if you do this the density matrix evolves in time according to the unitary time evolution:

rho(t) = U rho(0) U-dagger

U = e^(-i/hbar H t)

 

[denian]

I would suggest using the time-evolution operator to calculate the density matrix after time t. This operator, e^(-i/h Ht), acts on the initial density matrix to give the time-evolved density matrix. This method is commonly used in quantum mechanics to calculate the time evolution of a system described by a density matrix. It is important to note that using the time-evolved kets, as suggested in your question, may not accurately represent the density matrix after time t. Therefore, I highly recommend using the time-evolution operator for accurate results. Additionally, there may be other methods for calculating the time evolution of a density matrix, but using the time-evolution operator is a commonly accepted and reliable approach. I hope this helps in your calculation process.