Calculating Time Evolution of Density Matrix

In summary, the conversation discusses the calculation of the time evolution of a density matrix, specifically when a mixed state is involved. It is suggested to use kets that have undergone time evolution, taking into account the Hermitian conjugate. This results in the density matrix evolving according to the unitary time evolution.
  • #1
dg88
10
0
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?
 
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  • #2
dg88 said:
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)
 
  • #3


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.
 

1. How is the density matrix used to calculate time evolution?

The density matrix is a mathematical representation of the quantum state of a system. To calculate its time evolution, the density matrix is multiplied by the Hamiltonian operator, which describes the energy of the system. This results in a new density matrix, which represents the system at a later time.

2. What is the significance of calculating time evolution of the density matrix?

Calculating the time evolution of the density matrix allows us to understand how a quantum system changes over time. This is important for predicting and studying the behavior of quantum systems, which can have complex and unpredictable dynamics.

3. Can the time evolution of the density matrix be calculated for any quantum system?

Yes, the time evolution of the density matrix can be calculated for any quantum system. However, the complexity of the calculations may vary depending on the size and complexity of the system.

4. How does the time evolution of the density matrix relate to quantum decoherence?

Quantum decoherence is the process by which a quantum system becomes entangled with its surrounding environment, causing it to lose its quantum properties. The time evolution of the density matrix shows how this entanglement affects the system over time, leading to decoherence.

5. Are there any limitations to calculating the time evolution of the density matrix?

One limitation is the assumption that the system is isolated and not interacting with its environment. In reality, most quantum systems are subject to interactions with their surroundings, making the calculations more complex. Additionally, the accuracy of the calculations can be affected by the size and complexity of the system.

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