# Time Dependent Density Matrix

Given a wave function $\left|\psi\right>$ at time $t=0$ and an Hamiltonian, how does one find the time-dependent density matrix? So far I calculated the density matrix of the wave function, which gave me $\rho\left(t=0\right)$. Next, I know that

$$\rho\left(t\right)=\exp\left(\frac{i}{\hbar}Ht\right)\rho\left(t=0\right)\exp\left(-\frac{i}{\hbar}Ht\right)$$

and

$$\frac{\partial\rho}{\partial t}=\frac{i}{\hbar}[\rho,H]$$

where $H = \epsilon\begin{pmatrix} 1&i \\ -i&1 \end{pmatrix}$ is the Hamiltonian. So I also calculated the commutator. From here, I am not sure what to do next. Any suggestion is appreciated.

• What is your Hamiltonian? If you calculated the commutator, the second equation is an ODE you need to solve. – eranreches Nov 30 '17 at 12:24
• What do you mean by "find" the time-dependent density matrix? There's nothing more to be said about the density matrix beyond what you've already written, really, in a general setting ─ if you want to get something more specific, then you need to write in some more specific conditions, i.e. the hamiltonian, the state space, and the initial conditions. – Emilio Pisanty Nov 30 '17 at 12:24
• If you have $\rho(t=0)$ and the Hamiltonian $H$, then I don't understand why you can't just apply your first equation. What problem do you encounter when attempting to plug in your values? – ACuriousMind Nov 30 '17 at 12:24
• What is your initial state? What is the commutator? What difficulties have you encountered in applying the second equation you've stated? You should elaborate more on your calculations. – eranreches Nov 30 '17 at 12:40
• It seems I may have misunderstood the question and overcomplicated it in my head. I just didn't think that all I needed was to plug in H in the exponent. Thank you. – Kane Billiot Nov 30 '17 at 12:40