0
$\begingroup$

If I have a quantum system described by a time-independent Hamiltonian $\hat{H}$, then the Liouville-von Neumann equation is

\begin{align} i\hbar\frac{\partial\hat{\rho}}{\partial t}=[\hat{H},\hat{\rho}]\,, \end{align}

where $\hat{\rho}$ is the density matrix. What happens if the Hamiltonian is explicitly time-dependent, such that $\hat{H}=\hat{H}(t)$? Is the Liouville-von Neumann equation the same?

$\endgroup$
3
$\begingroup$

If the Hamiltonian is time-dependent the evolution of a pure state is $$ |\psi(t)\rangle = {\mathcal T} exp\left[ -\frac{i}{\hbar} \int_{t_0}^t d\tau H(\tau)\right] |\psi(0)\rangle = U(t; t_0) |\psi(0)\rangle $$ where $\mathcal T$ is the time-ordering operator and $$ i\hbar\dot{U} = H(t) U(t, t_0) $$ Then a density matrix evolves according to $$ \rho(t) = U(t; t_0) \rho(0) U^\dagger (t; t_0) $$ and you can check that taking the time derivative gives $$ i\hbar\dot{\rho} = \left[ H(t), \rho(t)\right] $$

$\endgroup$
  • $\begingroup$ I have $\frac{\partial}{\partial t}\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]=\frac{1}{i\hbar}H(t)\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]$, but how do I treat the time-ordering operator? Is it true that $\frac{\partial}{\partial t}\left(\mathcal{T}\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]\right)=\mathcal{T}\left(\frac{\partial}{\partial t}\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]\right)$? $\endgroup$ – user85503 Mar 1 '17 at 20:27
  • $\begingroup$ If this is so, then why would $\mathcal{T}\left(H(t)\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]\right)=\left(H(t)\mathcal{T}\exp\left[-\frac{i}{\hbar}\int_{t_0}^tH(\tau)d\tau\right]\right)$? $\endgroup$ – user85503 Mar 1 '17 at 20:27
  • 1
    $\begingroup$ No, that's not how it works. You may want to take a look at Sec.5.2.1 in ocw.mit.edu/courses/nuclear-engineering/…. The time ordered exponential ${\mathcal T}\exp\left[-(i/\hbar)\int_{t_0}^t{d\tau\; H(\tau)}\right]$ is a compact way of writing a series where each term is a nested integral of the form $$\int_{t_0}^t{d\tau_1\int_{t_0}^{t_1}{d\tau_2\; \dots \int_{t_0}^{t_{n-1}}{d\tau_n H(\tau_1) H(\tau_2)\dots H(\tau_n)}}}$$. $\endgroup$ – udrv Mar 2 '17 at 5:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.