Deducing the solution of the von Neumann equation

[xyver]

Homework Statement

$$\hat{\rho}(t)=?$$ $$|\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle$$ $$\imath\hbar\partial_{t}\hat{p}=[\hat{H},\hat{\rho}]$$

Homework Equations

$$\imath\hbar\partial_{t}\hat{p}=[\hat{H},\hat{\rho}] \Leftrightarrow\imath\hbar\partial_{t}\hat{p}=\hat{H}\hat{\rho}-\hat{\rho}\hat{H}$$

The Attempt at a Solution

I already know the solution: $$\hat{\rho}(t)=\hat{U}\hat{\rho}(0)\hat{U}^{+}$$
But where do I get this from? How do I know that I have to write the time evolution operator multiplied once in front of the density operator and once the Hermitian conjugate after it?

Also, I tried to verify the solution:
$$\Rightarrow\imath\hbar\partial_{t}\hat{U}\hat{\rho}(0)\hat{U}^{+}=\hat{H}\hat{U}\hat{\rho}(0)\hat{U}^{+}-\hat{U}\hat{\rho}(0)\hat{U}^{+}\hat{H}=[H,\hat{\rho}(t)]$$
Can't I take any other operator instead of the time evolution operator at this place, since in my attempt to verify the solution the $\hat{U}$ goes away again?

Or is this just guessing as one way to solve a differential equation. Then, still, how do you get the idea?

 

[dextercioby]

Why don't you use the definition of the von Neumann density operator ?

 

[xyver]

The definition should be $\hat{\rho}=\sum_{i}p_{n}|\psi(t)\rangle\langle\psi(t)|$
I can do with that:
$$\partial_{t}\hat{\rho}=\partial_{t}\sum_{i}p_{n}| \psi(t)\rangle\langle\psi(t)|+ \sum_{i} p_{n}|\psi(t) \rangle\partial_{t}\langle\psi(t)| \Leftrightarrow$$$$\partial_{t}\hat{\rho}=\frac{1}{\imath\hbar}Hp_{n}|\psi(t)\rangle\langle\psi(t)|+\sum_{i}p_{n}|\psi(t)\rangle\frac{1}{\imath\hbar}H\langle\psi(t)| \Leftrightarrow$$$$\partial_{t}\hat{\rho}=\frac{1}{\imath\hbar}\hat{H}p_{n}|\psi(t)\rangle\langle\psi(t)|+\frac{1}{ \imath\hbar}\sum_{i}p_{n}|\psi(t)\rangle\langle \psi(t)\hat{H}|$$

 

[xyver]

I found the solution here: http://books.google.de/books?id=0Yx5VzaMYm8C&pg=PA110&redir_esc=y#v=twopage&q&f=true .
That's Heinz-Peter Breuer, Petruccione, "The Theory of Open Quantum Systems", pp. 110-111.
Nearly every step ist explained.

 

[paranormal]

As a scientist, it is important to understand the underlying principles and reasoning behind a solution, rather than just memorizing it. In this case, the solution to the von Neumann equation can be deduced from the fact that it describes the time evolution of a quantum system. The time evolution operator, \hat{U}, is defined as the operator that transforms a state at time t_0 to a state at time t. Therefore, it makes sense that it would be involved in the solution to the von Neumann equation, as it represents the time evolution of the system.

Furthermore, the von Neumann equation is essentially a differential equation, and the solution to a differential equation involves finding an operator that satisfies the equation. In this case, the time evolution operator, \hat{U}, satisfies the equation and therefore can be used as the solution.

As for your attempt to verify the solution, it is important to remember that the time evolution operator, \hat{U}, is not just any operator, but specifically the operator that describes the time evolution of the system. Therefore, it is the only operator that can be used in the solution to the von Neumann equation.

In summary, the solution to the von Neumann equation can be deduced from the principles of quantum mechanics and the fact that it is a differential equation. The time evolution operator, \hat{U}, is an integral part of the solution as it represents the time evolution of the system.