# Deducing the solution of the von Neumann equation

1. May 27, 2012

### xyver

1. The problem statement, all variables and given/known data
$$\hat{\rho}(t)=?$$ $$|\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle$$ $$\imath\hbar\partial_{t}\hat{p}=[\hat{H},\hat{\rho}]$$

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

3. 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?

Last edited: May 27, 2012
2. May 27, 2012

### dextercioby

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

3. May 27, 2012

### xyver

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

4. Jun 20, 2012