Exploring Quantum Physics - Week 5 Bonus Question

Question:

Prove the Ehrenfest theorem.

Solution:

I could have write cool equations like this

$ \frac{d}{dt}\int{\psi^* \hat{A} \psi} = \int{\frac{d}{dt}\psi^* \hat{A} \psi + \psi^* \frac{d\hat{A}}{dt} \psi + \psi^* \hat{A} \frac{d}{dt}\psi } $ and then use the relation $ \hat{H} = ih\frac{d}{dt} $ to present a 'proof', just like the wikipedia page does.

The key difficulty is defining the derivative of an operator, otherwise the right hand side of the theorem just cannot be understood.

There is nothing magic in it, except this. The operator itself is a function of $ t $ in order for the differentiation to make sense

So we go back to the first principle and define the derivative of an operator to be a limit.

$ \frac{d\hat{A(t)}}{dt}\psi = \lim\limits_{\Delta t \to 0}{\frac{\hat{A(t + \Delta t)}\psi - \hat{A(t)}\psi}{\Delta t}} $

Then we can proof the product rule on operator differentiation just like one would prove it in analysis.

$ \begin{eqnarray*} \frac{d}{dt}{\hat{A(t)}\psi(t)} &=& \lim\limits_{\Delta t \to 0}{\frac{\hat{A(t + \Delta t)}\psi(t + \Delta t) - \hat{A(t)}\psi(t)}{\Delta t}} \\ &=& \lim\limits_{\Delta t \to 0}{\frac{\hat{A(t + \Delta t)}\psi(t + \Delta t) - \hat{A(t + \Delta t)}\psi(t) + \hat{A(t + \Delta t)}\psi(t) - \hat{A(t)}\psi(t)}{\Delta t}} \\ &=& \hat{A(t)}\frac{d\psi(t)}{dt} + \frac{d\hat{A(t)}}{dt}\psi(t) \end{eqnarray*} $

The rest should follows ...