Quantum Mechanics: Time dependence of an expectation value

Question

In Griffiths's Introduction to Quantum Mechanics, he says that the time dependence of an expectation value is $$\frac{d\langle Q\rangle}{dt}=\frac{i}{\hbar}\langle [H,Q]\rangle+\langle \frac{\partial Q}{\partial t}\rangle$$ And I also saw a lecture note saying that time dependence of an expectation value is: $$\frac{\partial}{\partial t}\langle \hat{Q}(t)\rangle=-\frac{i}{h}\langle [\hat{Q}, \hat{H}]\rangle$$

I know they are saying the same thing. For me the second one is quite easy to understand. But how can I understand the first one?

Answer 1

The first equation is quite elementary to derive. First, we define the expectation value of an operator: $$\langle O \rangle=\langle \psi(t)| O | \psi (t)\rangle = \langle \psi(t=0)|U^\dagger O \ U |\psi (t=0)\rangle \tag{1}$$ where $U$ is the time evolution operator: $U=\exp\left( -i\frac{Ht}{\hbar}\right)$ if $H$ is independent of time.

Now, we can take the time derivative of (1), which will clearly have three terms (by the product rule). For the time evolution operator, we have the Schrodinger equation: $$\frac{d}{dt} U=\frac{1}{i\hbar}HU$$ and its hermitian conjugate $$\frac{d}{dt} U^\dagger=-\frac{1}{i\hbar}U^\dagger H$$ where we used that $H=H^\dagger$. Applying these two equations we find: $$\frac{d}{dt} \langle O \rangle =\frac{i}{\hbar}\langle [H,O]\rangle+\left\langle \frac{\partial O}{\partial t}\right\rangle$$ where we have defined $$\left\langle \frac{\partial O}{\partial t}\right\rangle=\langle \psi(t=0)|U^\dagger \frac{d O}{dt} \ U |\psi (t=0)\rangle$$

Answer 2

The second one is a particular case of the first one, as I understand it. In the first one we are assuming that, in general, Q can depend on time explicitly and through other dynamical variables, as for example, position or momentum. That is,

$$ \hat{Q} = \hat{Q}(\hat{x}(t),\hat{p}(t),t) $$

So, if we want to know how it evolves with time, we must know how it evolves in time because of the dynamical variables (that's the commutator part) and how it evolves in time because of its explicit dependence (the time partial derivative).

About the second equation, I think it's using a lax notation for the time dependence that is often used in theoretical physics, that is, writing partial derivative when we mean total derivative. And it's the particular case in which the operator doesn't depend explicitly on the time, which is the most interesting case most of the time.