4
$\begingroup$

In Griffiths Intro to QM (2nd edition), he gives the equation

$$ \frac{d}{dt} \langle Q \rangle = \frac{i}{\hbar}\langle [\hat{H},\hat{Q}] \rangle + \left\langle \frac{\partial{\hat{Q}}}{\partial{t}} \right \rangle \tag{3.71} $$

and he goes on to state that $$\left \langle \frac{\partial\hat{Q}}{\partial{t}} \right \rangle =0 $$

for many operators $\hat{Q}$.

However, in problem 3.31 when deriving the virial theorem, we use $\hat{Q} = \hat{x}\hat{p}$. Why are they not considered functions of time, thus giving a non-zero value for $$\left \langle \frac{\partial (\hat{x}\hat{p})}{\partial{t}} \right \rangle \, ?$$

$\endgroup$
3
$\begingroup$

$\frac{\partial\hat{Q}}{\partial t}$ denotes the partial derivative of time, which is nonvanishing only when $\hat{Q}$ manifestly depends on time. Every operator $\hat{Q}$ can be time dependent in an implicit way such as $\hat{x}\hat{p}$ which can be time dependent when $\hat{x}$ or $\hat{p}$ depends on time.

$\endgroup$

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.