The Ehrenfest's theorem that we quoted in the text was that quantum expectation values satisfy the classical equations of motion. In Sec. 4.2.1, we had also stated that this isn't quite true. If the quantum expectation value of momentum satisfied the classical equations of motion, then its time derivative would be

\[\begin{equation*}\frac{d\langle\hat{p}angle}{d t}=-\frac{d V(\langle\hat{x}angle)}{d\langle\hat{x}angle} \tag{4.165}\end{equation*}\]

but this is not what we had found for its time dependence in Eq. (4.54). In this equation, we'll study the differences between these expressions.

(a) For what potential \(V(\hat{x})\) is the true time dependence of Eq. (4.54) equal to the simpler expression of Eq. (4.165)?

(b) Consider a power-law potential with

\[\begin{equation*}V(\hat{x})=k \hat{x}^{2 n} \tag{4.166}\end{equation*}\]

where \(k\) is a constant that has the correct units to make the potential have units of energy and \(n\) is an integer greater than 1. For a general state \(|\psiangle\), can you say how the time dependences of Eqs. (4.54) and (4.165) compare? Which expression produces a larger time derivative of the expectation value of momentum?

(c) For the form of the potential in part (b), what must the state \(|\psiangle\) be for Eqs. (4.54) and (4.165) to be equal?