Question 6.6: A particle of mass m moves in one dimension in a potential V...

From Equation 6.71 [\hat{U}(t)=\exp\left[-\frac{it}{\hbar }\hat{H} \right] . ],

\hat{U}(\delta) \approx 1-i\frac{\delta }{\hbar } \hat{H}

Applying Equation 6.72 [\hat{Q}_{H}(t)=\hat{U}^{\dagger} (t) \hat{Q}\hat{U}(t) ], we have

\hat{x}_{H}(\delta )\approx \left(1+i \frac {\delta }{\hbar } \hat{H}^{\dagger }\right) \hat{x} \left(1-i\frac{\delta }{\hbar } \hat{H}\right) \approx\hat{x} -i\frac{\delta }{\hbar } \left[ \hat {x} ,\hat{H} \right] \approx \hat{x} -i\frac{\delta }{\hbar}i\hbar \frac{\hat{p} }{m}

so

\hat{x} _{H}(\delta ) \approx \hat{x} _{H}(0)+\frac{1}{m}\hat{p} _{H}(0)\delta

(making use of the fact that the Heisenberg-picture operators at time 0 are just the untransformed operators). This looks exactly like classical mechanics: x(\delta) \approx x(0)+v(0) \delta. The Heisenberg picture illuminates the connection between classical and quantum mechanics: the quantum operators obey the classical equations of motion (see Problem 6.29).