Question 6.7: A particle of mass m moves in one dimension in a harmonic-os...

Consider the action of \hat{x}_H on a stationary state \psi_n . (Introducing \psi_n allows us to replace the operator e^{-i\hat{H}t/\hbar}  with the number e^{-iE_n t/\hbar} , since e^{-i\hat{H}t/\hbar}\psi_n= e^{-iE_nt/\hbar}\psi_n.) Writing \hat{x} in terms of raising and lowering operators we have (using Equations 2.62, 2.67 and 2.70)[\psi _{n}(x)=A_{n}(\hat{a} _{+}) ^{n} \psi _{0}(x), with E_{n}=\left(n+\frac{1}{2} \right ) \hbar w],  [\hat{a} _{+}\psi _{n}=\sqrt {n+1} \psi _{n+1}, \hat{a} _{-}\psi _{n}= \sqrt{n}\psi _{n-1}, ],  [x=\sqrt{\frac {\hbar }{2m\omega } } (\hat{a} _{+}+ \hat{a} _{-}) ,\hat{p} =i\sqrt{\frac{\hbar m\omega}{2 } } (\hat{a} _{+}- \hat{a} _{-}))]

\hat{x} _{H}(t)\psi _{n}(x)=\hat{U}^{\dagger } (t)\hat{x}\hat{U}(t)\psi _{n}(x)      (6.73)

=e^{i\hat{H}t/\hbar }\sqrt{\frac {\hbar }{2m \omega } } (\hat{a} _{+}+ \hat{a} _{-})e^{-i\hat{H} t/\hbar }\psi _{n}(x)

=\sqrt{\frac {\hbar }{2m\omega } }e^{-iE_{n}t/ \hbar }e^{i\hat{H}t/\hbar }(\hat{a} _{+}+ \hat{a} _{-})\psi _{n}(x)

=\sqrt{\frac {\hbar }{2m\omega } }e^{-iE_{n}t/ \hbar }e^{i\hat{H}t/\hbar }\left[\sqrt{n+1} \psi _{n+1}(x)+\sqrt{n} \psi _{n-1}(x)\right]

=\sqrt{\frac {\hbar }{2m\omega } }e^{-iE_{n}t/ \hbar }\left[\sqrt{n+1} e^{ iE_{n+1} t/\hbar }\psi _{n+1}(x)+\sqrt{n} e^{iE_{n-1}t/\hbar }\psi _{n-1}(x)\right]

=\sqrt{\frac {\hbar }{2m\omega } }\left[ \sqrt {n+1} e^{i\omega t}\psi _{n+1}(x)+\sqrt{n} e^{-i \omega t }\psi _{n-1}(x)\right].

Thus

\hat{x} _{H}(t)=\sqrt{\frac {\hbar }{2m\omega } }\left[e^{i\omega t }\hat{a} _{+}+ e^{-i\omega t }\hat{a} _{-}\right].

Or, using Equation 2.48 [\hat{a} _{\pm }\equiv \frac{1}{\sqrt{2\hbar m \omega } } (\mp i\hat{p} + m \omega x)] to express \hat{a} _ { \pm}  in terms of \hat{x} and \hat{p} ,

\hat{x} _{H}(t)=\hat{x} _{H}(0) \cos(\omega t)+\frac {1}{m\omega } \hat{p}_{H}(0)\sin(\omega t) .    (6.74)

As in Example 6.6 we see that the Heisenberg-picture operator satisfies the classical equation of motion for a mass on a spring.