Question

Spreading of a free wave packet Consider a free particle. a. Show, applying Ehrenfest's theorem, that $\langle X\rangle$ is a linear function of time, the mean value $\langle P\rangle$ remaining constant. $b$. Write the equations of motion for the mean values $\left\langle X^{2}\right\rangle$ and $\langle X P+P X\rangle$ Integrate these equations. c. Show that, with a suitable choice of the time origin, the root-meansquare deviation $\Delta X$ is given by : \[ (A X)^{2}=\frac{1}{m^{2}}(A P)_{0}^{2} t^{2}+(\Delta X)_{0}^{2} \] where $(A X)_{0}$ and $(A P)_{0}$ are the root-mean-square deviations at the initial time. How does the width of the wave packet vary as a function of time (see $\$ 3-c$ of complement $G_{1}$ )? Give a physical interpretation.

          Spreading of a free wave packet
Consider a free particle.
a. Show, applying Ehrenfest's theorem, that $\langle X\rangle$ is a linear function of time, the mean value $\langle P\rangle$ remaining constant.
$b$. Write the equations of motion for the mean values $\left\langle X^{2}\right\rangle$ and $\langle X P+P X\rangle$ Integrate these equations.
c. Show that, with a suitable choice of the time origin, the root-meansquare deviation $\Delta X$ is given by :
\[
(A X)^{2}=\frac{1}{m^{2}}(A P)_{0}^{2} t^{2}+(\Delta X)_{0}^{2}
\]
where $(A X)_{0}$ and $(A P)_{0}$ are the root-mean-square deviations at the initial time. How does the width of the wave packet vary as a function of time (see $\$ 3-c$ of complement $G_{1}$ )? Give a physical interpretation.

Added by Laura R.

Question

Please give Ace some feedback