Chapter 2.4, Problem 2.21P

(a)

To determine

Normalize $\Psi \left(x,0\right)=A{e}^{-a{x}^2}$

(b)

To determine

Find $\Psi \left(x,t\right)$

(c)

To determine

Find ${\left|\Psi \left(x,t\right)\right|}^2$ and express answer in terms of $w\equiv \sqrt{a/\left[1+{\left(2\hslash at/m\right)}^2\right]}$ sketch ${\left|\Psi \right|}^2$ as a function of x at $t=0$ and again for some very large t. What happens to ${\left|\Psi \right|}^2$, as time goes on?

(d)

To determine

Find $\langle x\rangle ,\langle p\rangle ,\langle x^2\rangle ,\langle p^2\rangle ,{\sigma }_x,$ and ${\sigma }_p$

(e)

To determine

Whether the uncertainty principle holds? At what time t does the system come closest to the uncertainty limit?